ginormous systems

GINORMOUS
SYSTEMS
April 30–May 1, 2013
Washington, D.C.
RECONNAISSANCE PAPERS
2013
INDEX TO PAPERS
Sure, Big Data Is Great. But So Is Intuition by Steve Lohr
Are the algorithms shaping our digital world too simple or too smart? There’s no easy answer; listening to data
is important, but so is experience and intuition.
http://www.nytimes.com/2012/12/30/technology/big-data-is-great-but-dont-forgetintuition.html?ref=technology& r=0
Looking to Industry for the Next Digital Disruption by Steve Lohr
This article looks at GE’s big bet on what it calls the “industrial Internet,” bringing digital intelligence to the
physical world of industry as never before.
http://www.nytimes.com/2012/11/24/technology/internet/ge-looks-to-industry-for-the-next-digitaldisruption.html?pagewanted=all
GE’s Industrial Internet and the Smart Grid in the Cloud by Jeff St. John
Building an Internet for the utility industry that ties together smart meters, grid sensors, and enterprise IT into a
cloud-hosted package may be technologically achievable, but who will manage and integrate the back end,
legacy billing, and customer service?
http://www.greentechmedia.com/articles/read/ges-industrial-internet-and-the-smart-grid-in-the-cloud
An Internet for Manufacturing by Michael Fitzgerald
Can the industrial Internet go from its current state of facility intranet to a global Internet, feeding information
back about weather conditions, supply and demand, and logistics?
http://www.technologyreview.com/news/509331/an-internet-for-manufacturing/
One Big Cluster: How CloudFlare Launched 10 Data Centers in 30 Days by Sean Gallagher
Content delivery provider CloudFare builds its data centers by never setting foot in them; they ship the
equipment with a how-to manual, use open-source software, storage tricks from the world of big data, a bit of
network peering arbitrage, and voila!
http://arstechnica.com/information-technology/2012/10/one-big-cluster-how-cloudflare-launched-10-datacenters-in-30-days/
Daniel Tunkelang Talks about LinkedIn’s Data Graph by Paul Miller
Speaker Daniel Tunkelang says not get too obsessed with building systems that are perfect. Instead, he
suggests we communicate with users and offer UIs that guide and help them explore.
http://semanticweb.com/daniel-tunkelang-talks-about-linkedins-data-graph b29699
Beyond MapReduce: Hadoop Hangs On by Matt Asay
Hadoop helped usher in the big-data movement. Yet the Web world is moving toward real-time, ad-hoc
analytics that batch-oriented Hadoop can't match.
http://www.theregister.co.uk/2012/07/10/hadoop_past_its_prime/
10 Predictions about Networking and Moore’s Law from Andy Bechtolsheim
Speaker Andy Bechtolsheim suggests that we are in the golden age of networking, and he predicts that the
economics of chips will change, architecture and flexibility will matter even more, and that Moore’s Law is alive
and well.
http://venturebeat.com/2012/10/11/bechtolsheims-10-predictions-about-networking-and-moores-law/
Internet Architects Warn of Risks in Ultrafast Networks by Quentin Hardy
The article profiles Arista and its two founders, including speaker Andy Bechtolsheim, both of whom say the
promise of having access to mammoth amounts of data instantly, anywhere, is matched by the threat of
catastrophe. The company was built with the 10-gigabit world in mind.
http://www.nytimes.com/2011/11/14/technology/arista-networks-founders-aim-to-alter-how-computersconnect.html?pagewanted=all
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A “Big Data” Freeway for Scientists by John Markoff
This article looks at a new advanced optical computer network that is intended to serve as a “Big Data
freeway system” for next-generation science projects in fields including genomic sequencing, climate science,
electron microscopy, oceanography and physics. The network is at the University of California, San Diego,
and was developed by Arista Networks.
http://bits.blogs.nytimes.com/2013/03/20/a-big-data-freeway-for-scientists/
A New Approach to Innovation Will Be Crucial in the Coming Era of Cognitive Systems by Bernard Meyerson
Speaker Bernie Meyerson argues that in the early stages of building cognitive systems, the benefits will arrive
sooner and stronger if companies, governments, and universities adopt a culture of innovation that includes
making big bets, fostering disruptive innovations, taking a long-term view, and collaborating across
institutional boundaries.
http://asmarterplanet.com/blog/2013/01/a-new-approach-to-innovation-will-be-needed-in-the-coming-era-ofcognitive-systems.html
Construction of a Chaotic Computer Chip by William Ditto, K. Murali, and Sudeshna Sinha
Speaker William Ditto and his colleagues discuss progress on the construction of a chaotic computer chip
consisting of large numbers of individual chaotic elements that can be individually and rapidly morphed to
become the full range of logic gates. Such a chip of arrays of morphing chaotic logic gates can then be
programmed to perform higher order functions and to rapidly switch among such functions.
http://www.imsc.res.in/~sudeshna/Ditto for ICAND.pdf
Panasas Kingpin: What's the Solid State State of Play? by Chris Mellor
Speaker Garth Gibson provides his perspective on what NAND flash can do now for high-performance
computing storage and how it will evolve.
http://www.theregister.co.uk/2012/03/29/panasas on ssd/
Storage at Exascale: Some Thoughts from Panasas CTO Garth Gibson
What kind of storage performance will need to be delivered to achieve exascale computing? Speaker Garth
Gibson answers that question, and others in this interview.
http://www.hpcwire.com/hpcwire/2011-0525/storage at exascale some thoughts from panasas cto garth gibson.html
Biff (Bloom Filter) Codes: Fast Error Correction for Large Data Sets by Michael Mitzenmacher and George
Varghese
Large data sets are increasingly common in cloud and virtualized environments. There is a need for fast errorcorrection or data reconciliation in such settings, even when the expected number of errors is small. The
authors, including speaker Michael Mitzenmacher, consider error correction schemes designed for large data.
http://cseweb.ucsd.edu/~varghese/PAPERS/biffcodes.pdf
Verifiable Computation with Massively Parallel Interactive Proofs by Justin Thaler, Mike Roberts, Michael
Mitzenmacher, and Hanspeter Pfister
In the cloud, the need for verifiable computation has grown increasingly urgent. The authors believe their
results with verifiable computation demonstrate the immediate practicality of using GPUs for such tasks, and
more generally, that protocols for verifiable computation have become sufficiently mature to deploy in real
cloud computing systems.
http://arxiv.org/pdf/1202.1350v3.pdf
Unreported Side Effects of Drugs Are Found Using Internet Search Data, Study Finds by John Markoff
Using data drawn from Web-wide search queries, scientists have been able to detect evidence of unreported
prescription drug side effects before they were found by the Food and Drug Administration’s warning system.
http://www.nytimes.com/2013/03/07/science/unreported-side-effects-of-drugs-found-using-internet-datastudy-finds.html?ref=technology& r=0
Six Provocations for Big Data by danah boyd and Kate Crawford
With the increased automation of data collection and analysis, as well as algorithms that can extract and
inform us of massive patterns in human behavior, it is necessary to ask which systems are driving these
practices and which are regulating them. In this essay, the authors offer six provocations that they hope can
spark conversations about the issues of big data.
http://softwarestudies.com/cultural_analytics/Six_Provocations_for_Big_Data.pdf
Why Hadoop Is the Future of the Database by Cade Metz
A revamped Hadoop, operating more like a relational database, can now store massive amounts of
information and answer questions using SQL significantly faster than before.
http://www.wired.com/wiredenterprise/2013/02/pivotal-hd-greenplum-emc/
Algorithms Get a Human Hand in Steering Web by Steve Lohr
Computers are being asked to be more humanlike in what they figure out. Although algorithms are growing
ever more powerful, fast, and precise, computers are not always up to deciphering the ambiguity of human
language and the mystery of reasoning.
http://www.nytimes.com/2013/03/11/technology/computer-algorithms-rely-increasingly-on-humanhelpers.html?hp& r=0
How Complex Systems Fail by Richard I. Cook
This essay shows us 18 ways in which we can look at the nature of failure, how failure is evaluated, and how
failure is attributed to proximate cause.
http://www.ctlab.org/documents/How%20Complex%20Systems%20Fail.pdf
What Data Can’t Do by David Brooks
Data can be used to make sense of mind-bogglingly complex situations, yet it can obscure values and
struggle with context and social cognition.
http://www.nytimes.com/2013/02/19/opinion/brooks-what-data-cant-do.html?hpw
In Mysterious Pattern, Math and Nature Converge by Natalie Wolchove
A universality model, coming from an underlying connection to mathematics, is helping to model complex
systems from the Internet to Earth’s climate.
http://www.wired.com/wiredscience/2013/02/math-and-nature-universality/all/
Embracing Complexity
We are beginning to understand that complex systems are even more complex than we first thought.
Complexity theorists are now studying how physical systems go through phase transitions to try and predict
when everyday networks will go through potentially catastrophic changes.
http://fqxi.org/community/articles/display/174
Optimization of Lyapunov Invariants in Verification of Software Systems by Mardavij Roozbehani, Alexandre
Megretski, and Eric Feron
The authors of this paper have developed a method for applying principles from control theory to formal
verification, a set of methods for mathematically proving that a computer program does what it’s supposed to
do.
http://arxiv.org/pdf/1108.0170v1.pdf
ADDITIONAL WEB SITES, REFERENCES, AND PAPERS
Roberto Rigobon Published Papers
This website contains papers on economics by speaker Roberto Rigobon
http://web.mit.edu/rigobon/www/Robertos Web Page/New.html
The Billion Prices Project
Co-led by speaker Roberto Rigobon, this project is an academic initiative that uses prices collected from
hundreds of online retailers around the world on a daily basis to conduct economic research.
http://bpp.mit.edu/
Research activities by Katherine Yelick
Speaker Katherine Yelick is involved in a number of HPC projects; the last link is to her publications.
http://upc.lbl.gov/
http://bebop.cs.berkeley.edu/
http://crd.lbl.gov/groups-depts/ftg/projects/current-projects/DEGAS/
http://parlab.eecs.berkeley.edu/
http://www.cs.berkeley.edu/~yelick/papers.html
The Cloudant Blog
Keep up on the latest developments at Cloudant and speaker Michael Miller.
https://cloudant.com/blog/
A Case for Redundant Arrays of Inexpensive Disks (RAID) by David Patterson, Garth Gibson, and Randy Katz
The original Berkeley Raid paper, co-authored by speaker Garth Gibson.
http://www.cs.cmu.edu/~garth/RAIDpaper/Patterson88.pdf
Michael Mitzenmacher: Publications by Year
Peruse speaker Michael Mitzenmacher’s papers on the general subject of verifiable computation.
http://www.eecs.harvard.edu/~michaelm/ListByYear.html
My Biased Coin
Speaker Michael Mitzenmacher’s take on computer science, algorithms, networking, information theory, and
related items.
http://mybiasedcoin.blogspot.com/
HPC Wire
HPCwire is a news and information portal covering the fastest computers in the world and the people who run
them.
http://www.hpcwire.com/
Datanami
Datanami is a news portal dedicated to providing insight, analysis, and up-to-the-minute information about
emerging trends and solutions in big data.
http://www.datanami.com/
REFERENCES FROM PREVIOUS TTI/VANGUARD CONFERENCES
Previous TTI/Vanguard Conferences have contained discussions and presentations on a number of topics related
to those being presented at our conference in Washington, D.C. These may be accessed from the Member
Archives section of our website (ttivanguard.com) as Reinforcement Papers and as the actual presentations.
Understanding Understanding – TTI/Vanguard Conference
October, 2012 – Pittsburgh, Pennsylvania
Taming Complexity – TTI/Vanguard Conference
October, 2011 – Washington, DC
Real Time – TTI/Vanguard Conference
July, 2011 – Paris, France
Matters of Scale – TTI/Vanguard Conference
July, 2010 – London, England
Ahead in the Clouds – TTI/Vanguard Conference
February, 2009 – San Diego, California
Smart(er) Data – TTI/Vanguard Conference
February, 2008
All That Data – TTI/Vanguard Conference
February, 2005 – Washington, D.C.
The Challenge of Complexity – TTI/Vanguard Conference
September, 2004 – Los Angeles, California
We Choose to Go Exascale, Not Because It’s Easy, but Because It’s Hard – Dr. Satoshi Matsuoka
Every year, the information technologies sector consumes an increasing share of global electricity production.
While there is the potential for this burden to lessen somewhat as more people turn from desktops and
laptops to their smaller screened cousins—smartphones and tablets—it is incumbent on the computing
sector to find ways to become as energy-miserly as possible. There are few places that recognize this as fully
as Japan, where the available generating capacity plunged precipitously in the wake of the 3.11 Tohoku
earthquake and the subsequent nuclear disaster. Not only is less electricity there to be had, but quick
replacements were of the dirty form, generated from coal-powered plants, which is antithetical to the nation’s
commitment to clean energy. Satoshi Matsuoka of the Tokyo Institute of Technology is looking to IT to help
solve the problem to which it otherwise contributes.
July, 2012 – Tokyo, Japan
Energy and Parallelism: The Challenge of Future Computing – Dr. William Dally
In terms of transistor size, there remain some generations of Moore’s Law progress ahead, but the
performance improvements that have traditionally accompanied the shrinking core are in technology’s
rearview mirror. The problem: The computational roadmap has hit the power wall. The development of
multicore chips has provided a brief respite from the inevitable, but even these undergo routine power
throttling to keep systems from overheating. This fundamental balance of power in vs. waste heat out is as
relevant to the world’s most powerful supercomputers as it is to smartphones—no digital technology is
immune. William Dally of NVIDIA makes it clear that none of this is news; he also makes it clear that not
enough is being done to ameliorate the problem.
December, 2011 – Miami, Florida
HPC Clouds: Raining Multicore Services – Dr. Daniel Reed
Over its history, science, in all its disciplines, has relied on two fundamental approaches for advancement:
empiricism and theory. The growth of the digital domain in the past decades has added simulation-based
computation to this short list, and more recently still has emerged a fourth paradigm of science: data-intensive
science. In this fourth paradigm, researchers dig into huge data stores and ongoing data streams to reveal
new patterns to drive science forward. To make progress requires extreme computing; to begin with, the data
sets are extreme, but also involved are an extreme number of processors working on the data with extreme
parallelism. The problems of high-performance computing (HPC) in science are closely aligned with those of
consumer-oriented data center computation and cloud computing, although the sociological underpinnings of
the two communities couldn’t be more different. Putting it succinctly, Microsoft Research’s Daniel Reed
compares HPC and cloud computing as “twins separated at birth.” Whereas the cloud model makes
processor- and data-intensive computation accessible to the masses, both with piecewise business models
and useful tools, HPC possesses huge resources but is hardly user-friendly.
December, 2009 – Salt Lake City, Utah
The Parallel Revolution Has Started – Dr. David Patterson
For decades, Moore’s Law has driven down the size of transistors and thereby driven up the speed and
capability of the devices that rely on the chips those transistors compose. As chips have gotten faster,
individuals and institutions have been eager to replace each current generation of computer, electronic
apparatus, or widget with its follow-on because of the new potential it promises, rather than because its
predecessor had ceased to perform as intended. Hiding within Moore’s Law was a nasty little secret,
however: More transistors demand more power, and more power generates more waste heat. In 2005, Intel’s
single-core chip clock rate maxed out at a scant handful of gigahertz and could go no higher without
exceeding the power a standard-issue computer could withstand (~100 W). The move to multiple-core chips
resulted not from a great technological breakthrough, but rather from the necessity of chip makers coming
face to face with a power wall. This new way to keep chip speed chugging along at Moore’s Law pace,
however, assumes a commensurate ability of programmers to make them perform to the collective potential
of the cores. Assumptions—as everyone learns in grade school—are fraught with danger, and in this case the
industry finds itself unprepared to switch on a dime from a sequential-programming mindset to a parallelprogramming mindset. David Patterson of the University of California-Berkeley’s Parallel Computing
Laboratory (Par Lab) is sounding the alarm.
December, 2008 – Phoenix, Arizona
Reinventing Computing – Mr. Burton Smith
Computers, as universal algorithm-executing devices, genuinely benefit from increases in speed. The steady
half-century-long fulfillment of the Moore’s Law promise of ever-increasing on-chip transistor density is nearing
the end of its viable lifecycle, however, and chip manufacturers are turning to other techniques to satisfy the
seemingly insatiable appetite for performance. The new paradigm adopted by Intel, AMD, and others is
multiple cores on the same chip; cores can be homogenous or imbued with specialized capabilities, such a
graphics processing unit (GPU) and central processing unit (CPU) on a single die with shared memory. A chip
with up to eight cores is dubbed multicore; raise the ante higher, and it is called manycore. As hardware
manufacturers forge ahead, the software community is scrambling to catch up. It takes more than parallel
hardware to create a parallel computer: it takes parallel code, appropriate languages with which to create it,
and a change in zeitgeist from which to approach the matter. Thus is the conundrum faced by Microsoft’s
Burton Smith.
December, 2007 – Santa Monica, California
On the Interaction of Life and Machines in Ultra-Large-Scale Systems – Dr. Richard Gabriel
Today’s Internet may seem immense, uncontrolled, and almost with a life of its own—just as its founders and
the end-to-end principle intended. Yet, Richard Gabriel of Sun Microsystems sees it as a diminutive precursor
of the ultra-large-scale (ULS) system—or systems—to come, the comprehension of which is beyond our ken.
Comprising trillions of lines of code and an untold number of components—large and small, coming and
going, well-defined and ephemeral, tightly integrated and discontinuous—a ULS system is too complex to
plan, analyze, or even build at the current juncture. Once conceived, this real-time, embedded, distributed
system will inherently exceed familiar management boundaries derived from the philosophical underpinnings
and tools of modern-day computer science, which indeed exist for the management of complexity through
abstraction.
December, 2005 – Washington, D.C.
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Sure, Big Data Is Great. But So Is Intuition.
John Hersey
By STEVE LOHR
Published: December 29, 2012
It was the bold title of a conference this month at the Massachusetts
Institute of Technology, and of a widely read article in The Harvard
Business Review last October: “Big Data: The Management
Revolution.”
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Andrew McAfee, principal research
scientist at the M.I.T. Center for
Bits Blog: Big Data: Rise of the
Digital Business, led off the
Machines (December 31, 2012)
conference by saying that Big Data
would be “the next big chapter of our business history.” Next on stage
was Erik Brynjolfsson, a professor and director of the M.I.T. center
and a co-author of the article with Dr. McAfee. Big Data, said
Professor Brynjolfsson, will “replace ideas, paradigms, organizations
and ways of thinking about the world.”
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These drumroll claims rest on the premise that data like Web-browsing trails, sensor
signals, GPS tracking, and social network messages will open the door to measuring and
monitoring people and machines as never before. And by setting clever computer
algorithms loose on the data troves, you can predict behavior of all kinds: shopping, dating
and voting, for example.
The results, according to technologists and business executives, will be a smarter world,
with more efficient companies, better-served consumers and superior decisions guided by
data and analysis.
I’ve written about what is now being called Big Data a fair bit over the years, and I think
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it’s a powerful tool and an unstoppable trend. But a year-end column, I thought, might be
a time for reflection, questions and qualms about this technology.
The quest to draw useful insights from business measurements is nothing new. Big Data is
a descendant of Frederick Winslow Taylor’s “scientific management” of more than a
century ago. Taylor’s instrument of measurement was the stopwatch, timing and
monitoring a worker’s every movement. Taylor and his acolytes used these time-andmotion studies to redesign work for maximum efficiency. The excesses of this approach
would become satirical grist for Charlie Chaplin’s “Modern Times.” The enthusiasm for
quantitative methods has waxed and waned ever since.
Big Data proponents point to the Internet for examples of triumphant data businesses,
notably Google. But many of the Big Data techniques of math modeling, predictive
algorithms and artificial intelligence software were first widely applied on Wall Street.
At the M.I.T. conference, a panel was asked to cite examples of big failures in Big Data. No
one could really think of any. Soon after, though, Roberto Rigobon could barely contain
himself as he took to the stage. Mr. Rigobon, a professor at M.I.T.’s Sloan School of
Management, said that the financial crisis certainly humbled the data hounds. “Hedge
funds failed all over the world,” he said.
The problem is that a math model, like a metaphor, is a simplification. This type of
modeling came out of the sciences, where the behavior of particles in a fluid, for example,
is predictable according to the laws of physics.
In so many Big Data applications, a math model attaches a crisp number to human
behavior, interests and preferences. The peril of that approach, as in finance, was the
subject of a recent book by Emanuel Derman, a former quant at Goldman Sachs and now a
professor at Columbia University. Its title is “Models. Behaving. Badly.”
Claudia Perlich, chief scientist at Media6Degrees, an online ad-targeting start-up in New
York, puts the problem this way: “You can fool yourself with data like you can’t with
anything else. I fear a Big Data bubble.”
The bubble that concerns Ms. Perlich is not so much a surge of investment, with new
companies forming and then failing in large numbers. That’s capitalism, she says. She is
worried about a rush of people calling themselves “data scientists,” doing poor work and
giving the field a bad name.
Indeed, Big Data does seem to be facing a work-force bottleneck.
“We can’t grow the skills fast enough,” says Ms. Perlich, who formerly worked for I.B.M.
Watson Labs and is an adjunct professor at the Stern School of Business at New York
University.
A report last year by the McKinsey Global Institute, the research arm of the consulting
firm, projected that the United States needed 140,000 to 190,000 more workers with
“deep analytical” expertise and 1.5 million more data-literate managers, whether retrained
or hired.
Thomas H. Davenport, a visiting professor at the Harvard Business School, is writing a
book called “Keeping Up With the Quants” to help managers cope with the Big Data
challenge. A major part of managing Big Data projects, he says, is asking the right
questions: How do you define the problem? What data do you need? Where does it come
from? What are the assumptions behind the model that the data is fed into? How is the
model different from reality?
Society might be well served if the model makers pondered the ethical dimensions of their
work as well as studying the math, according to Rachel Schutt, a senior statistician at
Google Research.
“Models do not just predict, but they can make things happen,” says Ms. Schutt, who
taught a data science course this year at Columbia. “That’s not discussed generally in our
field.”
Models can create what data scientists call a behavioral loop. A person feeds in data, which
is collected by an algorithm that then presents the user with choices, thus steering
behavior.
Consider Facebook. You put personal data on your Facebook page, and Facebook’s
software tracks your clicks and your searches on the site. Then, algorithms sift through
that data to present you with “friend” suggestions.
Understandably, the increasing use of software that microscopically tracks and monitors
online behavior has raised privacy worries. Will Big Data usher in a digital surveillance
state, mainly serving corporate interests?
Personally, my bigger concern is that the algorithms that are shaping my digital world are
too simple-minded, rather than too smart. That was a theme of a book by Eli Pariser, titled
“The Filter Bubble: What the Internet Is Hiding From You.”
It’s encouraging that thoughtful data scientists like Ms. Perlich and Ms. Schutt recognize
the limits and shortcomings of the Big Data technology that they are building. Listening to
the data is important, they say, but so is experience and intuition. After all, what is
intuition at its best but large amounts of data of all kinds filtered through a human brain
rather than a math model?
At the M.I.T. conference, Ms. Schutt was asked what makes a good data scientist.
Obviously, she replied, the requirements include computer science and math skills, but
you also want someone who has a deep, wide-ranging curiosity, is innovative and is guided
by experience as well as data.
“I don’t worship the machine,” she said.
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Looking to Industry for the Next Digital Disruption
Peter DaSilva for The New York Times
William Ruh, a vice president at General Electric, and Sharoda Paul, an expert in social computing.
By STEVE LOHR
Published: November 23, 2012
SAN RAMON, Calif. — When Sharoda Paul finished a postdoctoral
fellowship last year at the Palo Alto Research Center, she did what
most of her peers do — considered a job at a big Silicon Valley
company, in her case, Google. But instead, Ms. Paul, a 31-year-old
expert in social computing, went to work for General Electric.
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Ms. Paul is one of more than 250 engineers recruited in the last year
and a half to G.E.’s new software center here, in the East Bay of San
Francisco. The company plans to increase that work force of
computer scientists and software developers to 400, and to invest $1
billion in the center by 2015. The buildup is part of G.E’s big bet on
what it calls the “industrial Internet,” bringing digital intelligence to
the physical world of industry as never before.
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The concept of Internet-connected machines that collect data and
communicate, often called the “Internet of Things,” has been around for years.
Information technology companies, too, are pursuing this emerging field. I.B.M. has its
“Smarter Planet” projects, while Cisco champions the “Internet of Everything.”
But G.E.’s effort, analysts say, shows that Internet-era technology is ready to sweep
through the industrial economy much as the consumer Internet has transformed media,
communications and advertising over the last decade.
In recent months, Ms. Paul has donned a hard hat and safety boots to study power plants.
She has ridden on a rail locomotive and toured hospital wards. “Here, you get to work with
things that touch people in so many ways,” she said. “That was a big draw.”
G.E. is the nation’s largest industrial company, a producer of aircraft engines, power plant
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turbines, rail locomotives and medical imaging equipment. It makes the heavy-duty
machinery that transports people, heats homes and powers factories, and lets doctors
diagnose life-threatening diseases.
G.E. resides in a different world from the consumer Internet. But the major technologies
that animate Google and Facebook are also vital ingredients in the industrial Internet —
tools from artificial intelligence, like machine-learning software, and vast streams of new
data. In industry, the data flood comes mainly from smaller, more powerful and cheaper
sensors on the equipment.
Smarter machines, for example, can alert their human handlers when they will need
maintenance, before a breakdown. It is the equivalent of preventive and personalized care
for equipment, with less downtime and more output.
“These technologies are really there now, in a way that is practical and economic,” said
Mark M. Little, G.E.’s senior vice president for global research.
G.E.’s embrace of the industrial Internet is a long-term strategy. But if its optimism proves
justified, the impact could be felt across the economy.
The outlook for technology-led economic growth is a subject of considerable debate. In a
recent research paper, Robert J. Gordon, a prominent economist at Northwestern
University, argues that the gains from computing and the Internet have petered out in the
last eight years.
Since 2000, Mr. Gordon asserts, invention has focused mainly on consumer and
communications technologies, including smartphones and tablet computers. Such devices,
he writes, are “smaller, smarter and more capable, but do not fundamentally change labor
productivity or the standard of living” in the way that electric lighting or the automobile
did.
But others say such pessimism misses the next wave of technology. “The reason I think
Bob Gordon is wrong is precisely because of the kind of thing G.E. is doing,” said Andrew
McAfee, principal research scientist at M.I.T.’s Center for Digital Business.
Today, G.E. is putting sensors on everything, be it a gas turbine or a hospital bed. The
mission of the engineers in San Ramon is to design the software for gathering data, and
the clever algorithms for sifting through it for cost savings and productivity gains. Across
the industries it covers, G.E. estimates such efficiency opportunities at as much as $150
billion.
Some industrial Internet projects are already under way. First Wind, an owner and
operator of 16 wind farms in America, is a G.E. customer for wind turbines. It has been
experimenting with upgrades that add more sensors, controls and optimization software.
The new sensors measure temperature, wind speeds, location and pitch of the blades. They
collect three to five times as much data as the sensors on turbines of a few years ago, said
Paul Gaynor, chief executive of First Wind. The data is collected and analyzed by G.E.
software, and the operation of each turbine can be tweaked for efficiency. For example, in
very high winds, turbines across an entire farm are routinely shut down to prevent damage
from rotating too fast. But more refined measurement of wind speeds might mean only a
portion of the turbines need to be shut down. In wintry conditions, turbines can detect
when they are icing up, and speed up or change pitch to knock off the ice.
Upgrades on 123 turbines on two wind farms have so far delivered a 3 percent increase in
energy output, about 120 megawatt hours per turbine a year. That translates to $1.2
million in additional revenue a year from those two farms, Mr. Gaynor said.
“It’s not earthshaking, but it is meaningful,” he said. “These are real commercial
investments for us that make economic sense now.”
For the last few years, G.E. and Mount Sinai Medical Center have been working on a
project to optimize the operations of the 1,100-bed hospital in New York. Hospitals, in a
sense, are factories of health care. The challenge for hospitals, especially as cost pressures
tighten, is to treat more patients more efficiently, while improving the quality of care.
Technology, said Wayne Keathley, president of Mount Sinai, can play a vital role.
At Mount Sinai, patients get a black plastic wristband with a location sensor and other
information. Similar sensors are on beds and medical equipment. An important
advantage, Mr. Keathley said, is to be able to see the daily flow of patients, physical assets
and treatment as it unfolds.
But he said the real benefit was how the data could be used to automate and streamline
operations and then make better decisions. For example, in a typical hospital, getting a
patient who shows up in an emergency room into an assigned bed in a hospital ward can
take several hours and phone calls.
At Mount Sinai, G.E. has worked on optimization and modeling software that enables
admitting officers to see beds and patient movements throughout the hospital, to help
them more efficiently match patients and beds. Beyond that, modeling software is
beginning to make predictions about likely patient admission and discharge numbers over
the next several hours, based on historical patterns at the hospital and other
circumstances — say, in flu season.
The software, which Mount Sinai has been trying out in recent months, acts as an
intelligent assistant to admitting officers. “It essentially says, ‘Hold off, your instinct is to
give this bed to that guy, but there might be a better choice,’ ” Mr. Keathley explained.
At a hospital like Mount Sinai, G.E. estimates that the optimization and modeling
technologies can translate into roughly 10,000 more patients treated a year, and $120
million in savings and additional revenue over several years.
The origins of G.E.’s industrial Internet strategy date back to meetings at the company’s
headquarters in Fairfield, Conn., in May 2009. In the depths of the financial crisis, Jeffrey
R. Immelt, G.E.’s chief executive, met with his senior staff to discuss long-term growth
opportunities. The industrial Internet, they decided, built on G.E.’s strength in research
and could be leveraged across its varied industrial businesses, adding to the company’s
revenue in services, which reached $42 billion last year.
Now G.E. is trying to rally support for its vision from industry partners, academics,
venture capitalists and start-ups. About 250 of them have been invited to a conference in
San Francisco, sponsored by the company, on Thursday.
Mr. Immelt himself becomes involved in recruiting. His message, he says, is that if you
want to have an effect on major societal challenges like improving health care, energy and
transportation, consider G.E.
An early convert was William Ruh, who joined G.E. from Cisco, to become vice president
in charge of the software center in San Ramon. And Mr. Ruh is taking the same message to
high-tech recruits like Ms. Paul. “Here, they are working on things they can explain to
their parents and grandparents,” he said. “It’s not a social network,” even if the G.E.
projects share some of the same technology.
General Electric ties smart meters, grid sensors, and enterprise IT
into a cloud-hosted package. But will utilities buy in?
JEFF ST. JOHN: DECEMBER 12, 2012
Two weeks ago, General Electric made a big splash in the world of the Internet of
Things, or, as GE likes to call it, the “industrial internet.” In a series of high-profile
announcements, the global energy and engineering giant laid out its plan to add
networking and distributed intelligence capabilities to more and more of its devices,
ranging from aircraft engines to industrial and grid control systems, and start
analyzing all that data to drive big new gains in efficiency across the industries it
serves.
That includes the smart grid, of course. GE is a massive grid player, alongside such
competitors as Siemens, ABB, Alstom, Schneider Electric and Eaton/Cooper
Power. But in terms of scope, GE and Siemens stand apart in that they make
everything from natural gas and wind turbines, to the heavy transmission and
distribution gear -- transformers, sensors, switches and the like -- that delivers it to
end users.
GE and its competitors also have their own lines of industrial communication,
networking and control gear for distribution automation (DA) tasks on the grid, of
course. Unlike most of the above-named competitors, however, GE is also a big
maker of smart meters – although the networking technology that links up all those
meters tends to come from other partners.
So we’ve got the technological underpinnings for a true Internet of things
environment on the smart grid. But who’s managing it all on the back end? Right
now, utilities tend to run their own data centers and back-office control rooms. But
legacy billing, customer service and enterprise resource planning systems don’t
easily integrate with the new breed of data coming at them from the smart grid.
Indeed, we’ve got a host of IT giants like Cisco, IBM, Microsoft, Oracle, SAP,
Infosys, Wipro and many more offering smart grid software services and
integration, aimed at making sure data from smart meters, grid sensors and other
formerly siloed technologies can be freely shared across the enterprise.
Perhaps the most important stepping stone for GE in moving its smart grid
business into the “industrial internet” age is to capture its own share of this future
market in smart grid integration. GE’s “Grid IQ Solutions as a Service” business,
launched last year, represents that effort. In a move increasingly being rolled out by
grid giants and startups alike, GE is moving the smart grid to the cloud -- in this
case, dedicated servers in its GE Digital Energy data center in Atlanta, Ga. -- and
offering utilities the opportunity to choose from a list of products and functions
they’d like to deploy, all for a structured fee.
In the year since it launched, GE’s smart grid service has landed two city utilities,
Norcross, Ga. and Leesburg, Fla., as named customers for its first SaaS product
line, the Grid IQ Connect platform. That’s essentially a smart meter deployment run
and managed by GE working with unnamed AMI partners, Todd Jackson, SaaS
product line leader for GE Digital Energy, said in a Tuesday interview.
GE has lined up partners to provide a host of AMI networking flavors, including the
mesh networking that dominates in U.S. smart meter deployments to date, as well
as point-to-multipoint and cellular solutions, Jackson said. That’s not unlike GE’s
current smart metering business model, in which it works with partners such as
Silver Spring Networks, Trilliant, and others that add their own communications
gear to GE’s core meters.
GE’s new role as back-end IT services provider to its Grid IQ Connect customers
means that GE is also bringing a lot more software expertise to the fore, Jackson
noted. While its AMI partners tend to provide the networking and meter data
management aspects of the deployment, GE is providing about half of the
remaining IT functionality, he said -- including the core task of hosting all its
partners’ software on its own dedicated servers. GE has also been rolling out new
feature sets for its smart-grid-as-a-service platform, including prepay options for
smart meters, as well as its Grid IQ Restore, which adds outage detection and
management to the array of options for its customers.
Earlier this year, GE also took a step beyond the utility and into the homes and
businesses that they serve, launching its Grid IQ Respond platform. Essentially, it’s
a version of GE’s demand response technology offered over the cloud, and is
currently being rolled out with three unnamed utilities, two in the United States and
one in Europe, Jackson said.
Right now the projects are mostly focused on homes, he explained, and most of
those are connecting to load control switches, attached to major household loads
like pool pumps, water heaters and air conditioners, that the utility can switch off
and on to help manage peak power demands. A few million homes across the U.S.
have these kinds of radio or pager-operated load control switches installed, usually
in exchange for rebates or cheaper power rate offers from utilities desperate to curb
their customers’ appetite for expensive peak power.
At the same time, competitors in this business, such as Honeywell, Eaton/Cooper
Power, Comverge and others, have been busy working on their own softwareas-a-service models, complete with cloud-hosted applications and increasing
options for networking end-devices in homes and businesses. And of course, we’ve
got literally dozens of startups competing for the still-nascent market for in-home
energy management devices and the networks that can connect them to utilities, as
well as the internet at large.
GE, which is a huge appliance maker, has its own version of a home energy
management device, called the Nucleus. But it hasn’t rolled it out to market yet,
preferring to keep it in pilot projects so far, and Jackson said there aren’t any
immediate plans to include it in GE’s Grid IQ Respond offerings.
As for target markets, GE is largely looking at municipal utilities and cooperatives,
which tend to lack the big budgets and capital expenditure recovery mechanisms of
larger investor-owned utilities (IOUs), Jackson said. At the same time, GE does
offer its smart grid platform in a so-called “boosted model,” in which utilities can put
the capital equipment on their balance sheets, as well as a managed service model
where GE owns the hardware, he said. So far, utilities are about evenly split in their
interest between the two business models, he said.
So how does this tie into the Internet of Things concept? Well, “Once the network is
deployed, there are other things that municipal utilities can tie in there and benefit
from,” Jackson noted. Some examples include the ability to connect streetlights or
traffic cameras to the same network that supports smart meters, he said. That’s a
concept that we’ve seen deployed by such smart grid players as Sensus and Santa
Clara, Calif.-based startup Tropos Networks, which was bought by grid giant ABB
earlier this year.
On the backend IT side, GE is also tackling challenges like connecting smart meter
data to customer service platforms and other utility business software platforms,
Jackson said. That’s led to integration that allows customer service reps to tie
directly into an individual customer’s smart meter during an outage, to figure out
whether or not it’s a utility problem or a blown fuse, for example -- the kind of
incremental improvement that only comes when data is freely shared.
Whether or not utilities will catch on to the smart-grid-as-a-service model remains to
be seen. Jackson said that GE has been talking to multiple utilities that haven't
announced themselves yet. Amidst a general slowdown in North American smart
meter deployments expected next year, smaller municipal and cooperative utilities
stand out as a relatively untapped sector -- and one that will need some help in
managing the IT behind an AMI or DA deployment at a cost commensurate with the
smaller scale of their projects, in the tens or hundreds of thousands of meters,
rather than millions.
Utilities do face some regulatory challenges and uncertainties in turning over key
parts of their operations to a third party. At the same time, they're under pressure to
meet a whole new array of requirements, including smart grid security and data
privacy, that may well be better managed by a big central provider like GE than by
each small utility. In the end, services will be the key to unlocking the small utility
smart grid market, to be sure. But GE faces plenty of competition in establishing
itself as the platform to trust -- and as with every shift in the way utilities do
business, it's going to take years to develop.
With high-performance computing, "pixie-booting" servers a half-world away.
by Sean Gallagher - Oct 18 2012, 3:13pm PDT
The inside of Equinix's co-location facility in San Jose—the home of CloudFlare's primary data center.
Photo: Peter McCollough/Wired.com
On August 22, CloudFlare, a content delivery network, turned on a brand new data center in Seoul,
Korea—the last of ten new facilities started across four continents in a span of thirty days. The Seoul
data center brought CloudFlare's number of data centers up to 23, nearly doubling the company's
global reach—a significant feat in itself for a company of just 32 employees.
But there was something else relatively significant about the Seoul data center and the other 9 facilities
set up this summer: despite the fact that the company owned every router and every server in their
racks, and each had been configured with great care to handle the demands of CloudFlare's CDN and
security services, no one from CloudFlare had ever set foot in them. All that came from CloudFlare
directly was a six-page manual instructing facility managers and local suppliers on how to rack and
plug in the boxes shipped to them.
"We have nobody stationed in Stockholm or Seoul or Sydney, or a lot of the places that we put these
new data centers," CloudFlare CEO Matthew Prince told Ars. "In fact, no CloudFlare employees have
stepped foot in half of the facilities where we've launched." The totally remote-controlled data center
approach used by the company is one of the reasons that CloudFlare can afford to provide its services
for free to most of its customers—and still make a 75 percent profit margin.
In the two years since its launch, the content delivery network and denial-of-service protection
company has helped keep all sorts of sites online during global attacks, both famous and infamous
—including recognition from both Davos and LulzSec. And all that attention has amounted to
Yahoo-sized traffic—the CloudFlare service has handled over 581 billion pageviews since its launch.
Yet CloudFlare does all this without the sort of Domain Name Service "black magic" that Akamai and
that level of efficiency, CloudFlare has done some black magic of a different sort, relying on
open-source software from the realm of high-performance computing, storage tricks from the world of
"big data," a bit of network peering arbitrage and clever use of a core Internet routing technology.
In the process, it has created an ever-expanding army of remote-controlled service points around the
globe that can eat 60-gigabit-per-second distributed denial of service attacks for breakfast.
CloudFlare's CDN is based on Anycast, a standard defined in the Border Gateway Protocol—the
routing protocol that's at the center of how the Internet directs traffic. Anycast is part of how BGP
supports the multi-homing of IP addresses, in which multiple routers connect a network to the Internet;
through the broadcasts of IP addresses available through a router, other routers determine the shortest
path for network traffic to take to reach that destination.
Using Anycast means that CloudFlare makes the servers it fronts appear to be in many places, while
only using one IP address. "If you do a traceroute to Metallica.com (a CloudFlare customer), depending
on where you are in the world, you would hit a different data center," Prince said. "But you're getting
back the same IP address."
That means that as CloudFlare adds more data centers, and those data centers advertise the IP
addresses of the websites that are fronted by the service, the Internet's core routers automatically
re-map the routes to the IP addresses of the sites. There's no need to do anything special with the
Domain Name Service to handle load-balancing of network traffic to sites other than point the
hostname for a site at CloudFlare's IP address. It also means that when a specific data center needs to
be taken down for an upgrade or maintenance (or gets knocked offline for some other reason), the
routes can be adjusted on the fly.
That makes it much harder for distributed denial of service attacks to go after servers behind
CloudFlare's CDN network; if they're geographically widespread, the traffic they generate gets spread
across all of CloudFlare's data centers—as long as the network connections at each site aren't
overcome.
In September, Prince said, "there was a brand new botnet out there launching big attacks, and it
targeted one of our customers. It generated 65 gigabits per second of traffic hitting our network. But
none of that traffic was focused in one place—it was split fairly evenly across our 23 data centers, so
each of those facilities only had to deal with about 3 gigs of traffic. That's much more manageable."
Making CloudFlare's approach work requires that it put its networks as close as possible to the core
routers of the Internet—at least in terms of network hops. While companies like Google, Facebook,
Microsoft, and Yahoo have gone to great lengths to build their own custom data centers in places
where power is cheap and where they can take advantage of the economies of scale, CloudFlare looks
to use existing facilities that "your network traffic would be passing through even if you weren't using
our service," Prince said.
As a result, the company's "data centers" are usually at most a few racks of hardware, installed at
co-location facilities that are major network exchange points. Prince said that most of his company's
data centers are set up at Equinix IBX co-location facilities in the US, including CloudFlare's primary
facility in San Jose—a facility also used by Google and other major cloud players as an on-ramp to the
Internet.
CloudFlare looks for co-location facilities with the same sort of capabilities wherever it can. But these
sorts of facilities tend to be older, without the kind of power distribution density that a custom-built data
center would have. "That means that to get as much compute power as possible into any given rack,
we're spending a lot of time paying attention to what power decisions we make," Prince said.
The other factor driving what goes into those racks is the need to maximize the utilization of
CloudFlare's outbound Internet connections. CloudFlare buys its bandwidth wholesale from network
transit providers, committing to a certain level of service. "We're paying for that no matter what," Prince
said, "so it's optimal to fill that pipe up."
That means that the computing power of CloudFlare's servers is less of a priority than networking and
cache input/output and power consumption. And since CloudFlare depends heavily on the facility
providers overseas or other partners to do hardware installations and swap-outs, the company needed
to make its servers as simple as possible to install—bringing it down to that six-page manual. To make
that possible, CloudFlare's engineering team drew on experience and technology from the
high-performance computing world.
"A lot of our team comes from the HPC space," Prince said. "They include people who built HPC
networks for the Department of Energy, where they have an 80 thousand node cluster, and had to
figure out how to get 80,000 computers, fit them into one space, cable them in a really reliable way, and
make sure that you can manage them from a single location."
One of the things that CloudFlare brought over from the team's DoE experience was the Perceus
Provisioning System, an open-source provisioning system for Linux used by DoE for its HPC
environments. All of CloudFlare's servers are "pixie-booted" (using a Preboot eXecution Environment,
or PXE) across a virtual private network between data centers; servers are delivered with no operating
system or configuration whatsoever, other than a bootloader that calls back to Perceus for provisioning.
"The servers come from whatever equipment vendor we buy them from completely bare," Prince said.
"All we get from them is the MAC address."
CloudFlare's servers run on a custom-built Linux distribution based on Debian. For security purposes,
the servers are "statelessly" provisioned with Perceus—that is, the operating system is loaded
completely in RAM. The mass storage on CloudFlare servers (which is universally based on SSD
drives) is used exclusively for caching data from clients' sites.
The gear deployed to data centers that gets significant pre-installation attention from CloudFlare's
engineers is the routers—primarily supplied by Juniper Networks, which works with CloudFlare to
preconfigure them before being shipped to new data centers. Part of the configuration is to create
virtual network connections over the Internet to the other CloudFlare data centers, which allows each
data center to use its nearest peer to pull software from during provisioning and updating.
"When we booted up Vienna, for example," said Prince, "the closest data center was Frankfurt, so we
used the Frankfurt facility to boot the new Vienna facility." One server in Vienna was booted first as the
"master node," with provisioning instructions for each of the other machines. Once the servers are all
provisioned and loaded, "they call back to our central facility (in San Jose) and say, 'Here are our MAC
addresses, what do you need us to do?'"
Once the machines have passed a final set of tests, each gets designated with an operational
responsibility: acting as a proxy for Web requests to clients' servers, managing the cache of content to
speed responses, DNS and logging services. Each of those services can be run on any server in the
stack and step up to take over a service if one of its comrades fails.
Caching is part of the job for every server in each CloudFlare facility, and being able to scale up the
size of the cache is another reason for the modular nature of how the company thinks of servers.
Rather than storing cached webpage objects in a traditional database or file system, CloudFlare uses a
hash-based database that works in a fashion similar to "NoSQL" databases like 10gen's MongoDB and
Amazon's Dynamo storage system.
When a request for a webpage comes in for the first time, CloudFlare retrieves the site contents. A
consistent hashing algorithm in CloudFlare's caching engine then converts the URL used to call each
element into a value, which is used as the key under which the content is stored locally at each data
center. Each server in the stack is assigned a range of hashes to store content for, and subsequent
requests for the content are routed to the appropriate server for that cache.
Unlike most database applications, the cache stored at each CloudFlare facility has an undefined
expiration date—and because of the nature of those facilities, it isn't a simple matter to add more
storage. To keep the utilization level of installed storage high, the cache system simply purges older
cache data when it needs to store new content.
The downside of the hash-based cache's simplicity is that it has no built-in logging system to track
content. CloudFlare can't tell customers which data centers have copies of which content they've
posted. "A customer will ask me, 'Tell me all of the files you have in cache,'" Prince said. "For us, all we
know is there are a whole bunch of hashes sitting on a disk somewhere—we don't keep track of which
object belongs to what site."
The upside, however, is that the system has a very low overhead as a result and can retrieve site
content quickly and keep those outbound pipes full. And when you're scaling a 32-person company to
fight the speed of light worldwide, it helps to keep things as simple as possible.
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2012 SEMTECHBIZ WEST
By Paul Miller on June 7, 2012 1:30 PM
Daniel Tunkelang, Principal Data Scientist at LinkedIn, delivered the final
keynote at SemTechBiz in San Francisco this morning, exploring the way in
which “semantics emerge when we apply the right analytical techniques to
a sufficient quality and quantity of data.”
Daniel began by offering his key takeaways for the presentation;
Communication trumps knowledge representation.
Communication is the problem and the solution.
Knowledge representation, and the systems that support it, are possibly over-rated. We get too
obsessed, Tunkelang suggested, with building systems that are ‘perfect,’ and in setting out to
assemble ‘comprehensive’ sets of data.
On the flip side, Computation is underrated – machines can do a lot to help us cope with
incomplete or messy data, especially at scale.
We have a communication problem.
Daniel goes back to the dream of AI, referencing Arthur C Clarke’s HAL 9000 and Star Trek’s
android, Data. Both, he suggests, were “constructed by their authors as intelligent computers.”
Importantly, they “supported natural language interfaces” to communicate with humans. Their
creators, Tunkelang suggested, believed that the computation and the access to knowledge were
the hard part – communication was an ‘easy’ after-thought.
Moving on, we reach Vannevar Bush‘s Memex from the 1940′s.
And in the 1980s we reach Cyc. Loaded with domain-specific knowledge and more, but “this
approach did not and will not get us” anywhere particularly useful.
Moving closer to the present, Freebase. “One of the best practical examples of semantic
technologies in the semantic web sense… doing relations across a very large triple store… and
making the result available in an open way.” But Freebase has problems, and “they are
fundamental in nature.” When you’re dealing with structured data acquired from all over the world,
it is difficult to ensure consistency or completeness. “We’re unlikely to achieve perfection, so we
shouldn’t make perfection a requirement for success.”
Wolfram Alpha, starting from a proprietary collection of knowledge, is enabling reasoning and
analysis over a growing collection of data. Wolfram Alpha is very good when it’s good, but
extremely weak when it comes to guiding users toward ‘appropriate’ sources; there is a breakdown
in communication, and a failure to manage or guide user expectations.
“Today’s knowledge repositories are incomplete, inconsistent, and inscrutable.”
“They are not sustained by economic incentives.”
Computation is under-rated. IBM’s Deep Blue, for example. A feat of brute-force computation
rather than semantics, intelligence or cleverness. “Chess isn’t that hard.”
Also IBM – Watson and its win at Jeopardy. “A completely different ball of wax to playing chess”
that is far more freeform and unpredictable than rules-based chess. Although Stephen Wolfram’s
blog post from 2011 suggests that internet search engines can also actually do pretty well in
identifying Jeopardy answers.
Google’s Alon Halevy, Peter Norvig and Fernando Pereira suggested in 2009 that “more data
beats clever algorithms.”
Where can we go from here? “We have a glut of semi-structured data.”
LinkedIn has a lot of semi-structured data from 160 million members, predominantly in the form of
free-text descriptive profile text;
marked-up (but typically incomplete and ambiguous) statements of employment, education,
promotion etc;
(also typically incomplete) graph data representing the relationships between people and roles.
Semi-structured search is a killer app. Faceted search UI on LinkedIn, letting the user explore and
follow connections, without the need for data to be entered in accordance with a strict classification
system or data model.
There is no perfect schema or vocabulary. And even if there were, not everyone would ue it.
Knowledge representation only tends to exceed in narrowly scoped areas. Brute force computation
can be surprisingly successful.
Machines don’t have to be perfect. Structure doesn’t have to be perfect. We don’t have to be
perfect. Communicate with the user. Offer a UI that guides them and helps them explore. Don’t aim
for perfection. Offer just enough to help the user move forward.
“More data beats clever algorithms, but better data beats more data.” Computation isn’t the enemy.
Make sure ‘better’ data – from SemTech community and others – is available to these machines
and then we’ll see something remarkable.
For more from Daniel, listen to April’s episode of the Semantic Link podcast in which he was our
guest.
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Beyond MapReduce: Hadoop hangs on
Tooling up
By Matt Asay • Get more from this author
Posted in Developer, 10th July 2012 13:03 GMT
Free whitepaper – EMA advanced performance analytics report
Open ... and Shut Hadoop is all the rage in enterprise computing, and has become the poster child for
the big-data movement. But just as the enterprise consolidates around Hadoop, the web world, including
Google – which originated the technology ideas behind Hadoop – is moving on to real-time, ad-hoc
analytics that batch-oriented Hadoop can't match.
Is Hadoop already outdated?
As Cloudant chief scientist Mike Miller points out,
Google's MapReduce approach to big data analytics
may already be passé. It certainly is at Google:
[Google's MapReduce] no longer holds such
prominence in the Google stack... Google seems
to be moving past it. In fact, many of the
technologies [Google now uses like Percolator
for incremental indexing and analysis of
frequently changing datasets and Dremel for
ad-hoc analytics] aren’t even new; they date
back the second half of the last decade, mere
years after the seminal [MapReduce] paper was
in print.
By one estimate, Hadoop, which is an open-source implementation of Google's MapReduce technology,
hasn't even caught up to Google's original MapReduce framework. And now people like Miller are
arguing that a MapReduce approach to Big Data is the wrong starting point altogether.
For a slow-moving enterprise, what to do?
The good news is that soon most enterprises likely won't have to bother with Hadoop at all, as Hadoop
will be baked into the cloud applications that enterprises buy. And as those vendors figure out better
technologies to handle real-time (like Storm) or ad hoc analysis (like Dremel), they, too, will be baked into
cloud applications.
As an interim step to such applications, big-data tools vendors like Datameer and Karmasphere are
already releasing cloud-based tools for analyzing Hadoop data. This is critical to Hadoop's short-term
success as Forrester notes that Hadoop is still "an immature technology with many moving parts that are
neither robust nor well integrated." Good tooling helps.
But is Hadoop the right place to start, good tooling or no?
Cloudscale chief executive Bill McColl, writing back in 2010, says "definitely not." He argues:
Simple batch processing tools like MapReduce and Hadoop are just not powerful enough in
any one of the dimensions of the big data space that really matters. Sure, Hadoop is great for
simple batch processing tasks that are “embarrassingly parallel”, but most of the difficult big
data tasks confronting companies today are much more complex than that.
McColl isn't a neutral observer of Hadoop: his company competes with vanilla Hadoop deployments. My
own company, Nodeable, offers a real-time complement to Hadoop, based on the open-source Storm
project, but I'm much more sanguine about Hadoop's medium-term prospects than either McColl or
Miller. But his point is well-taken, especially in light of Miller's observation that even the originator of
MapReduce, Google, has largely moved on for faster, more responsive analytical tools.
Does it matter?
Probably not. At least, not anytime soon. It has long been the case that web giants like Facebook and
Google have moved faster than enterprise IT, which tends to be much more risk-averse and more prone
to hanging onto technology once it's made to work. So it's a Very Good Thing, as Businessweek
highlights, that the web's technology of today is being open sourced to fuel the enterprise technology of
tomorrow.
Hadoop still has several kinks to work out before it can go truly mainstream in the enterprise. It's not as if
enterprises are going to go charging ahead into Percolator or other more modern approaches to big data
when they have yet to squeeze Hadoop for maximum value. Enterprise IT managers like to travel in
packs, and the pack is currently working on Hadoop. There may be better options out there, but they're
going to need to find ways to complement Hadoop, not displace it. Hadoop simply has too much
momentum going for it.
I suspect we'll see Hadoop continue forward as the primary engine of big data analytics. We're looking at
many years of dominance for Hadoop. However, I think we'll also see add-on technologies offered by
cloud vendors to augment the framework. Hadoop is never going to be a real-time system, so things like
Storm will come to be viewed as must-have tools to provide real-time insight alongside Hadoop's timely,
deep analytics.
Some early adopters will figure these tools out on their own without help from cloud application vendors.
But for most, they're going to buy, not build, and that "buy" decision will include plenty of Hadoop,
whether from Cloudera or Metamarkets or Hortonworks or EMC or anybody else. That's why Forrester
pegs today's Hadoop ecosystem at $1bn, a number that is only going to grow, no matter what Google
thinks is a better approach to big data. ®
Matt Asay is senior vice president of business development at Nodeable, offering systems management for managing and analysing
cloud-based data. He was formerly SVP of biz dev at HTML5 start-up Strobe and chief operating officer of Ubuntu commercial
operation Canonical. With more than a decade spent in open source, Asay served as Alfresco's general manager for the Americas
and vice president of business development, and he helped put Novell on its open source track. Asay is an emeritus board member
of the Open Source Initiative (OSI). His column, Open...and Shut, appears three times a week on The Register.
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17
Sun Microsystems cofounder and networking guru Andy Bechtholsheim
predicted that networking chips — which determine how quickly you can surf
the Internet — will keep following the path of progress that it has for decades.
Moore’s Law, the prediction in 1965 by Intel’s Gordon Moore that the number
of transistors on a chip will double every two years, is still holding up.
In the next 20 years,
Bechtolsheim
expects an
improvement of
1,000 times in chip
performance. We
should all greet that
with relief, since the
$1 trillion-plus
electronics economy
7
T
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Share
depends on the continuous efficiencies gained from making chips
smaller, faster, and cheaper.
“We are in the Golden Age of networking, driven by Moore’s
Law,” said Bechtolsheim in a keynote speech at the Linley Tech Processor conference in San Jose, Calif.
Bechtolsheim is worth listening to. He is the founder, chairman, and chief development officer at networking hardware firm
Arista Networks, which builds cloud-based appliances for large data centers. He was the chief system architect at Sun and
became famous as the angel who funded Google.
He also started Granite Systems, which Cisco acquired in 1996. He developed a series of switches at Cisco and also founded
Kealia, which Sun acquired in 2004.
Bechtolsheim talked a lot about leaf switches and buffers and spines and other stuff that was way over my head. But he
closed his talk with a series of predictions about the future of Moore’s Law and its relevance to the future of networking,
which depends on data centers with tons of servers, each with lots of chips, powered by multiple computing brains (or
cores). In data centers, keeping the flow of data moving as fast as possible between the outside world, through memory, into
processors and into long-term storage is of paramount concern. It’s a realm in which nanoseconds matter.
Today’s networking chips transfer data at rates of 10 gigabits a second, 40 gigabits a second, or 100 gigabits a second. Part of
that depends on chips, but it also depends on optical components that transfer data using laser components, which are
harder to improve compared to silicon chip technology.
“Optics, unfortunately, is not on Moore’s Law,” said Bechtolsheim. But he remained optimistic about progress in the future.
Bechtolsheim predicted:
1. Moore’s Law is alive and well. By doubling the number of
transistors per chip every two years, chip designers will be able to
keep feeding faster and cheaper chips to networking-gear
designers. In the next 12 years, the path ahead is clear,
Bechtolsheim said. That will give us almost 100 times more
transistors — the basic on-off switches of digital computers — on
every chip.
2. The economics of chips are changing. Each generation of chip
design is getting more expensive as it takes more engineers to
craft designs from the greater number of transistors available.
Designing a new complex switch chip for networks can cost $20
million. Making chips that sell in low volumes is no longer viable.
Chip startups often can’t afford to do this anymore. And in-house designs make less sense.
3. Merchant-network silicon vendors will gain market share. Those who design chips that many different system companies
use will likely prevail over those who design in-house chips for just one vendor. Moreover, the differentiation now happens
in the software that runs on the chip, not in the hardware. And internal design teams often can’t keep up with advances in
silicon design tools on the merchant market.
4. Custom designs lead the way. Custom designs can get more bang for the buck out of the available transistors. So even the
merchant silicon vendors will have to modify solutions for each customer.
5. Using the best available silicon manufacturing technology is the key. With each new manufacturing generation, chips
become faster, smaller, and cheaper. Today’s silicon chip designs have to be built in 28-nanometer technology or better.
Those designs must use less power, access more memory, and perform faster.
“No one wants to roll the clock back, and the silicon march is relentless,” Bechtolsheim said.
6. Product life cycles are shorter. Each new silicon chip has a shorter life, but it can ship in higher volumes. The days of
10-year product life cycles are gone and will never come back. Chip designers and system makers can count on frequent
upgrade cycles, but they’re face more competition.
7. Architecture matters. Having a faster internal engine makes a car run faster. That’s also true for a chip. With better design
at the component level, the overall chip and system run better. This requires rethinking approaches that worked in the past
for a more modern technology. Keeping the data flowing within the chip is critical.
8. Flexibility matters. Chips are becoming more versatile and programmable. They can support a variety of protocols and
usage cases. Flexibility allows for reuse over generations and expansion to new markets.
9. Building blocks matter. In the age of multitasking, multiple components matter. Replicating cores, or brains, across a chip
is the way to faster, more reliable, and lower-power chips. Every component is reusable.
10. The system is the chip. In the future, with more efficient manufacturing technology, future switch chips will be
single-chip designs. That requires close communication between makers of systems, software designers, and chip vendors.
Anyone who tries to lock out any of the other parties will likely be doomed.
“In conclusion, Moore’s Law is alive and well,” Bechtolsheim said.
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Internet Architects Warn of Risks in Ultrafast Networks
By QUENTIN HARDY
Published: November 13, 2011
SANTA CLARA, Calif. — If nothing else, Arista Networks proves that
two people can make more than $1 billion each building the Internet
and still be worried about its reliability.
Enlarge This Image
Jim Wilson/The New York Times
The Arista Networks founders,
Andreas Bechtolsheim, left, David
Cheriton and Kenneth Duda, with a
data-routing switch at the company's
headquarters in Santa Clara, Calif.
Enlarge This Image
David Cheriton, a computer science
professor at Stanford known for his
skills in software design, and Andreas
Bechtolsheim, one of the founders of
Sun Microsystems, have committed
$100 million of their money, and
spent half that, to shake up the
business of connecting computers in
the Internet’s big computing centers.
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As the Arista founders say, the promise of having access to
mammoth amounts of data instantly, anywhere, is matched
by the threat of catastrophe. People are creating more data
and moving it ever faster on computer networks. The fast
networks allow people to pour much more of civilization
online, including not just Facebook posts and every book
ever written, but all music, live video calls, and most of the
information technology behind modern business, into a
worldwide “cloud” of data centers. The networks are
designed so it will always be available, via phone, tablet,
personal computer or an increasing array of connected
devices.
Statistics dictate that the vastly greater number of
transactions among computers in a world 100 times faster
than today will lead to a greater number of unpredictable
Jim Wilson/The New York Times
accidents,
with less time in between them. Already,
Lorenz Redlefsen, an Arista engineer,
with a data-routing switch.
Amazon’s cloud for businesses failed for several hours in
April, when normal computer routines faltered and the
system overloaded. Google’s cloud of e-mail and document collaboration software has
been interrupted several times.
“We think of the Internet as always there. Just because we’ve become dependent on it, that
doesn’t mean it’s true,” Mr. Cheriton says. Mr. Bechtolsheim says that because of the
Internet’s complexity, the global network is impossible to design without bugs. Very
dangerous bugs, as they describe them, capable of halting commerce, destroying financial
information or enabling hostile attacks by foreign powers.
Both were among the first investors in Google, which made them billionaires, and, before
that, they created and sold a company to the networking giant Cisco Systems for $220
million. Wealth and reputations as technology seers give their arguments about the risks
ARTS
STYLE
TRAVEL
JOBS
REAL ESTATE
AUTOS
of faster networks rare credibility.
More transactions also mean more system attacks. Even though he says there is no turning
back on the online society, Mr. Cheriton worries most about security hazards. “I’ve made
the claim that the Chinese military can take it down in 30 seconds, no one can prove me
wrong,” he said. By building a new way to run networks in the cloud era, he says, “we have
a path to having software that is more sophisticated, can be self-defending, and is able to
detect more problems, quicker.”
The common connection among computer servers, one gigabit per second, is giving way to
10-gigabit connections, because of improvements in semiconductor design and software.
Speeds of 40 gigabits, even 100 gigabits, are now used for specialty purposes like
consolidating huge data streams among hundreds of thousands of computers across the
globe, and that technology is headed into the mainstream. An engineering standard for a
terabit per second, 1,000 gigabits, is expected in about seven years.
Arista, which is based here, was built with the 10-gigabit world in mind. It now has 250
employees, 167 of them engineers, building a fast data-routing switch that could isolate
problems and fix them without ever shutting down the network. It is intended to run on
inexpensive mass-produced chips. In terms of software and hardware, it was a big break
from the way things had been done in networking for the last quarter-century.
“Companies like Cisco had to build their own specialty chips to work at high speed for the
time,” Mr. Bechtolsheim said. Because of improvements in the quality and capability of the
kind of chips used in computers, phones and cable television boxes, “we could build a
network that is a lot more software-enabled, something that is a lot easier to defend and
modify,” he said.
For Mr. Cheriton, who cuts his own hair despite his great wealth, Arista was an
opportunity to work on a new style of software he said he had been thinking about since
1989.
No matter how complex, software is essentially a linear system of commands: Do this, and
then do that. Sometimes it is divided into “objects” or modules, but these tend to operate
sequentially.
From 2004 to 2008, when Arista shipped its first product, Mr. Cheriton developed a five
million-line system that breaks operations into a series of tasks, which when completed,
other parts of the program can check on and pick up if everything seems fine. If it does
not, the problem is rapidly isolated and addressed. Mr. Bechtolsheim worked with him to
make the system operate with chips that were already on the market.
The first products were sold to financial traders looking to shave 100 nanoseconds off their
high-frequency trades. Arista has more than 1,000 customers now, including
telecommunications companies and university research laboratories.
“They have created something that is architecturally unique in networking, with a lot of
value for the industry,” says Nicholas Lippis, who tests and evaluates switching
equipment. “They built something fast that has a unique value for the industry.”
Kenneth Duda, another founder, said, “What drives us here is finding a new way to do
software.” Mr. Duda also worked with Mr. Cheriton and Mr. Bechtolsheim at Granite
Systems, the company they sold to Cisco. “The great enemy is complexity, measured in
lines of code, or interactions,” he said. In the world of cloud computing, “there is no
person alive who can understand 10 percent of the technology involved in my writing and
printing out an online shopping list.”
Not surprisingly, Cisco, which dominates the $5 billion network switching business,
disagrees.
“You don’t have to reinvent the Internet,” says Ram Velaga, vice president for product
management in Cisco’s core technology group. “These protocols were designed to work
even if Washington is taken out. That is in the architecture.”
Still, Cisco’s newest data center switches have rewritten software in a way more like
Arista’s. A few products are using so-called merchant silicon, instead of its typical custom
chips. “Andy made a bet that Cisco would never use merchant silicon,” Mr. Velaga says.
Mr. Cheriton and Mr. Bechtolsheim have known each other since 1981, when Mr. Cheriton
arrived from his native Canada to teach at Stanford. Mr. Bechtolsheim, a native of
Germany, was studying electrical engineering and building what became Sun’s first
product, a computer workstation.
The two became friends and intellectual compatriots, and in 1994 began Granite
Networks, which made one of the first gigabit switches. Cisco bought the company two
years later.
With no outside investors in Arista, they could take as long as they wanted on the product,
Mr. Bechtolsheim said.
“Venture capitalists have no patience for a product to develop.” he said. “Pretty soon they
want to bring in their best buddy as the C.E.O. Besides, this looked like a good
investment.”
Mr. Cheriton said, “Not being venture funded was definitely a competitive advantage.”
Besides, he said, “Andy never told me it would be $100 million.”
A 'Big Data' Freeway for Scientists - NYTimes.com
1 of 2
http://bits.blogs nytimes.com/2013/03/20/a-big-data-freeway-for-scientist...
MARCH 20, 2013, 1:29 PM
A ‘Big Data’ Freeway for Scientists
By JOHN MARKOFF
The University of California, San Diego, this week plans to announce that it has installed an
advanced optical computer network that is intended to serve as a “Big Data freeway system” for
next-generation science projects in fields including genomic sequencing, climate science,
electron microscopy, oceanography and physics.
The new network, which is funded in part by a $500,000 grant from the National Science
Foundation and based on an optical switch developed by Arista Networks, a start-up firm
founded by the legendary Silicon Valley computer designer Andreas Bechtolsheim, is intended
to move from an era where networks moved billions of bits of data each second to the coming
age of trillion-bit-per-second data flows. (A terabit network has the capacity to move roughly the
equivalent of 2.5 Blu-ray videodiscs each second.)
However, the new ultrahigh speed networks are not just about moving files more quickly, or
even moving larger files. Increasingly, computers used by scientific researchers are starting to
escape the boundaries of a single box or even cluster and spread out to become “virtual,” in
some cases across thousands of miles.
The new network, known as Prism, is intended for a new style of scientific computing
characterized both by “big data” data sets and optical networks that make it possible to compute
on data that is stored at a distant location from the computer’s processor, said Philip M.
Papadopoulos, program director for computing systems at the San Diego Supercomputer
Center, and the principal investigator for the new network.
The Prism network “enables users to simply not care where their data is located on campus,” he
said.
The Prism network is targeted at speeds of 100 billion bits a second and is intended as a bypass
network that allows scientists to move data without affecting the performance of the normal
campus network, which is based on a 10 billion-bit capacity and is near saturation.
There is a range of scientific users with requirements that have easily outstripped the capacity of
current-day computer networks, he said.
For example he pointed to work being done in medicine by the National Center for Microscopy
Imaging Research, with both light and electron microscopes that now generate threedimensional images that may range up to 10 terabytes of data. The laboratory stores several
petabytes (a petabyte is one thousand terabytes) and will require Prism to move data between
different facilities on campus.
3/21/2013 8:13 AM
A 'Big Data' Freeway for Scientists - NYTimes.com
2 of 2
http://bits.blogs nytimes.com/2013/03/20/a-big-data-freeway-for-scientist...
A previous optical network, known as Quartzite, was installed at San Diego beginning in 2004.
That network was built on an earlier, less powerful, model of the Arista switch. The new version
of the switch will handle up to 576 simultaneous 10 billion-bit connections. In some cases the
links can be “bonded” to support even higher capacity data flows.
During an event last month to introduce the event on campus, Larry Smarr, an astrophysicist
who is the director of the California Institute for Telecommunications and Information
Technology, a U.C.S.D. laboratory that is the focal point for the new network, demonstrated the
ability to share data and scientific visualization information with other scientists by holding a
videoconference with researchers at the Electronic Visualization Laboratory at the University of
Illinois at Chicago.
At one point he showed a three-dimensional image created from an M.R.I. of his own abdomen,
demonstrating how it was possible to view and manipulate the digital image remotely.
“The radiologists are used to reading the two dimensional scans and turning it into 3-D in their
heads, but the doctors and surely the patients have never been able to see what is in their
bodies,” he said. “I’m turning the insides of my body into a video game.”
This post has been revised to reflect the following correction:
Correction: March 20, 2013
An earlier version of this post misstated the name of the organization where Philip M.
Papadopoulos works as a program director. It is the San Diego Supercomputer Center, not the
San Diego Computing Center. It also misstated the location of the University of Illinois'
Electronic Visualization Laboratory. It is at the university's Chicago campus, not the UrbanaChampaign campus.
Copyright 2013 The New York Times Company
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NYTimes.com 620 Eighth Avenue New York, NY 10018
3/21/2013 8:13 AM
January, 10th 2013
8:15
Bernard Meyerson,
IBM Chief
Innovation Officer
By Bernard Meyerson
As IBM’s chief innovation officer, I’m especially proud to reveal today that the company has accomplished a remarkable
achievement: It has been awarded the largest number of United States patents for the 20th year in a row. IBM’s scientists
and engineers racked up 6,478 patents last year, and nearly 67,000 patents over the past two decades.
The sheer number and diversity of these patents matters. It shows that a lot of truly novel thinking is going on at IBM’s
global research and development labs in a wide variety of fields—from nanotechnology and computer systems design to
business analytics and artificial intelligence, and beyond.
Yet volume alone doesn’t tell the whole story. What good are a pile of patents if they don’t change the world? That’s why
we set our research priorities and make our investments with the goal of producing maximum global impact.
Today, we’re focused on a new era in Information Technology that is now in its early stages, but one that will continue to
roll out over the next two decades. We call it the era of cognitive systems. We believe that the benefits of this new era will
arrive sooner and stronger if companies, governments and universities adopt a culture of innovation that includes making
big bets, fostering disruptive innovations, taking a long-term view and collaborating across institutional boundaries. That
last part is crucial. What’s needed is radical collaboration—large-scale efforts to find common cause and share resources,
expertise and ideas across the borders between companies and institutions.
Innovation isn’t about “me” anymore—one person, one company, or even one country. It’s about “we.”
First, a little bit about the new era. Today’s computers are programmed by humans to perform specific tasks. They are
designed to calculate rapidly. In contrast, cognitive systems will learn from their interactions with data and humans
—continuously reprogramming themselves to become ever more accurate and efficient in their outputs. They optimize
their functions with each cycle of learning. They will be uniquely designed to analyze vast quantities of information.
Today’s computers are processing centric; tomorrow’s will be data centric.
Because of these improvements, the machines of the future will be able draw insights from data to help us learn how the
world really works, making sense of all of its complexity. They will provide trusted advice to humans—whether heads of
state or individuals trying to manage their careers or finances.
At IBM, we’re producing some of the scientific advances that will enable the era of cognitive systems. Our early work has
already shown up in patents granted last year.
Consider Watson, the groundbreaking computer that defeated two former grand champions on the TV quiz show
Jeopardy!. Watson’s creators in IBM Research programmed the machine to read millions of pages of information about all
manner of things, and then, during the game, dig into that huge database for potential answers and come up with the most
likely answer in less than 3 seconds. The U.S. Patent & Trademark Office has awarded several patents for elements of
Watson, including U.S. Patent #8,275,803 – System and method for providing answers to questions.
The scientists whose names are listed on that patent were all IBMers, but Watson was by no means an IBM-only effort.
The project managers enlisted help from researchers at eight universities. They included natural language processing
specialists Eric Nyberg of Carnegie Mellon University and Boris Katz of MIT.
You can think of Watson as the left brain of cognitive computing. It’s where a lot of language-related work takes place.
Separately, IBM Researchers in California, Texas and New York are working on a right-brain project, code-name
SyNAPSE. They’re designing a cognitive chip that’s really good at taking sensory input from the world around us and
turning it into insights.
The team has received a number of patents, including U.S. Patent #8,311,965 – Area efficient neuromorphic circuits using
field effect transistors (FET) and variable resistance material.
The SyNAPSE project has had even more input from outside IBM than did Watson. Team leader Dharmendra Modha
formed a collaborative partnership with university faculty members who brought expertise that IBM doesn’t possess
internally. The first phases of the project included circuit designer Rajit Manohar of Cornell University, psychiatrist
Giulio Tononi of University of Wisconsin-Madison, neuroscientist Stefano Fusi of Columbia University and robotics
specialist Christopher Kello of University of California-Merced.
Interesting, but not yet radical. To understand what radical collaboration is consider the workings of America’s largest
“smart grid” research initiative, the Pacific Northwest Smart Grid Project. Participants include the Battelle Memorial
Institute Pacific Northwest Division, the U.S. Department of Energy, eleven utilities, five technology partners (including
IBM Research) and 60,000 customers across the states of Idaho, Montana, Oregon, Washington and Wyoming. The $178
million cost is being split between the government and private sector. The project is deploying and testing a two-way data
communications system that is designed to lower costs, improve reliability, reduce emissions, and increase integration of
renewable energy sources like wind and solar. The secret sauce is getting everybody involved to focus on common goals
and a shared outcome, rather than on their own parochial interests. This project provides a model for how public-private
partnerships can address large and complex problems to the benefit of consumers, companies, and society.
In the past, science and technology researchers typically took on problems one piece at a time. Each expert or group
developed solutions to address one aspect of the problem. We can’t do things that way anymore. There are simply too
many interrelationships and interdependencies to work things independently. Coming up with solutions to today’s biggest
problems requires a lot of different skills from a lot of different people and organizations. Think of this as an innovation
mashup.
Some people think that the process of teaming runs the risk of producing a mediocre, consensus result. I believe the
opposite to be true, as teams more often build on one another’s expertise to create solutions neither could have created on
its own. I see these new radical collaborations as an opportunity for talented teams to grapple with the large and
hard-to-solve problems that defied solutions before. Sure, these projects are going to be difficult to structure and
coordinate, and they will require leaders with clear visions and strong management skills. But innovating the old
fashioned way has become unsustainable, so we’ve got to try something new.
Construction of a Chaotic Computer Chip
William L. Ditto, K. Murali and Sudeshna Sinha
abstract
Chaotic systems are great pattern generators and their defining feature, sensitivity to initial conditions, allows them to switch between patterns exponentially
fast. We exploit such pattern generation by “tuning” representative continuous and
discrete chaotic systems to generate all logic gate functions. We then exploit exponential sensitivity to initial conditions to achieve rapid switching between all the
logic gates generated by each representative chaotic element. With this as a starting
point we will present our progress on the construction of a chaotic computer chip
consisting of large numbers of individual chaotic elements that can be individually
and rapidly morphed to become all logic gates. Such a chip of arrays of morphing
chaotic logic gates can then be programmed to perform higher order functions (such
as memory, arithmetic logic, input/output operations, . . .) and to rapidly switch between such functions. Thus we hope that our reconfigurable chaotic computer chips
will enable us to achieve the flexibility of field programmable gate arrays (FPGA),
the optimization and speed of application specific integrated circuits (ASIC) and the
general utility of a central processing unit (CPU) within the same computer chip architecture. Results on the construction and commercialization of the ChaoLogixTM
chaotic computer chip will also be presented to demonstrate progress being made
towards the commercialization of this technology ( http://www.chaologix.com ).
William L. Ditto
J. Crayton Pruitt Family Department of Biomedical Engineering, University of Florida,
Gainesville, FL 32611-6131, USA, and
ChaoLogix, Inc. 101 S.E. 2nd Place, Suite 201 - A, Gainesville, FL 32601, USA
e-mail: william.ditto@bme.ufl.edu
K. Murali
Department of Physics, Anna University, Chennai 600 025, INDIA
e-mail: kmurali@annauniv.edu
Sudeshna Sinha
Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, INDIA
e-mail: sudeshna@imsc.res.in
1
2
William L. Ditto, K. Murali and Sudeshna Sinha
1 Introduction
It was proposed in 1998 that chaotic systems may be utilized to design computing
devices [1]. In the early years the focus was on proof-of-principle schemes that
demonstrated the capability of chaotic elements to do universal computing. The
distinctive feature of this alternate computing paradigm was that they exploited the
sensitivity and pattern formation features of chaotic systems.
In subsequent years, it was realized that one of the most promising direction of
this computing paradigm is its ability to exploit a single chaotic element to reconfigure into different logic gates through a threshold based morphing mechanism [2, 3].
In contrast to a conventional field programmable gate array element, where reconfiguration is achieved through switching between multiple single purpose gates, reconfigurable chaotic logic gates (RCLGs) are comprised of chaotic elements that
morph (or reconfigure) logic gates through the control of the pattern inherent in
their nonlinear element. Two input RCLGs have recently been realized and shown
to be capable of reconfiguring between all logic gates in discrete circuits [4, 5, 6].
Additionally such RCLGs have been realized in prototype VLSI circuits (0.13µ m
CMOS, 30Mhz clock cycles) that employ two input reconfigurable chaotic logic
gates arrays (RCGA) to morph between higher order functions such as those found
in a typical arithmetic logic unit (ALU) [7].
In this article we first recall the theoretical scheme for flexible implementation of
all these fundamental logical operations utilizing low dimensional chaos [2], and the
specific realisation of the theory in a discrete-time and a continuous-time chaotic circuit. Then we will present new results on the design of reconfigurable multiple input
gates. Note that multiple input logic gates are preferred mainly for reasons of space
in circuits and also many combinational and sequential logic operations can be realized with these logic gates, in which one can minimize the propagation delay. Such
a multiple input CGA would make RCLGs more power efficient, increase their performance and widen their range of applications. Here we specifically demonstrate
a three input RCLG by implementing representative fundamental NOR and NAND
gates with a continuous-time chaotic system.
2 Concept
In order to use the rich temporal patterns embedded in a nonlinear time series efficiently one needs a mechanism to extract different responses from the system, in a
controlled manner, without much run-time effort. Here we employ a threshold based
scheme to achieve this [8].
Consider the discrete-time chaotic map, with its state represented by a variable x,
as our chaotic chip or chaotic processor. In our scheme all the basic logic gate operations (AND, OR, XOR, NAND, NOR, NOT) involve the following simple steps:
Construction of a Chaotic Computer Chip
3
1. Inputs:
x → x0 + I1 + I2 for 2-input gates such as the AND, OR, XOR, NAND and NOR
operations, and
x → x0 + I for the 1-input gate such as the NOT operation.
Here x0 is the initial state of the system, and the input value I = 0 when logic
input is 0 and I = Vin when logic input is 1 (where Vin is a positive constant).
2. Dynamical update, i.e. x → f (x)
where f (x) is a strongly nonlinear function.
3. Threshold mechanism to obtain output V0 :
V0 = 0 if f (x) ≤ E, and
V0 = f (x) − E if f (x) > E
where E is the threshold.
This is interpretated as logic output 0 if V0 = 0 and Logic Ouput 1 if V0 ∼ Vin .
Since the system is chaotic, in order to specify the inital x0 accurately one needs
a controlling mechanism. Here we will employ a threshold controller to set the inital x0 . So in this example we use the clipping action of the threshold controller to
achieve the initialization, and subsequently to obtain the output as well.
Note that in our implementation we demand that the input and output have equivalent definitions (i.e. 1 unit is the same quantity for input and output), as well as
among various logical operations. This requires that constant Vin assumes the same
value throughout a network, and this will allow the output of one gate element to
easily couple to another gate element as input, so that gates can be “wired” directly
into gate arrays implementing compounded logic operations.
In order to obtain all the desired input-output responses of the different gates,
we need to satisfy the conditions enumerated in Table 1 simultaneously. So given a
dynamics f (x) corresponding to the physical device in actual implementation, one
must find values of threshold and initial state satisfying the conditions derived from
the Truth Tables to be implemented.
For instance, Table 2 shows the exact solutions of the initial x0 and threshold E
which satisfy the conditions in Table 1 when
f (x) = 4x(1 − x)
The constant Vin =
1
4
is common to both input and output and to all logical gates.
4
William L. Ditto, K. Murali and Sudeshna Sinha
Logic Operation
AND
OR
XOR
NOR
NAND
NOT
Input Set (I1 , I2 ) Output
Necessary and Sufficient Condition
(0,0)
0
f (x0 ) < E
(0,1)/(1,0)
0
f (x0 +Vin ) < E
(1,1)
1
f (x0 + 2Vin ) − E = Vin
(0,0)
0
f (x0 ) < E
(0,1)/(1,0)
1
f (x0 +Vin ) − E = Vin
(1,1)
1
f (x0 + 2Vin ) − E = Vin
(0,0)
0
f (x0 ) < E
(0,1)/(1,0)
1
f (x0 +Vin ) − E = Vin
(1,1)
0
f (x0 + 2Vin ) < E
(0,0)
1
f (x0 ) − E = Vin
(0,1)/(1,0)
0
f (x0 +Vin ) < E
(1,1)
0
f (x0 + 2Vin ) < E
(0,0)
1
f (x0 ) − E = Vin
(0,1)/(1,0)
1
f (x0 +Vin ) − E = vin
(1,1)
0
f (x0 + 2Vin ) < E
0
1
f (x0 ) − E = Vin
1
0
f (x0 +Vin ) < E
Table 1 Necessary and sufficient conditions, derived from the logic truth tables, to be satisfied
simultaneously by the nonlinear dynamical element, in order to have the capacity to implement the
logical operations AND, OR, XOR, NAND, NOR and NOT with the same computing module.
Construction of a Chaotic Computer Chip
5
Operation
AND
OR
XOR
NAND
NOT
x0
0
1/8
1/4
3/8
1/2
E
3/4
11/16
3/4
11/16
3/4
Table 2 One specific solution of the conditions in Table 1 which yields the logical operations
AND, OR, XOR, NAND and NOT, with Vin = 41 . Note that these theoretical solutions have been
fully verified in a discrete electrical circuit emulating a logistic map [4].
3 Continuous-time Nonlinear System
We now present a somewhat different scheme for obtaining logic responses from a
continuous-time nonlinear system. Our processor is now a continuous time system
described by the evolution equation d x /dt = F (x,t), where x = (x1 , x2 , . . . xN ) are
the state variables and F is a nonlinear function. In this system we choose a variable,
say x1 , to be thresholded. Whenever the value of this variable exceeds a threshold E
it resets to E, i.e. when x1 > E then (and only then) x1 = E.
Now the basic 2-input 1-output logic operation on a pair of inputs I1 , I2 in this
scheme simply involves the setting of an inputs-dependent threshold, namely the
threshold voltage
E = VC + I1 + I2
where VC is the dynamic control signal determining the functionality of the processor. By switching the value of VC one can switch the logic operation being performed.
Again I1 /I2 has value 0 when logic input is 0 and has value Vin when logic input
is 1. So the theshold E is equal to VC when logic inputs are (0, 0), VC + Vin when
logic inputs are (0, 1) or (1, 0), and VC + 2Vin when logic inputs are (1, 1).
The output is interpreted as logic output 0 if x1 < E, i.e. the excess above
threshold V0 = 0. The logic output is 1 if x1 > E, and the excess above threshold
V0 = (x1 − E) ∼ Vin . The schematic diagram of this method is displayed in Fig. 1.
Now for a NOR gate implementation (VC = VNOR ) the following must hold true:
(i) when input set is (0, 0), output is 1, which implies that for threshold E = VNOR ,
output V0 = (x1 − E) ∼ Vin
(ii) when input set is (0, 1) or (1, 0), output is 0, which implies that for threshold
E = VNOR + Vin, x1 < E so that output V0 = 0.
(iii) when input set is (1, 1), output is 0, which implies that for threshold E =
VNOR + 2Vin, x1 < E so that output V0 = 0.
For a NAND gate (VC = VNAND ) the following must hold true:
(i) when input set is (0, 0), output is 1, which implies that for threshold E =
VNAND , output V0 = (x1 − E) ∼ Vin
6
William L. Ditto, K. Murali and Sudeshna Sinha
Chaotic
Circuit
X
Control with
threshold level
E = Vc + I1 + I2
Output V 0
Fig. 1 Schematic diagram for implementing a morphing 2 input logic cell with a continuous time
dynamical system. Here VC determines the nature of the logic response, and the 2 inputs are I1, I2.
(ii) when input set is (0, 1) or (1, 0), output is 1, which implies that for threshold
E = Vin + VNAND , output V0 = (x1 − E) ∼ Vin
(iii) when input set is (1, 1), output is 0, which implies that for threshold E =
VNAND + 2Vin, x1 < E so that output V0 = 0.
In order to design a dynamic NOR/NAND gate one has to find values of VC that
will satisfy all the above input-output associations in a robust and consistent manner.
A proof-of-principle experiment of the scheme was realized with the double
scroll chaotic Chua’s circuit given by the following set of (rescaled) 3 coupled ODEs
[9]
x˙1 = α (x2 − x1 − g(x1 ))
(1)
x˙2 = x1 − x2 + x3
x˙3 = −β x2
(2)
(3)
where α = 10. and β = 14.87 and the piecewise linear function g(x) = bx + 12 (a −
b)(|x + 1| − |x − 1|) with a = −1.27 and b = −0.68. We used the ring structure
configuration of the classic Chua’s circuit [9].
In the experiment we implemented minimal thresholding on variable x1 (this is
the part in the “control” box in the schematic figure). We clipped x1 to E, if it exceeded E, only in Eqn. 2. This has very easy implementation, as it avoids modifying
the value of x1 in the nonlinear element g(x1 ), which is harder to do. So then all we
need to do is to implement x˙2 = E − x2 + x3 instead of Eqn. 2, when x1 > E, and
there is no controlling action if x1 ≤ E.
A representative example of a dynamic NOR/NAND gate can be obtained in
this circuit implementation with parameters: Vin = 2V . The NOR gate is realized
Construction of a Chaotic Computer Chip
7
around VC = 0V . At this value of control signal, we have the following: for input
(0,0) the threshold level is at 0, which yields V0 ∼ 2V ; for inputs (1,0) or (0,1) the
threshold level is at 0, which yields V0 ∼ 0V ; and for input (1,1) the threshold level
is at 2V , which yields V0 = 0 as the threshold is beyond the bounds of the chaotic
attractor. The NAND gate is realized around VC = −2V . The control signal yields
the following: for input (0,0) the threshold level is at −2V , which yields V0 ∼ 2V ;
for inputs (1,0) or (0,1) the threshold level is at 2V , which yields V0 ∼ 2V ; and for
input (1,1) the threshold level is at 4V , which yields V0 = 0 [5].
So the knowledge of the dynamics allowed us to design a control signal that can
select out the temporal patterns emulating the NOR and NAND gates [6]. For instance in the example above, as the dynamic control signal VC switches between 0V
to −2V , the module first yields the NOR and then a NAND logic response. Thus one
has obtained a dynamic logic gate capable of switching between two fundamental
logic reponses, namely the NOR and NAND.
4 Design and Construction of a Three-Input Reconfigurable
Chaotic Logic Gate
As in Section 3, consider a single chaotic element (for inclusion into a RCLG) to
be a continuous time system described by the evolution equation: d x/dt = F (
x;t) where x = (x1 , x2 , . . . , xN ) are the state variables, and F is a strongly nonlinear
function. Again in this system we choose a variable, say x1 , to be thresholded. So
whenever the value of this variable exceeds a critical threshold E (i.e. when x1 > E),
it re-sets to E.
In accordance to our basic scheme, the logic operation on a set of inputs I1 , I2 and
I3 simply involves the setting of an inputs-dependent threshold, namely the threshold
voltage E = VC + I1 + I2 + I3 , where VC is the dynamic control signal determining
the functionality of the processor. By switching the value of VC , one can switch the
logic operation being performed.
I1,2,3 has value ∼ 0V when logic input is zero, and I1,2,3 has value Vin when logic
input is one. So for input (0,0,0) the threshold level is at VC ; for inputs (0,0,1) or
(0,1,0) or (1,0,0) the threshold level is at VC + Vin ; for input (0,1,1) or (1,1,0) or
(1,1,0) the threshold level is at VC + 2Vin and for input (1,1,1) the threshold level is
VC + 3Vin.
As before, the output is interpreted as logic output 0 if x1 < E, and the excess
above threshold V0 ∼ 0. The logic output is 1 if x1 > E, and V0 = (x1 − E) ∼ Vin .
Now for the 3-inputs NOR and the NAND gate implementations the input-output
relations given in Tables 3 and 4 must hold true. Again, in order to design the NOR
or NAND gates, one has to use the knowledge of the dynamics of the nonlinear system to find the values of VC and V0 that will satisfy all the input-output associations
in a consistent and robust manner.
Consider again the simple realization of the double-scroll chaotic Chua’s attractor represented by the set of (rescaled) 3-coupled ODEs given in Eqns. 1-3. This
8
William L. Ditto, K. Murali and Sudeshna Sinha
Input Set (I1 , I2 , I3 )
Threshold E
Output
Logic Output
(0,0,0)
VNOR
V0 = (x1 − E) ∼ Vin
1
(0,0,1) or (1,0,0) or (0,1,0)
VNOR +Vin
V0 ∼ 0V as x1 < E
0
(0,1,1) or (1,1,0) or (1,0,1)
VNOR + 2Vin
V0 ∼ 0V as x1 < E
0
(0,0,0)
VNOR + 3Vin
V0 ∼ 0V as x1 < E
0
Table 3 Truth table for NOR gate implementation (Vin = 1.84V , VNOR = 0V )
Input Set (I1 , I2 , I3 )
Threshold E
Output
Logic Output
(0,0,0)
VNAND
V0 = (x1 − E) ∼ Vin
1
(0,0,1) or (1,0,0) or (0,1,0)
VNAND +Vin
V0 = (x1 − E) ∼ Vin
1
(0,1,1) or (1,1,0) or (1,0,1)
VNAND + 2Vin
V0 = (x1 − E) ∼ Vin
1
(0,0,0)
VNAND + 3Vin
V0 ∼ 0V as x1 < E
0
Table 4 Truth table for NAND gate implementation (Vin = 1.84V , VNOR = −3.68V )
system was implemented by the circuit shown in Fig.3, with circuit component values: [ L = 18mH, R = 1710Ω , C1 = 10nF, C2 = 100nF, R1 = 220Ω , R2 = 220Ω ,
R3 = 2.2kΩ , R4 = 22kΩ , R5 = 22kΩ , R3 = 3.3kΩ , D = IN4148, B1 , B2 = Buffers,
OA1 - OA3 : opamp µ A741]. The x1 dynamical variable (corresponding to the voltage V1 across the capacitor C1) is thresholded by a control circuit shown in the
dotted box in Fig. 3, with voltage E setting varying thresholds. In the circuit, V T
corresponds to the output signal from the threshold controller. Note that, as in the
implementation of 2-input gates, we are only replacing dx2 /dt = x1 − x2 + x3 by
dx2 /dt = E − x2 + x3 in Eq.(2), when x1 > E, and there is no controlling action if
x1 ≤ E.
The schematic diagram for the NAND/NOR gate implementation is depicted in
Fig.2. In the representative example shown here, Vin = 1.84V . The NOR gate is
Construction of a Chaotic Computer Chip
9
Fig. 2 Symbolic diagram for dynamic 3-Input NOR/NAND logic cell. Dynamic control signal VC
determines the logic operation. In our example, VC can switch between VNAND giving a NAND
gate, and VNOR giving a NOR gate.
realized around VC = VNOR = 0V and the NAND gate is realized with VC = VNAND =
−3.68V (See Tables 3 and 4).
Fig. 3 Circuit module implementing a RCLG that morphs between NAND and NOR logic gates.
The diagram represented in the dotted region is the threshold controller. Here E = VC + I1 + I2 + I3
is the dynamically varying threshold voltage. V T is the output signal from the threshold controller
and V0 is the difference voltage signal.
10
William L. Ditto, K. Murali and Sudeshna Sinha
Fig. 4 Voltage timing sequences from top to bottom (PSPICE simulation): (a) First input I1, (b)
Second input I2, (c) Third input I3, (d) Dynamic control signal VC , where VC switches between
VNAND = −3.68V and VNOR = 0V (e) Output signal V 1 (corresponding to x1 (t)) from the Chua’s
circuit, (f) Recovered logic output signal from V 0. The fundamental period of oscillation of this
circuit is 0.33mS.
Thus the nonlinear evolution of the element has allowed us to obtain a control
signal that selects out temporal patterns corresponding to NOR and NAND gates.
For instance in Fig.4, as the dynamic control signal VC switches between −3.68V
to 0V , the element yields first a NAND gate and then morphs into a NOR gate.
The fundamental period of oscillation of the Chua’s circuit is 0.33ms. The average
latency of morphing between logic gates is 48% of this period.
Construction of a Chaotic Computer Chip
11
5 VLSI Implementation of Chaotic Computing Architectures –
Proof of Concept
Recently ChaoLogix Inc. designed and fabricated a proof of concept chip that
demonstrates the feasibility of constructing reconfigurable chaotic logic gates,
henceforth ChaoGates, in standard CMOS based VLSI (0.18µ m TSMC process operating at 30Mhz with a 3.1 × 3.1mm die size and a 1.8V digital core voltage). The
basic building block ChaoGate is shown schematically in Fig. 5. ChaoGates were
then incorporated into a ChaoGate Array in the VLSI chip to demonstrate higher
order morphing functionality including:
1. A small Arithmetic Logic Unit (ALU) that morphs between higher order arithmetic functions (multiplier and adder/accumulator) in less than one clock cycle.
An ALU is a basic building block of computer architectures.
2. A Communications Protocols (CP) Unit that morphs between two different complex communications protocols in less than one clock cycle: Serial Peripheral
Interface (SPI, a synchronous serial data link) and an Inter Integrated Circuit
Control bus implementation (I2C, a multi-master serial computer bus).
While the design of the ChaoGates and ChaoGate Arrays in this proof of concept
VLSI chip was not optimized for performance, it clearly demonstrates that ChaoGates can be constructed and organized into reconfigurable chaotic logic gate arrays
capable of morphing between higher order computational building blocks. Current
efforts are focused upon optimizing the design of a single ChaoGate to levels where
they are comparable or smaller to a single NAND gate in terms of power and size yet
are capable of morphing between all gate functions in under a single computer clock
cycle. Preliminary designs indicate that this goal is achievable and that all gates currently used to design computers may be replaced with ChaoGates to provide added
flexibility and performance.
Acknowledgments
We acknowledge the support of the Office of Naval Research [N000140211019].
References
1. Sinha, S. and Ditto, W.L. Phys. Rev. Lett. 81 (1998) 2156.
2. Sinha, S., Munakata, T. and Ditto, W.L, Phys. Rev. E 65 (2002) 036214; Munakata, T., Sinha,
S. and Ditto, W.L, IEEE Trans. Circ. and Systems 49 (2002) 1629.
3. Sinha, S. and Ditto, W.L. Phys. Rev. E 59 (1999) 363; Sinha, S., Munakata, T. and Ditto, W.L
Phys. Rev. E 65 036216.
12
William L. Ditto, K. Murali and Sudeshna Sinha
Input
Input
IN1
IN2
VT1
Select
Out
Output
ChaoGate
Element
VT2
VT3
Analog Analog Analog
Select Select Select
Global
Thresholds
Fig. 5 (Left) Schematic of a two-input, one output morphable ChaoGate. The gate logic functionality (NOR, NAND, XOR, ) is controlled (morphed), in the current VLSI design, by global
thresholds connected to VT1, VT2 and VT3 through analog multiplexing circuitry and (Right)
a size comparison between the current ChaoGate circuitry implemented in the ChaoLogix VLSI
chaotic comuting chip and a typical NAND gate circuit (Courtesy of ChaoLogix Inc.)
4. Murali, K., Sinha, S. and Ditto, W.L., Proceedings of the STATPHYS-22 Satellite conference
Perspectives in Nonlinear Dynamics Special Issue of Pramana 64 (2005) 433
5. Murali, K., Sinha, S. and Ditto, W.L., Int. J. Bif. and Chaos (Letts) 13 (2003) 2669; Murali,
K., Sinha S., and I. Raja Mohamed, I.R., Phys. Letts. A 339 (2005) 39.
6. Murali, K., Sinha, S., Ditto, W.L., Proceedings of Experimental Chaos Conference (ECC9),
Brazil (2006) published in Philosophical Transactions of the Royal Society of London (Series
A) (2007)
7. W. Ditto, S. Sinha and K. Murali, US Patent Number 07096347 (August 22, 2006).
8. Sinha, S., Nonlinear Systems, Eds. R. Sahadevan and M.L. Lakshmanan, (Narosa, 2002) 309328; Murali, K. and Sinha, S., Phys. Rev. E 68 (2003) 016210 ; Ditto, W.L. and Sinha, S.,
Philosophical Transactions of the Royal Society of London (Series A) 364 (2006) 2483-2494.
9. Dimitriev, A.S. et al, J. Comm. Tech. Electronics, 43 (1998) 1038.
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11
Panasas kingpin: What's the solid state state of play?
Garth Gibson on HPC - buffering the brute-force burst
By Chris Mellor • Get more from this author
Posted in Storage , 29th March 2012 16:28 GMT
Free whitepaper – Enabling efficient data center monitoring
Interview What can NAND flash do now for high-performance computing (HPC) storage and how will it
evolve? Garth Gibson, the co-founder and chief technology officer for Panasas, the (HPC) storage
provider, has definite views on it. Here's a snapshot of them.
El Reg: How can solid state technology benefit HPC in general?
Garth Gibson: The most demanding science done in
HPC is high resolution simulation. This manifests as
distributed shared memory -- the science is limited my
memory size. Memory is typically 40 per cent of capital
costs and 40 per cent of power costs, and can be said
to be the limiting technology in future HPC systems.
Solid state promises new choices in larger, lower
power memory systems, possibly enabling advances in
science better and faster. More narrowly, solid state
technology does not have mechanical positioning
delays, so small random accesses can have latencies
that are two orders of magnitude shorter.
El Reg: Does Panasas have any involvement with NAND in its products? If so, how and why?
Garth Gibson: Panasas uses NAND flash to accelerate small random accesses. In HPC storage, the
bulk of the data is sequentially accessed, so this means that the primary use of small random access
Forums
acceleration is file system metadata (directories, allocation maps, file attributes) and small files.
But we also use this space for small random accesses into large files, which, although rare, can lead to
disproportionately large fragmentation and read access slowdown.
El Reg: What are your views on SLC NAND and MLC NAND in terms of speed (IOPS, MB/sec),
endurance, form factor and interfaces?
Garth Gibson: Our experience is that the NAND flash technologies are becoming more mature, and we
can increasingly trust in the reliability mechanisms provided. This means that enterprise MLC is
sufficiently durable and reliable to be used, although SLC continues to be faster when that extra speed
can be fully exploited.
El Reg: Where in the HPC server-to-storage 'stack' could NAND be used and why?
Garth Gibson: The driving use of NAND flash in HPC by the end of this decade is likely to be so called
"burst buffers". These buffers are the target of memory to memory copies, enabling checkpoints
(defensive IO enabling a later "restart from checkpoint" after a failure) to be captured faster.
The compute can then resume when the burst buffer drains to less expensive storage, typically on
magnetic hard disk. But shortly after that use is established I expect scientists to want to do data
analytics on multiple sequential checkpoints while these are still held in the burst buffer, because the low
latency random access of NAND flash will allow brute-force analysis computations not effective in main
memory or on magnetic disk.
El Reg: Does Panasas develop its own NAND controller technology? If yes or no - why?
Garth Gibson: Panasas is using best-in-class NAND flash controller technology today. But changes in
NAND flash technology and vendors are rapid and important and we continue to track this technology
closely, with an open mind to changing the way we use solid state.
El Reg: What does Panasas think of the merits and demerits of TLC NAND (3-bit MLC)?
Garth Gibson: TLC NAND flash is a new technology, not yet ready for use in Panasas equipment. As it
evolves, it might become appropriate for burst buffers ... hard to say now.
El Reg: How long before NAND runs out of steam?
Garth Gibson: As usual, technologists can point to challenges with current technology that seem to
favor alternative technologies in a timeframe of 2 to 4 generations in the future. I'm told in such
discussions that 2024 looks quite challenging for NAND flash, and much better for its competitors.
However, with that much time, the real issue is how much quality investment is made in the technology.
The market impact of NAND flash is large enough now to ensure that significant effort will go into
continued advances in NAND flash. This is not as clear for its competitors.
El Reg: What do you think of the various post-NAND technology candidates such as Phase Change
Memory, STT-RAM, memristor, Racetrack and the various Resistive-RAMs?
Garth Gibson: I am totally enamored of STT-RAM because it promises infinite rewrite and DRAM-class
speeds. Totally magic! I just hope the technology pans out, because it has a long way to go. Phase
change is much more real, and suffering disappointing endurance improvement so far.
El Reg: Any other pertinent points?
Garth Gibson: Magnetic disk bits are small compared to solid state bits, and solid directions are
available to continue to make them smaller. As long as society's appetite for online data continues to
grow, I would expect magnetic disk to continue to play an important role. However, I would expect that
the memory hierarchy - on-chip to RAM to disk will become deeper, with NAND flash and its competitors
between RAM and disk.
Not such good news in his views on memristor technology. Maybe HP will surprise us all. &reg
Storage at Exascale: Some Thoughts from Panasas CTO Garth Gibson
May 25, 2011
Exascale computing is not just about FLOPS. It will also require a new breed of external storage capable of feeding these exaflop
beasts. Panasas co-founder and chief technology officer Garth Gibson has some ideas on how this can be accomplished and we
asked him to expound on the topic in some detail.
HPCwire: What kind of storage performance will need to be delivered for exascale computing?
Garth Gibson: The top requirement for storage in an exascale supercomputer is the
capability to store a checkpoint in approximately 15 minutes or less so as to keep
the supercomputer busy with computational tasks most of the time. If you do a
checkpoint in 15 minutes, your compute period can be as little as two and a half
hours and you still spend only 10 percent of your time checkpointing. The size of the
checkpoint data is determined by the memory sizing; something that some experts
expect will be approximately 64 petabytes based on the power and capital costs
involved. Based on that memory size, we estimate the storage system must be
capable of writing at 70 terabytes per second to support a 15 minute checkpoint.
HPCwire: Given the slower performance slope of disk compared to compute,
what types of hardware technologies and storage tiering will be required to
provide such performance?
Gibson: While we have seen peak rates of throughput on the hundreds of gigabytes
per second range today, we have to scale 1000x to get to the required write speed
for exascale compute. The challenge with the 70 terabyte-per-second write
requirement is that traditional disk drives will not get significantly faster over the coming decade so it will require almost 1000x the
number of spindles to sustain this level of write capability.
After all, we can only write as fast as the sum of the individual disk drives. We can look at other technologies like flash storage -such as SSDs -- with faster write capabilities. The challenge with this technology, however, is the huge cost delta between
flash-based solutions compared to ones based on traditional hard drives. Given that the scratch space will likely be at least 10 times
the size of main memory, we are looking at 640 petabytes of scratch storage which translates to over half a billion dollars of cost in
flash based storage alone.
The solution is a hybrid approach where the data is initially copied to flash at 70 terabytes per second but the second layer gets 10
times as much time to write from flash to disk, lowering storage bandwidth requirements to 7 terabytes per second, and storage
components to only about 100x today’s systems. You get the performance out of flash and the capacity out of spinning disk. In
essence, the flash layer is really temporary “cheap memory,” possibly not part of the storage system at all, with little of no use of its
non-volatility, and perhaps not using a disk interface like SATA.
HPCwire: What types of software technologies will have to be developed?
Gibson: If we solve the performance/capacity/cost issue with a hybrid model using flash as a temporary memory dump before data
is written off to disk, it will require a significant amount of intelligent copy and tiering software to manage the data movement between
main memory and the temporary flash memory and from there on to spinning disks. It is not even clear what layers of the application,
runtime system, operating system or file system manage this flash memory.
Perhaps more challenging, there will have to be a significant amount of software investment in building reliability into the system. An
exascale storage system is going to have two orders of magnitude more components than current systems. With a lot more
components comes a significantly higher rate of component failure. This means more RAID reconstructions needing to rebuild bigger
drives and more media failures during these reconstructions.
Exascale storage will need higher tolerance for failure as well as the capability for much faster reconstruction, such as is provided by
Panasas’ parallel reconstruction, in addition to improved defense against media failures, such as is provided by Panasas’ vertical
parity. And more importantly, end to end data integrity checking of stored data, data in transit, data in caches, data pushed through
servers and data received at compute nodes, because there is just so much data flowing that detection of the inevitable flipped bit is
going to be key. The storage industry is started on this type of high reliability feature development, but exascale computing will need
exascale mechanisms years before the broader engineering marketplaces will require it.
HPCwire: How will metadata management need to evolve?
Gibson: At Carnegie Mellon University we have already seen with tests run at Oak Ridge National Laboratory that it doesn’t take a
very big configuration before it starts to take thousands of seconds to open all the files, end-to-end. As you scale up the
supercomputer size, the increased processor count puts tremendous pressure on your available metadata server concurrency and
throughput. Frankly, this is one of the key pressure points we have right now – just simply creating, opening and deleting files can
really eat into your available compute cycles. This is the base problem with metadata management.
Exascale is going to mean 100,000 to 250,000 nodes or more. With hundreds to thousands of cores per node and many threads per
core -- GPUs in the extreme -- the number of concurrent threads in exascale computing can easily be estimated in the billions. With
this level of concurrent activity, a highly distributed, scalable metadata architecture is a must, with dramatically superior performance
over what any vendor offers today. While we at Panasas believe we are in a relatively good starting position, it will nevertheless
require a very significant software investment to adequately address this challenge.
HPCwire: Do you believe there is a reasonable roadmap to achieve all this? Do you think the proper investments are being
made?
Gibson: I believe that there is a well reasoned and understood roadmap to get from petascale to exascale. However it will take a lot
more investment than is currently being put into getting to the roadmap goals. The challenge is the return on investment for vendors.
When you consider that the work will take most of the time running up to 2018, when the first exascale systems will be needed, and
that there will barely be more than 500 publicly known petascale computers at that time, based on TOP500.org’s historical 7-year lag
on the scale of the 500th largest computer.
It is going to be hard to pay for systems development on that scale now, knowing that there is going to be only a few
implementations to apportion the cost against this decade and that it will take most of the decade after that for the exascale installed
base to grow to 500. We know that exascale features are a viable program at a time far enough down the line to spread the
investment cost across many commercial customers such as those in the commercial sector doing work like oil exploration or design
modeling.
However, in the mean time, funding a development project like exascale storage systems could sink a small company and it would
be highly unattractive on the P&L of a publicly traded company. What made petascale storage systems such as Panasas and Lustre
a reality was the investment that the government made with DARPA in the 1990’s and with the DOE Path Forward program this past
decade. Similar programs are going to be required to make exascale a reality. The government needs to share in this investment if it
wants production quality solutions to be available in the target exascale timeframe.
HPCwire: What do you think is the biggest hurdle for exascale storage?
Gibson: The principal challenge for this type of scale will be the software capability. Software that can manage these levels of
concurrency, streaming at such high levels of bandwidth without bottlenecking on metadata throughput, and at the same time ensure
high levels of reliability, availability, integrity, and ease-of-use, and in a package that is affordable to operate and maintain is going to
require a high level of coordination and cannot come from stringing together a bunch of open-source modules. Simply getting the
data path capable of going fast by hooking it together with bailing wire and duct tape is possible but it gives you a false confidence
because the capital costs look good and there is a piece of software that runs for awhile and appears to do the right thing.
But in fact, having a piece of software that maintains high availability, doesn’t lose data, and has high integrity and a manageable
cost of operation is way harder than many people give it credit for being. You can see this tension today in the Lustre open source
file system which seems to require a non-trivial, dedicated staff trained to keep the system up and effective.
HPCwire: Will there be a new parallel file system for exascale?
Gibson: The probability of starting from scratch today and building a brand new production file system deployable in time for 2018 is
just about zero. There is a huge investment in software technology required to get to exascale and we cannot get there without
significant further investment in the parallel file systems available today. So if we want to hit the timeline for exascale, it is going to
have to take investment in new ideas and existing implementations to hit the exascale target.
Biff (Bloom Filter) Codes :
Fast Error Correction for Large Data Sets
Michael Mitzenmacher and George Varghese
Abstract—Large data sets are increasingly common in cloud
and virtualized environments. For example, transfers of multiple
gigabytes are commonplace, as are replicated block of such sizes.
There is a need for fast error-correction or data reconciliation in
such settings even when the expected number of errors is small.
Motivated by such cloud reconciliation problems, we consider
error-correction schemes designed for large data, after explaining
why previous approaches appear unsuitable. We introduce Biff
codes, which are based on Bloom filters and are designed for large
data. For Biff codes with a message of length L and E errors,
the encoding time is O(L), decoding time is O(L + E) and the
space overhead is O(E). Biff codes are low-density parity-check
codes; they are similar to Tornado codes, but are designed for
errors instead of erasures. Further, Biff codes are designed to be
very simple, removing any explicit graph structures and based
entirely on hash tables. We derive Biff codes by a simple reduction
from a set reconciliation algorithm for a recently developed data
structure, invertible Bloom lookup tables. While the underlying
theory is extremely simple, what makes this code especially
attractive is the ease with which it can be implemented and
the speed of decoding, which we demonstrate with a prototype
implementation.
I. I NTRODUCTION
Motivated by the frequent need to transfer and reconcile
large data sets in virtualized and cloud envinonments, we
provide a very simple and fast error-correcting code designed
for very large data streams. For example, consider the specific
problem of reconciling two memories of 2 Gbytes whose
contents may differ by a small number of 32-bit words.
Alternatively, one can picture transferring a memory of this
size, and needing to check for errors after it is written to the
new storage. We assume errors are mutation errors; data order
remains intact.
Other possible applications include deduplication, as exemplified by the Difference Engine [6]. While storage may
seem cheap, great cost savings can be effected by replacing
redundant copies of data with a single copy and pointers in
other locations. For example, in virtualized environments, it
is not surprising that two virtual machines might have virtual
memories with a great deal of redundancy. For example, both
VMs may include similar copies of the operating system. More
generally, we are concerned with any setting with large data
transfers over networks.
In this setting, our primary notion of efficiency differs
somewhat from standard coding. While we still want the redundancy added for the code to be as small as possible, speed
appears to be a more important criterion for large data sets. In
particular, for a message of length L and E errors, while we
may want close to the minimum overhead of E words, O(E)
words with a reasonably small constant should suffice. More
importantly, we require very fast encoding and decoding times;
encoding should be O(L) and decoding should be O(L + E),
with very small constant factors implied in the asymptotic
notation. Typically, E will be very small compared to L; we
expect very small error rates, or even subconstant error rates
(such as a bounded number of errors).
In this paper, we describe new codes that are designed
for large data. We also show why other approaches (such
as Reed-Solomon Codes or Tornado codes with with blockbased checksum) are unsuitable. Our codes are extremely
attractive from the point of view of engineering effectiveness:
our software prototype implementation is very fast, decoding
messages of 1 million words with thousands of errors in under
a second.
We call our codes Biff codes, where Biff denotes how we
pronounce BF, for Bloom filter. Biff codes are motivated by
recent Bloom filter variations, the invertible Bloom filter [3]
and invertible Bloom lookup table [5], and their uses for set
reconciliation [4], as explained below. Alternatively, Biff codes
are similar to Tornado codes [1], [9], and can be viewed as a
practical, randomized low-density parity-check code with an
especially simple structure designed specifically for word-level
mutation errors. Also, while Tornado codes were designed
using multiple levels of random graphs with carefully chosen
degree distributions, Biff codes reduce this structure to its
barest elements; our basic structure is single-layer, and regular,
in that each message symbol takes part in the same number of
encoded symbols. As a result, programming efficient encoding
and decoding routines can easily be done in a matter of
hours. We expect this simplicity will be prized as a virtue
in practical settings; indeed, we believe Biff codes reflect the
essential ideas behind Tornado codes and subsequent related
low-density parity-check (LDPC) codes, in their simplest form.
We also provide a simple (and apparently new) general
reduction from error correcting codes to set reconciliation.
While reductions from erasure correcting codes to set reconciliation are well known [7], [10], our reduction may be useful
independent of Biff Codes.
II. F ROM S ET R ECONCILIATION TO E RROR C ORRECTING
O MISSION E RRORS
We now describe how to construct Biff codes from invertible
Bloom lookup tables (IBLTs). The source of the stream of
ideas we exploit is a seminal paper called Invertible Bloom
Filters by Eppstein and Goodrich that invented a streaming
data structure for the so-called straggler problem [3]. The
basic idea was generalized for set reconciliation by Eppstein,
Goodrich, Uyeda, and Varghese in [4] and generalized and
improved further by Goodrich and Mitzenmacher to IBLTs [5].
We choose to use the framework of IBLTs in the exposition
that follows.
We start by reviewing the main aspects of IBLTs that we
require from [5]. We note that we do not require the full IBLT
structure for our application, so we discuss only the elements
that we need, and refer readers to [5] for further details on
IBLT performance.
A. IBLTs via Hashing
Our IBLT construction uses a table T of m cells, and a set of
k random hash functions, h1 , h2 , . . ., hk , to store a collection
of key-value pairs. In our setting, keys will be distinct, and
each key will have a value determined by the key. On an
insertion, each key-value pair is placed into cells T [h1 (x)],
T [h2 (x)], . . . T [ht (x)]. We assume the hash functions are fully
random; in practice this assumption appears suitable (see, e.g.,
[11], [13] for related work on this point). For technical reasons,
we assume that distinct hash functions yield distinct locations.
This can be accomplished in various ways, such as by splitting
the m cells into k subtables each of size m/k, and having each
hash function choose one cell (uniformly) from each subtable.
Such splitting does not affect the asymptotic behavior in our
analysis.
In a standard IBLT, each cell contains three fields: a keySum
field, which is the exclusive-or (XOR) of all the keys that
have been inserted that map to this cell; a valueSum field,
which is the XOR of all the values of the keys that have been
inserted that map to this cell; and a count field, which counts
the number of keys that have been inserted into the cell.
As all operations are XORs, deletions are handled in an
equivalent manner: on deletion of a previously inserted keyvalue pair, the IBLT XORs the key and value with the fields in
the appropriate cells, and the count is reversed. This reverses
a corresponding insertion. We will discuss later how to deal
with deletions without corresponding insertions, a case that
can usefully occur in our setting.
B. Listing Set Entries
We now consider how to list the entries of the IBLT. The
approach is straightforward. We do a first pass through the
cells to find cells with a count of 1, and construct a list of
those cells. We recover the key and corresponding value from
this cell, and then delete the corresponding pair from the table.
In the course of performing deletions, we check the count of
the relevant cells. If a cell’s count becomes 1, we add it to the
list; if it drops from 1 to 0, we can remove it from the list.
This approach can easily be implemented O(m) time.
If at the end of this process all the cells have a count of
0, then we have succeeded in recovering all the entries in the
IBLT. Otherwise, the method only outputs a partial list of the
key-value pairs in B.
This “peeling process” is well known in the context of
random graphs and hypergraphs as the process used to find
k
ck
3
1.222
T HRESHOLDS FOR
THE
4
1.295
5
1.425
TABLE I
2- CORE ROUNDED
6
1.570
7
1.721
TO FOUR DECIMAL PLACES .
the 2-core of a random hypergraph (e.g., see [2], [12]). This
peeling process is similarly used for various codes, including
Tornado codes and their derivatives (e.g., see [9]). Previous
results therefore give tight thresholds: when the number of
hash values k for each pair is at least 2, there are constants
ck > 1 such that if m > (ck + )n for any constant > 0, the
listing process succeeds with probability 1 − o(1); similarly, if
m < (ck − )n for any constant > 0, the listing process fails
with probability o(1). As shown in [2], [12], these values are
given by
n
o
−kαxk−1
c−1
=
sup
α
:
0
<
α
<
1;
∀x
∈
(0,
1),
1
−
e
<
x
.
k
Numerical values for k ≥ 3 are given in Table I.
The choice of k affects the probability of the listing process
failing. By choosing k sufficiently large and m above the
2-core threshold, standard results give that the bottleneck
is the possibility of having two kye-value pairs with the
same collection of hash values, giving a failure probability
of O(m−k+2 ).
We note that, with some additional effort, there are various
ways to save space with the IBLT structure that are known in
the literature. including using compressed arrays, quotienting,
and irregular constructions (where different keys can utilize a
different number of hash values, as in irregular LDPC codes).
In practice the constant factors are small, and such approaches
may interfere with the simplicity we aim for with the IBLT
approach; we therefore do not consider them further here.
C. Set Reconciliation with IBLTs
We consider two users, Alice and Bob, referred to as A and
B. Suppose Alice and Bob hold distinct but similar sets of
keys, and they would like to reconcile the differences. This
is the well known set reconciliation problem. To achieve such
a reconciliation with low overhead [4], Alice constructs an
IBLT. The value associated with each key is a fingerprint (or
checksum) obtained from the key. In what follows, we assume
the value is taken by hashing the key, yielding an uniform
value over all b-bit values for an appropriate b, and that the
hash function is shared between Alice and Bob. Alice sends
Bob her IBLT, and he deletes the corresponding key-value
pairs from his set.
In this setting, when Bob deletes a key-value pair not held
by Alice, it is possible for a cell count to become negative.
The remaining key-value pairs left in the IBLT correspond
exactly to items that exactly one of Alice or Bob has. Bob
can use the IBLT structure to recover these pairs efficiently.
For lack of space, we present the argument informally; the
IBLT properties we use were formally derived in [5].
We first note that, in this setting, because deletions may
reduce the count of cells, it is possible that a cell can have a
count of 1 but not contain a single key-value pair. For example,
if two pairs are inserted by Alice into a cell, and Bob deletes
a pair that does not match Alice’s in that cell, the count will
be 1. Hence, in this setting, the proper way to check if a
cell contains a valid pair is to test that the checksum for the
keySum field matches the valueSum. In fact, because the value
corresponds to a checksum, the count field is extraneous. (It
can be a useful additional check, but strictly is unnecessary.
Moreover, it may not be space-effective; since the counts will
depend on the number of items inserted, not on the size of the
difference between the sets.) Instead, the list of cells that allow
us to recover a value in our listing process are determined by
a match of the key and checksum value. Importantly, because
Bob’s deletion operation is symmetric to Alice’s insertion
operation, this holds true for cells containing a pair deleted
by Bob as well as cells containing a pair inserted by Alice.
(In this case, the corresponding count, if used, should be −1
for cells with a deleted pair.)
Bob can therefore use the IBLT to recover these pairs efficiently. (Strictly speaking, Bob need only recover Alice’s keys,
but this simplification does not make a noticeable difference
in our context.) If ∆ is an upper bound on the number of keys
not shared between Alice and Bob, then from the argument
sketched above, an IBLT with only O(∆) cells is necessary,
with the constant factor dependent on the success probability
desired.
III. E RROR -C ORRECTING C ODES WITH IBLT S
We now show how to use the above scheme to obtain
a computationally efficient error-correcting code. Our errorcorrecting code can be viewed as a reduction using set reconciliation. Let B have a message for A corresponding to the sequence of values x1 , x2 , . . . , xn . Then B sends A the message
along with set reconciliation information – in our case, the
IBLT – for the set of ordered pairs (x1 , 1), (x2 , 2), . . . , (xn , n).
For now we assume the set reconciliation information A
obtains is without error; errors only occur in message values.
When A obtains the sequence y1 , y2 , . . . , yn , she constructs
her own set of pairs (y1 , 1), (y2 , 2), . . . , (yn , n), and reconciles the two sets to find erroneous positions. Notice that
this approach requires random symbol errors as opposed to
adversarial errors for our IBLT approach, as we require the
checksums to accurately determine when key-value pairs are
valid. However, there are standard approaches that overcome
this problem that would make it suitable for adversarial errors
with a suitably limited adversary (by applying a pseudorandom permutation on the symbols that is secret from the
adversary; see, for example, [8]). Also, the positions of the
errors can be anywhere in the message (as long as the positions
are chosen independently of the method used to generate the
set reconciliation information).
If there are no errors in the data for the IBLT structure, then
this reduction can be directly applied. However, assuming the
IBLT is sent over the same channel as the data, then some
cells in the IBLT will have erroneous keySum or valueSum
fields. If errors are randomly distributed and the error rate
is sufficiently small, this is not a concern; as shown in [5],
IBLT listing is quite robust against errors in the IBLT structure.
Specifically, an error will cause the keySum and valueSum
fields of an IBLT cell not to match, and as such it will not
be used for decoding; this can be problematic if all the cells
hashed to be an erroneous message cell are themselves in error,
as the value cannot then be recovered, but under appropriate
parameter settings this will be rare in practice. As a summary,
using the 1-layer scheme, where errors can occur in the IBLT,
the main contribution to the failure probability is when an
erroneous symbol suffers from all k of its hash locations in
the IBLT being in error. If z is the fraction of IBLT cells in
error, the expected number of such symbols is Ez k , and the
distribution of such failures is binomial (and approximately
Poisson, when the expectation is small). Hence, when such
errors occur, there is usually only one of them, and instead of
using recursive error correction on the IBLT one could instead
use a very small amount of error correction in the original
message.
For bursty errors or other error models, we may need to
randomly intersperse the IBLT structure with the original
message; note, however, that the randomness used in hashing
the message values protects us from bursty errors over the
message.
Basic pseudocode for encoding and decoding of Biff codes
is given below (using C-style notation in places); the code
is very simple, and is written entirely in terms of hash table
operations.
• ENCODE
for i = 1 . . . n do
for j = 1 . . . k do
Tj [hj ((xi , i))].keySum = (xi , i).
Tj [hj ((xi , i))].valueSum = Check((xi , i)).
• DECODE
for i = 1 . . . n do
for j = 1 . . . k do
Tj [hj ((yi , i))].keySum ˆ= (yi , i).
Tj [hj ((yi , i))].valueSum ˆ= Check((yi , i)).
while ∃ a, j with (Tj [a].keySum 6= 0) and
(Tj [a].valueSum == Check(Tj [a].keySum)) do
(z, i) = Tj [a].keySum
if z 6= yi then
set yi to z when decoding terminates
for j = 1 . . . k do
Tj [hj ((z, i))].keySum ˆ= (z, i).
Tj [hj ((z, i))].valueSum ˆ= Check((z, i)).
In our pseudocode, there is some leeway in how one implements the while statement. One natural implementation
would keep a list (such as a linked list) of pairs a, j that
satisfy the conditions. This list can be initialized by a walk
through the arrays, and then updated as the while loop modifies
the contents of the table. The total work will clearly be
proportional to the size of the tables, which will be O(E)
when the table size is chosen appropriately.
We may also recursively apply a further IBLT, treating the
first IBLT as data,; or we can use a more expensive error-
correcting code, such as a Reed-Solomon code, to protect the
much smaller IBLT. This approach is similar to that used under
the original scheme for Tornado codes, but appears unnecessary for many natural error models. For ease of exposition, we
assume random locations for errors henceforth.
The resulting error-correcting code is not space-optimal, but
the overhead in terms of the space required for the errorcorrection information is small when the error-rate is small.
If there are e errors, then there will be 2e key-value pairs in
the IBLT; the overhead with having 3, 4, or 5 choices, as seen
from Table I, will then correspond to less than 3e cells. Each
cell contains both a keySum or valueSum, each of which will
be (depending on the implementation) roughly the same size
as the original key. Note here the key in our setting includes
a position as well as the original message symbol, so this is
additional overhead. Putting these together, we can expect that
the error-correction overhead is roughly a factor of 6 over the
optimal amount of overhead, which would be e times the size
of a message symbol.
While this is a non-trivial price, it is important to place
it in context. For large keys, with a 1% error rate, even an
optimal code for a message of length M bytes would require
at least (1/0.99)M ≈ 1.01M bytes to be sent, and a standard
Reed-Solomon code (correcting E errors with 2E additional
values) would require at least 1.02M bytes. Biff codes would
require about 1.06M bytes. The resulting advantages, again,
are simplicity and speed. We expect that in many engineering
contexts, the advantages of the IBLT approach will outweigh
the small additional space cost.
For very large messages, parallelization can speed things
up further; key-value pairs can be inserted or deleted in
parallel easily, with the bottleneck being atomic writes when
XORing into a cell. The listing step also offers opportunities
for parallelization, with threads being based on cells, and cells
becoming active when their checksum value matches the key.
We don’t explore parallelization further here, but we note the
simple, regular framework at the heart of Biff codes.
We also note that, naturally, the approach of using IBLTs
can be applied to design a simple erasure-correcting code. This
corresponds to a set reconciliation problem where one set is
slightly larger than the other; nothing is inserted at A’s end
for missing elements. Other error models may also be handled
using the same technique.
IV. I SSUES WITH OTHER A PPROACHES
Other natural approaches fail to have both fast encoding and
decoding, and maintain O(E) overhead. While asymptotically
faster algorithms exist, the computational overhead of ReedSolomon codes is generally Θ(EL) in practice, making a
straightforward implementation infeasible in this setting, once
the number of errors is non-trivial. Breaking the data into
blocks and encoding each would be ineffective with bursty
errors. One could randomly permute the message data before
breaking it into blocks, to randomize the position of errors
and thereby spread them among blocks. In practice, however,
taking a large memory block and then permuting it is extremely expensive as it destroys natural data locality. Once a
memory or disk page is read it is almost “free” to read the
remaining words in sequence; randomizing positions becomes
hugely expensive. Finally, there are issues in finding a suitable
field size to compute over, particularly for large messages.
The problems we describe above are not original; similar
discussions, for example, appear with the early work on
Tornado codes [1]. Experiments comparing Reed-Solomon
codes for erasures with Tornado codes from the original paper
demonstrate that Reed-Solomon codes are orders of magnitude
slower at this scale.
An alternative approach is to use Tornado codes (or similar
LDPC codes) directly, using checksums to ensure that suitably
sized blocks are accurate. For example, we could divide the
message of length L into L/B blocks of B symbols and add
an error-detection checksum of c bits to each block. If we
assume blocks with detected errors are dropped, then E errors
could result in EB symbols being dropped, requiring the code
to send at least an additional kEB bits for a suitably small
constant k. The total overhead would then be Lc/B + kEB;
simple
calculus yields the minimum
p overhead is when B =
√
block sizes of O( L/E) and resulting space
2 ckLE, with √
overhead of O( LE).
On the other hand, for Biff codes the redundancy overhead is
O(E) with small constants hidden in the O notation, because
only the values in the cells of the hash table, and not the
original data, require checksums. This is a key benefit of
the Biff code approach; only the hash table cells need to be
protected with checksums.
V. E XPERIMENTAL R ESULTS
In order to test our approach, we have implemented Biff
codes in software. Our code uses pseudorandom hash values
generated from the C drand function (randomly seeded using
the clock), and therefore our timing information does not
include the time to hash. However, we point out that hashing
is unlikely to be a major bottleneck. For example, even if for
each one wants 4 hash locations for each key into 4 subtables
of size 1024, and an additional 24 bit hash for the checksum
for each key, all the necessary values can be obtained with a
single 64-bit hash operation.
Setup: Our has table is split into k equal subtables. As
mentioned, to determine locations in each subtable, we use
pseudorandom hash values. For convenience we use random
20 bit keys as our original message symbols and 20 bits to
describe the location in the sequence. While these keys are
small, it allows us to do all computation with 64-bit operations.
For a checksum, we use a simple invertible function: the pair
(xi , i) gives a checksum of (2i + 1) ∗ xi + i2 .
One standard test case uses 1 million 20-bit message symbols and an IBLT of 30000 cells, with errors introduced in
10000 messsage symbols and 600 IBLT cells. Note that with
20 bit keys and 20 bits to record the length, an IBLT cell is
actually 4 times the size of a message cell; however, we use a
2% error rate in the IBLT as we expect message symbols will
generally be much longer. For example, in practice a key might
be a 1KB packet, in which case 1 million message symbols
would correspond to a gigabyte.
Timing: Our results show Biff codes to be extremely fast.
There are two decoding stages, as can be seen in the previously
given pseudocode. First, the received sequence values must
be placed into the hash table. Second, the hash table must
be processed and the erroneous values recovered. Generally,
the bulk of the work will actually be in the first stage, when
the number of errors are small. We had to utilize messages
of 1 million symbols in order to obtain suitable timing
data; otherwise processing was too fast. On our standard test
case over 1000 trials, using 4 hash functions the first stage
took 0.0561 seconds on average and the second took 0.0069
seconds on average. With 5 hash functions, the numbers were
0.0651 second and 0.0078 seconds.
Thresholds: Our threshold calculations are very accurate. For
example, in a setting where no errors are introduced in the
IBLT, with 4 hash functions and 10000 errors we would expect
to require approximately 26000 cells in order to recover fully.
(Recall that 10000 errors means 20000 keys are placed into the
IBLT.) Our experiments yielded that with and IBLT of 26000
cells, complete recovery occured in 803 out of 1000 trials; for
26500 cells, complete recovery occured in 10000 out of 10000
trials.
Failure probabilities: We have purposely chosen parameters
that would lead to failures, in order to check our analysis.
Under our standard test case with four hash functions, we
estimate the probability of failure during any single trial as
10000 · (600/30000)4 = 1.6 × 10−3 . Over an experiment
with 10000 trials, we indeed found 16 trials with failures,
and in each failure, there was just one unrecovered erroneous
message symbol. Reducing to 500 errors in the IBLT reduces
the failure probability to 10000 · (500/30000)4 ≈ 7.7 × 10−4 ;
an experiment with 10000 trials led to a seven failures, each
with just one unrecovered erroneous message symbol. Finally,
with 5 hash functions and 600 IBLT errors, we would estimate
the failure probability as 10000 · (600/30000)5 = 3.2 × 10−5 ;
a run of 10000 trials yielded no failures.
VI. C ONCLUSIONS
Our goal was to design an error-correcting code that would
be extremely fast and simple for use in networking applications
such as large-scale data transfer and reconciliation in cloud
computing systems. While not optimal in terms of rate, the
amount of redundancy used is a small constant factor more
than optimal; we expect this will be suitable for many applications, given the other advantages. Although we have focused
on error correction of large data, Biff codes may also be
useful for smaller messages, in settings where computational
efficiency is paramount and where small block sizes were
introduced at least partially to reduce Reed-Solomon decoding
overheads.
We note that in the large data setting we can adapt the
sampling technique described in [4] to estimate the number
of errors E in O(log L) time. This allows the Biff code to be
sized correctly to O(E) without requiring any a priori bound
on E to be known in advance. For example, when two large
virtual memories are to be reconciled it is difficult to have a
reasonable bound on the number of errors or differences. In the
communications setting this is akin to estimating the channel
error rate and adapting the code. However, such error rate
estimation in the communication setting is done infrequently
to reduce overhead. In our large data setting, the cost of
estimation is so cheap that it can be done on each large data
reconciliation.
Finally, we note that modern low-density parity-check codes
are sufficiently complex that they are difficult to teach without
without going through a number of preliminaries. By contrast,
Biff codes are sufficiently simple that we believe they could
be taught in an introductory computer science class, and
even introductory level programmers could implement them.
Beyond their practical applications, Biff codes might prove
worthwhile as a gateway to modern coding techniques.
R EFERENCES
[1] J.W. Byers, M. Luby, and M. Mitzenmacher. A digital fountain
approach to asynchronous reliable multicast. IEEE Journal on Selected
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[2] M. Dietzfelbinger, A. Goerdt, M. Mitzenmacher, A. Montanari,
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Verifiable Computation with Massively Parallel Interactive Proofs
Justin Thaler∗
Mike Roberts †
Michael Mitzenmacher‡
Hanspeter Pfister §
arXiv:1202.1350v3 [cs.DC] 22 Feb 2012
Abstract
As the cloud computing paradigm has gained prominence, the need for verifiable computation has grown increasingly urgent. The concept of verifiable computation enables a weak client to outsource difficult computations
to a powerful, but untrusted, server. Protocols for verifiable computation aim to provide the client with a guarantee
that the server performed the requested computations correctly, without requiring the client to perform the requested
computations herself. By design, these protocols impose a minimal computational burden on the client. However,
existing protocols require the server to perform a very large amount of extra bookkeeping, on top of the requested
computations, in order to enable a client to easily verify the results. Verifiable computation has thus remained a
theoretical curiosity, and protocols for it have not been implemented in real cloud computing systems.
In this paper, our goal is to leverage GPUs to reduce the server-side slowdown for verifiable computation. To this
end, we identify abundant data parallelism in a state-of-the-art general-purpose protocol for verifiable computation,
originally due to Goldwasser, Kalai, and Rothblum [10], and recently extended by Cormode, Mitzenmacher, and
Thaler [8]. We implement this protocol on the GPU, and we obtain 40-120× server-side speedups relative to a stateof-the-art sequential implementation. For benchmark problems, our implementation thereby reduces the slowdown
of the server to within factors of 100-500× relative to the original computations requested by the client. Furthermore,
we reduce the already small runtime of the client by 100×. Similarly, we obtain 20-50× server-side and clientside speedups for related protocols targeted at specific streaming problems. We believe our results demonstrate the
immediate practicality of using GPUs for verifiable computation, and more generally, that protocols for verifiable
computation have become sufficiently mature to deploy in real cloud computing systems.
1
Introduction
A potential problem in outsourcing work to commercial cloud computing services is trust. If we store a large dataset
with a server, and ask the server to perform a computation on that dataset – for example, to compute the eigenvalues
of a large graph, or to compute a linear program on a large matrix derived from a database – how can we know the
computation was performed correctly? Obviously we don’t want to compute the result ourselves, and we might not
even be able to store all the data locally. Despite these constraints, we would like the server to not only provide us
with the answer, but to convince us the answer is correct.
Protocols for verifiable computation offer a possible solution to this problem. The ultimate goal of any such protocol is to enable the client to obtain results with a guarantee of correctness from the server much more efficiently than
performing the computations herself. Another important goal of any such protocol is to enable the server to provide
results with guarantees of correctness almost as efficiently as providing results without guarantees of correctness.
Interactive proofs are a powerful family of protocols for establishing guarantees of correctness between a client
and server. Although they have been studied in the theory community for decades, there had been no significant efforts
∗ Harvard University, School of Engineering and Applied Sciences, jthaler@seas.harvard.edu. Supported by the Department of Defense (DoD)
through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program, and in part by NSF grants CCF-0915922 and
IIS-0964473.
† Harvard University, School of Engineering and Applied Sciences, mroberts@seas.harvard.edu. This work was partially supported by the Intel
Science and Technology Center for Visual Computing, NVIDIA, and the National Science Foundation under Grant No. PHY-0835713.
‡ Harvard University, School of Engineering and Applied Sciences, michaelm@eecs.harvard.edu. This work was supported by NSF grants
CCF-0915922 and IIS-0964473.
§ Harvard University, School of Engineering and Applied Sciences, pfister@seas.harvard.edu. This work was partially supported by the Intel
Science and Technology Center for Visual Computing, NVIDIA, and the National Science Foundation under Grant No. PHY-0835713.
1
to implement or deploy such proof systems until very recently. A recent line of work (e.g., [5, 6, 7, 8, 9, 10, 19]) has
made substantial progress in advancing the practicality of these techniques. In particular, prior work of Cormode,
Mitzenmacher, and Thaler [8] demonstrates that: (1) a powerful general-purpose methodology due to Goldwasser,
Kalai and Rothblum [10] approaches practicality; and (2) special-purpose protocols for a large class of streaming
problems are already practical.
In this paper, we clearly articulate this line of work to researchers outside the theory community. We also take
things one step further, leveraging the parallelism offered by GPUs to obtain significant speedups relative to stateof-the-art implementations of [8]. Our goal is to invest the parallelism of the GPU to obtain correctness guarantees
with minimal slowdown, rather than to obtain raw speedups, as is the case with more traditional GPU applications.
We believe the insights of our GPU implementation could also apply to a multi-core CPU implementation. However,
GPUs are increasingly widespread, cost-effective, and power-efficient, and they offer the potential for speedups in
excess of those possible with commodity multi-core CPUs [17, 14].
We obtain server-side speedups ranging from 40-120× for the general-purpose protocol due to Goldwasser et
al. [10], and 20-50× speedups for related protocols targeted at specific streaming problems. Our general-purpose
implementation reduces the server-side cost of providing results with a guarantee of correctness to within factors
of 100-500× relative to a sequential algorithm without guarantees of correctness. Similarly, our implementation of
the special-purpose protocols reduces the server-side slowdown to within 10-100× relative to a sequential algorithm
without guarantees of correctness.
We believe the additional costs of obtaining correctness guarantees demonstrated in this paper would already be
considered modest in many correctness-critical applications. For example, at one end of the application spectrum is
Assured Cloud Computing for military contexts: a military user may need integrity guarantees when computing in
the presence of cyber attacks, or may need such guarantees when coordinating critical computations across a mixture
of secure military networks and insecure networks owned by civilians or other nations [1]. At the other end of the
spectrum, a hospital that outsources the processing of patients’ electronic medical records to the cloud may require
guarantees that the server is not dropping or corrupting any of the records. Even if every computation is not explicitly
checked, the mere ability to check the computation could mitigate trust issues and stimulate users to adopt cloud
computing solutions.
Our source code is available at [20].
2
2.1
Background
What are interactive proofs?
Interactive proofs (IPs) were introduced within the computer science theory community more than a quarter century
ago, in seminal papers by Babai [11] and Goldwasser, Micali and Rackoff [3]. In any IP, there are two parties: a
prover P, and a verifier V. P is typically considered to be computationally powerful, while V is considered to be
computationally weak.
In an IP, P solves a problem using her (possibly vast) computational resources, and tells V the answer. P and
V then have a conversation, which is to say, they engage in a randomized protocol involving the exchange of one or
more messages between the two parties. The term interactive proofs derives from the back-and-forth nature of this
conversation. During this conversation, P’s goal is to convince V that her answer is correct.
IPs naturally model the problem of a client (whom we model as V) outsourcing computation to an untrusted server
(who we model as P). That is, IPs provide a way for a client to hire a cloud computing service to store and process
data, and to efficiently check the integrity of the results returned by the server. This is useful whenever the server
is not a trusted entity, either because the server is deliberately deceptive, or is simply buggy or inept. We therefore
interchange the terms server and prover where appropriate. Similarly, we interchange the terms client and verifier
where appropriate.
Any IP must satisfy two properties. Roughly speaking, the first is that if P answers correctly and follows the
prescribed protocol, then P will convince V to accept the provided answer. The second property is a security guarantee,
which says that if P is lying, then V must catch P in the lie and reject the provided answer with high probability. A
trivial way to satisfy this property is to have V compute the answer to the problem herself, and accept only if her answer
2
Figure 1: High-level depiction of an execution of the GKR protocol.
matches P’s. But this defeats the purpose of having a prover. The goal of an interactive proof system is to allow V to
check P’s answer using resources considerably smaller than those required to solve the problem from scratch.
At first blush, this may appear difficult or even impossible to achieve. However, IPs have turned out to be surprisingly powerful. We direct the interested reader to [2, Chapter 8] for an excellent overview of this area.
2.2
How do interactive proofs work?
At the highest level, many interactive proof methods (including the ones in this paper) work as follows. Suppose the
goal is to compute a function f of the input x.
First, the verifier makes a single streaming pass over the input x, during which she extracts a short secret s. This
secret is actually a single (randomly chosen) symbol of an error-corrected encoding Enc(x) of the input. To be clear,
the secret does not depend on the problem being solved; in fact, for many interactive proofs, it is not necessary that
the problem be determined until after the secret is extracted.
Next, P and V engage in an extended conversation, during which V sends P various challenges, and P responds
to the challenges (see Figure 1 for an illustration). The challenges are all related to each other, and the verifier checks
that the prover’s responses to all challenges are internally consistent.
The challenges are chosen so that the prover’s response to the first challenge must include a (claimed) value for the
function of interest. Similarly, the prover’s response to the last challenge must include a claim about what the value of
the verifier’s secret s should be. If all of P’s responses are internally consistent, and the claimed value of s matches
the true value of s, then the verifier is convinced that prover followed the prescribed protocol and accepts. Otherwise,
the verifier knows that the prover deviated at some point, and rejects. From this point of view, the purpose of all
intermediate challenges is to guide the prover from a claim about f (x) to a claim about the secret s, while maintaining
V’s control over P.
Intuitively, what gives the verifier surprising power to detect deviations is the error-correcting properties of Enc(x).
Any good error-correcting code satisfies the property that if two strings x and x0 differ in even one location, then
Enc(x) and Enc(x0 ) differ in almost every location. In the same way, interactive proofs ensure that if P flips even a
single bit of a single message in the protocol, then P either has to make an inconsistent claim at some later point, or
else has to lie almost everywhere in her final claim about the value of the secret s. Thus, if the prover deviates from
the prescribed protocol even once the verifier will detect this with high probability and reject.
3
2.3
Previous work
Unfortunately, despite their power, IPs have had very little influence on real systems where integrity guarantees on outsourced computation would be useful. There appears to have been a folklore belief that these methods are impractical
[19]. As previously mentioned, a recent line of work (e.g., [5, 6, 7, 8, 9, 10, 19]) has made substantial progress in advancing the practicality of these techniques. In particular, Goldwasser et al. [10] described a powerful general-purpose
protocol (henceforth referred to as the GKR protocol) that achieves a polynomial-time prover and nearly linear-time
verifier for a large class of computations. Very recently, Cormode, Mitzenmacher, and Thaler [8] showed how to
significantly speed up the prover in the GKR protocol [10]. They also implemented this protocol, and demonstrated
experimentally that their implementation approaches practicality. Even with their optimizations, the bottleneck in the
implementation of [8] is the prover’s runtime, with all other costs (such as verifier space and runtime) being extremely
low.
A related line of work has looked at protocols for specific streaming problems. Here, the goal is not just to save
the verifier time (compared to doing the computation without a prover), but also to save the verifier space. This is
motivated by cloud computing settings where the client does not even have space to store a local copy of the input, and
thus uses the cloud to both store and process the data. The protocols developed in this line of work do not require the
client to store the input, but rather allow the client to make a single streaming pass over the input (which can occur, for
example, while the client is uploading data to the cloud). Throughout this paper, whenever we mention a streaming
verifier, we mean the verifier makes a single pass over the input, and uses space significantly sublinear in the size of
the data.
The notion of a non-interactive streaming verifier was first put forth by Chakrabarti et al. [6] and studied further by
Cormode et al. [7]. These works allow the prover to send only a single message to the verifier (e.g., as an attachment
to an email, or posted on a website), with no communication in the reverse direction. Moreover, these works present
protocols achieving provably optimal tradeoffs between the size of the proof and the space used by the verifier for a
variety of problems, ranging from matrix-vector multiplication to graph problems like bipartite perfect matching.
Later, Cormode, Thaler, and Yi extended the streaming model of [6] to allow an interactive prover and verifier,
who actually have a conversation. They demonstrated that interaction allows for much more efficient protocols in
terms of client space, communication, and server running time than are possible in the one-message model of [6, 7]. It
was also observed in this work that the general-purpose GKR protocol works with just a streaming verifier. Finally, the
aforementioned work of Cormode, Thaler, and Mitzenmacher [8] also showed how to use sophisticated Fast Fourier
Transform (FFT) techniques to drastically speed up the prover’s computation in the protocols of [6, 7].
Also relevant is work by Setty et al. [19], who implemented a protocol for verifiable computation due to Ishai et al.
[13]. To set the stage for our results using parallelization, in Section 6 we compare our approach with [19] and [8] in
detail. As a summary, the implementation of the GKR protocol described in both this paper and in [8] has several advantages over [19]. For example, the GKR implementation saves space and time for the verifier even when outsourcing
a single computation, while [19] saves time for the verifier only when batching together several dozen computations at
once and amortizing the verifier’s cost over the batch. Moreover, the GKR protocol is unconditionally secure against
computationally unbounded adversaries who deviate from the prescribed protocol, while the Ishai et al. protocol relies
on cryptographic assumptions to obtain security guarantees. We present experimental results demonstrating that that
the prover in the sequential implementation of [8] based on the GKR protocol runs significantly faster than the prover
in the implementation of [19] based on the Ishai et al. protocol [13].
Based on this comparison, we use the sequential implementation of [8] as our baseline. We then present results
that our new GPU-based implementation runs 40-120× faster than the sequential implementation in [8].
3
Our interactive proof protocols
In this section, we give an overview of the methods implemented in this paper. Due to their highly technical nature,
we seek only to convey a high-level description of the protocols relevant to this paper, and deliberately avoid rigorous
definitions or theorems. We direct the interested reader to prior work for further details [6, 7, 8, 10].
4
Figure 2: A small arithmetic circuit.
3.1
GKR protocol
The prover and verifier first agree on a layered arithmetic circuit of fan-in two over a finite field F computing the
function of interest. An arithmetic circuit is just like a boolean circuit, except that the inputs are elements of F rather
than boolean values, and the gates perform addition and multiplication over the field F, rather than computing AND,
OR, and NOT operations. See Figure 2 for an example circuit. In fact, any boolean circuit can be transformed into an
arithmetic circuit computing an equivalent function over a suitable finite field, although this approach may not yield
the most succinct arithmetic circuit for the function.
Suppose the output layer of the circuit is layer d, and the input layer is layer 0. The protocol of [10] proceeds in
iterations, with one iteration for each layer of the circuit. The first iteration follows the general outline described in
Section 2.2, with V guiding P from a claim about the output of the circuit to a claim about a secret s, via a sequence
of challenges and responses. The challenges sent by V to P are simply random coins, which are interpreted as random
points in the finite field F. The prescribed responses of P are polynomials, where each prescribed polynomial depends
on the preceding challenge. Such a polynomial can be specified either by listing its coefficients, or by listing its
evaluations at several points.
However, unlike in Section 2.2, the secret s is not a symbol in an error-corrected encoding of the input, but rather
a symbol in an error-corrected encoding of the gate values at layer d − 1. Unfortunately, V cannot compute this secret
s on her own. Doing so would require evaluating all previous layers of the circuit, and the whole point of outsourcing
is to avoid this. So V has P tell her what s should be. But now V has to make sure that P is not lying about s.
This is what the second iteration accomplishes, with V guiding P from a claim about s, to the claim about a new
secret s0 , which is a symbol in an encoding of the gate values at layer d − 2. This continues until we get to the input
layer. At this point, the secret is actually a symbol in an error-corrected encoding of the input, and V can compute this
secret in advance from the input easily on her own. Figure 1 illustrates the entirety of the GKR protocol at a very high
level.
We take this opportunity to point out an important property of the protocol of [10], which was critical in allowing
our GPU-based implementation to scale to large inputs. Namely, any iteration of the protocol involves only two layers
of the circuit at a time. In the ith iteration, the verifier guides the prover from a claim about gate values at layer d − i
to a claim about gate values at layer d − i − 1. Gates at higher or lower layers do not affect the prescribed responses
within iteration i.
3.2
Special-purpose protocols
As mentioned in Section 2.3, efficient problem-specific non-interactive verifiable protocols have been developed for a
variety of problems of central importance in streaming and database processing, ranging from linear programming to
5
graph problems like shortest s − t path. The central primitive in many of these protocols is itself a protocol originally
due to Chakrabarti et al. [6], for a problem known as the second frequency moment, or F2 . P
In this problem, the input
is a sequence of m items from a universe U of size n, and the goal is to compute F2 (x) = i∈U fi2 , where fi is the
number of times item i appears in the sequence. As explained in [8], speeding up this primitive immediately speeds
up protocols for all of the problems that use the F2 protocol as a subroutine.
The aforementioned F2 protocol of Chakrabarti et al. [6] achieves provably optimal tradeoffs between the length
of the proof and the space used by the verifier. Specifically, for any positive integer h, the protocol can achieve a proof
length of just h machine words,
as long as the verifier uses v = O(n/h) words of space. For example, we may set
√
both h and v to be roughly n, which is substantially sublinear in the input size n.
Very roughly speaking, this protocol follows the same outline as in Section 2.2, except that in order to remove
the interaction from the protocol, the verifier needs to compute a more complicated secret. Specifically, the verifier’s
secret s consists of v symbols in an error-corrected encoding of the input, rather than a single symbol. To compute the
prescribed proof, the prover has to evaluate 2n symbols in the error-corrected encoding of the input. The key insight of
[8] is that these 2n symbols need not be computed independently (which would require substantially superlinear time),
but instead can be computed in O(n log n) time using FFT techniques. More specifically, the protocol of [8] partitions
the universe into a v × h grid, and it performs a sophisticated FFT variant known as the Prime Factor Algorithm [4] on
each row of the grid. The final step of P’s computation is to compute the sum of the squared entries for each column
of the (transformed) grid; these values form the actual content of P’s prescribed message.
4
Parallelizing our protocols
In this section, we explain the insights necessary to parallelize the computation of both the prover and the verifier for
the protocols we implemented.
4.1
4.1.1
GKR protocol
Parallelizing P’s computation
In every one of P’s responses in the GKR protocol, the prescribed message from P is defined via a large sum over
roughly S 3 terms, where S is the size of the circuit, and so computing this sum naively would take Ω(S 3 ) time.
Roughly speaking, Cormode et al. in [8] observe that each gate of the circuit contributes to only a single term of this
sum, and thus this sum can be computed via a single pass over the relevant gates. The contribution of each gate to the
sum can be computed in constant time, and each gate contributes to logarithmically many messages over the course
of the protocol. Using these observations carefully reduces P’s runtime from Ω(S 3 ), to O(S log S), where again S is
the circuit size.
The same observation reveals that P’s computation can be parallelized: each gate contributes independently to the
sum in P’s prescribed response. Therefore, P can compute the contribution of many gates in parallel, save the results
in a temporary array, and use a parallel reduction to sum the results. We stress that all arithmetic is done within the
finite field F, rather than over the integers. Figure 3 illustrates this process.
4.1.2
Parallelizing V’s computation
The bulk of V’s computation (by far) consists of computing her secret, which consists of a single symbol s in a
particular error-corrected encoding of the input x. As observed in prior work [9], each symbol of the input contributes
independently to s. Thus, V can compute the contribution of many input symbols in parallel, and sum the results via
a parallel reduction, just as in the parallel implementation of P’s computation. This speedup is perhaps of secondary
importance, as V runs extremely quickly even in the sequential implementation of [8]. However, parallelizing V’s
computation is still an appealing goal, especially as GPUs are becoming more common on personal computers and
mobile devices.
6
Figure 3: Illustration of parallel computation of the server’s message to the client in the GKR protocol.
4.2
Special-purpose protocols
4.2.1
Parallelizing P’s computation
Recall that the prover in the special-purpose protocols can compute the prescribed message by interpreting the input
as a v × h grid, where h is roughly the proof length and v is the amount of space used by the verifier. The prover
then performs a sophisticated FFT on each row of the grid independently. This can be parallelized by transforming
multiple rows of the grid in parallel. Indeed, Cormode et al. [8] achieved roughly a 7× speedup for this problem by
using all eight cores of a multicore processor. Here, we obtain a much larger 20-50× speedup using the GPU. (Note
that [8] did not develop a parallel implementation of the GKR protocol, only of the special-purpose protocols).
4.2.2
Parallelizing V’s computation
Recall that in the special-purpose protocols, the verifier’s secret s consists of v symbols in an error-corrected encoding
of the input, rather than a single symbol. Just as in Section 3.1, this computation can be parallelized by noting that
each input symbol contributes independently to each entry of the encoded input. This requires V to store a large buffer
of input symbols to work on in parallel. In some streaming contexts, V may not have the memory to accomplish this.
Still, there are many settings in which this is feasible. For example, V may have several hundred megabytes of memory
available, and seek to outsource processing of a stream that is many gigabytes or terabytes in length. Thus, parallel
computation combined with buffering can help a streaming verifier keep up with a live stream of data: V splits her
memory into two buffers, and at all times one buffer will be collecting arriving items. As long as V can process the
full buffer (aided by parallelism) before her other buffer overflows, V will be able to keep up with the live data stream.
Notice this discussion applies to the client in the GKR protocol as well, as the GKR protocol also enables a streaming
verifier.
5
5.1
Architectural considerations
GKR protocol
The primary issue with any GPU-based implementation of the prover in the GKR protocol is that the computation is
extremely memory-intensive: for a circuit of size S (which corresponds to S arithmetic operations in an unverifiable
7
algorithm), the prover in the GKR protocol has to store all S gates explicitly, because she needs the values of these gates
to compute her prescribed messages. We investigate three alternative strategies for managing the memory overhead
of the GKR protocol, which we refer to as the no-copying approach, the copy-once-per-layer approach, and the copyevery-message approach.
5.1.1
The no-copying approach
The simplest approach is to store the entire circuit explicitly on the GPU. We call this the no-copying approach. However, this means that the entire circuit must fit in device memory, a requirement which is violated even for relatively
small circuits, consisting of roughly tens of million of gates.
5.1.2
The copy-once-per-layer approach
Another approach is to keep the circuit in host memory, and only copy information to the device when it is needed.
This is possible because, as mentioned in Section 3.1, at any point in the protocol the prover only operates on two
layers of the circuit at a time, so only two layers of the circuit need to reside in device memory. We refer to this as the
copy-once-per-layer approach. This is the approach we used in the experiments in Section 6.
Care must be taken with this approach to prevent host-to-device copying from becoming a bottleneck. Fortunately,
in the protocol for each layer there are several dozen messages to be computed before the prover moves on to the next
layer, and this ensures that the copying from host to device makes up a very small portion of the runtime.
This method is sufficient to scale to very large circuits for all of the problems considered in the experimental
section of [8], since no single layer of the circuits is significantly larger than the problem input itself. However, this
method remains problematic for circuits that have (one or several) layers which are particularly wide, as an explicit
representation of all the gates within a single wide layer may still be too large to fit in device memory.
5.1.3
The copy-every-message approach
In the event that there are individual layers which are too large to reside in device memory, a third approach is to
copy part of a layer at a time from the host to the device, and compute the contribution of each gate in the part to
the prover’s message before swapping the part back to host memory and bringing in the next part. We call this the
copy-every-message approach. This approach is viable, but it raises a significant issue, alluded to in its name. Namely,
this approach requires host-to-device copying for every message, rather than just once per layer of the circuit. That
is, in any iteration i of the protocol, P cannot compute her jth message until after the (j − 1)th challenge from V is
received. Thus, for each message j, the entirety of the ith layer must be loaded piece-by-piece into device memory,
swapping each piece back to host memory after the piece has been processed. In contrast, the copy-once-per-layer
approach allows P to copy an entire layer i to the device and leave the entire layer in device memory for the entirety
of iteration i (which will consist of several dozen messages). Thus, the slowdown inherent in the copy-every-message
approach is not just that P has to break each layer into parts, but that P has to do host-to-device and device-to-host
copying for each message, instead of copying an entire layer and computing several messages from that layer.
We leave implementing the copy-once-per-message approach in full for future work, but preliminary experiments
suggest that this approach is viable in practice, resulting in less than a 3× slowdown compared to the copy-once-perlayer approach. Notice that even after paying this slowdown, our GPU-based implementation would still achieve a
10-40× speedup compared to the sequential implementation of [8].
5.1.4
Memory access
Recall that for each message in the ith iteration of the GKR protocol, we assign a thread to each gate g at the ith layer
of the circuit, as each gate contributes independently to the prescribed message of the prover. The contribution of gate
g depends only on the index of g, the indices of the two gates feeding into g, and the values of the two gates feeding
into g.
Given this data, the contribution of gate g to the prescribed message can be computed using roughly 10-20 additions
and multiplications within the finite field F (the precise number of arithmetic operations required varies over the course
8
of the iteration). As described in Section 6, we choose to work over a field which allows for extremely efficient
arithmetic; for example, multiplying two field elements requires three machine multiplications of 64-bit data types,
and a handful of additions and bit shifts.
In all of the circuits we consider, the indices of g’s in-neighbors can be determined with very little arithmetic
and no global memory accesses. For example, if the wiring pattern of the circuit forms a binary tree, then the first
in-neighbor of g has index 2 · index(g), and the second in-neighbor of g has index 2 · index(g) + 1. For each message,
the thread assigned to g can compute this information from scratch without incurring any memory accesses.
In contrast, obtaining the values of g’s in-neighbors requires fetching 8 bytes per in-neighbor from global memory.
Memory accesses are necessary because it is infeasible to compute the value of each gate’s in-neighbors from scratch
each message, and so we store these values explicitly. As these global memory accesses can be a bottleneck in the
protocol, we strive to arrange the data in memory to ensure that adjacent threads access adjacent memory locations. To
this end, for each layer i we maintain two separate arrays, with the j’th entry of the first (respectively, second) array
storing the first (respectively, second) in-neighbor of the j’th gate at layer i. During iteration i, the thread assigned to
the jth gate accesses location j of the first and second array to retrieve the value of its first and second in-neighbors
respectively. This ensures that adjacent threads access adjacent memory locations.
For all layers, the corresponding arrays are populated with in-neighbor values when we evaluate the circuit at the
start of the protocol (we store each layer i’s arrays on the host until the i’th iteration of the protocol, at which point
we transfer the array from host memory to device memory as described in Section 5.1.2). Notice this methodology
sometimes requires data duplication: if many gates at layer i share the same in-neighbor g1 , then g1 ’s value will
appear many times in layer i’s arrays. We feel that slightly increased space usage is a reasonable price to pay to ensure
memory coalescing.
5.2
5.2.1
Special-purpose protocols
Memory access
Recall that the prover in our special-purpose protocols views the input as a v × h grid, and performs a sophisticated
FFT on each row of the grid independently. Although the independence of calculations in each row offers abundant
opportunities for task-parallelism, extracting the data-parallelism required for high performance on GPUs requires
care due to the irregular memory access pattern of the specific FFT algorithm used.
We observe that although each FFT has a highly irregular memory access pattern, this memory access pattern
is data-independent. Thus, we can convert abundant task-parallelism into abundant data-parallelism by transposing
the data grid into column-major rather than row-major order. This simple transformation ensures perfect memory
coalescing despite the irregular memory access pattern of each FFT, and improves the performance of our specialpurpose prover by more than 10×.
6
6.1
Evaluation
Implementation details
Except where noted, we performed our experiments on an Intel Xeon 3 GHz workstation with 16 GB of host memory.
Our workstation also has an NVIDIA GeForce GTX 480 GPU with 1.5 GB of device memory. We implemented all
our GPU code in CUDA and Thrust [12] with all compiler optimizations turned on.
Similar to the sequential implementations of [8], both our implementation of the GKR protocol and the specialpurpose F2 protocol due to [6, 8] work over the finite field Fp with p = 261 − 1. We chose this field for a number
of reasons. Firstly, the integers embed naturally within it. Secondly, the field is large enough that the probability the
verifier fails to detect a cheating prover is tiny (roughly proportional to reciprocal of the field size). Thirdly, arithmetic
within the field can be performed efficiently with simple shifts and bit-wise operations [21]. We remark that we used
no floating point operations were necessary in any of our implementations, because all arithmetic is done over finite
fields.
9
Finally, we stress that in all reported costs below, we do count the time taken to copy data between the host and
the device, and all reported speedups relative to sequential processing take this cost into account. We do not count the
time to allocate memory for scratch space because this can be done in advance.
6.2
Experimental methodology for the GKR protocol
We ran our GPU-based implementation of the GKR protocol on four separate circuits, which together capture several
different aspects of computation, from data aggregation to search, to linear algebra. The first three circuits were
described and evaluated in [8] using the sequential implementation of the GKR protocol. The fourth problem was
described and evaluated in [19] based on the Ishai et al. protocol [13]. Below, [n] denotes the integers {0, 1, . . . , n−1}.
P
• F2 : Given a stream of m elements from [n], compute i∈[n] a2i where ai is the number of occurrences of i in
the stream.
• F0 : Given a stream of m elements from [n], compute the number of distinct elements (i.e., the number of i with
ai 6= 0, where again ai is the number of occurrences of i in the stream).
• PM: Given a stream representing text T = (t0 , . . . , tn−1 ) ∈ [n]n and pattern P = (p0 , . . . , pq−1 ) ∈ [n]q , the
pattern P is said to occur at location i in t if, for every position j in P , pj = ti+j . The pattern-matching problem
is to determine the number of locations at which P occurs in T .
2
• M AT M ULT: Given three matrices A, B, C ∈ [n]m , determine whether AB = C. (In practice, we do not expect
C to truly be part of the input data stream. Rather, prior work [9, 8] has shown that the GKR protocol works
even if A and B are specified from a stream, while C is given later by P.)
The first two problems, F2 and F0 , are classical data aggregation queries which have been studied for more than
a decade in the data streaming community. F0 is also a highly useful subroutine in more complicated computations,
as it effectively allows for equality testing of vectors or matrices (by subtracting two vectors and seeing if the result is
equal to the zero vector). We make use of this subroutine when designing our matrix-multiplication circuit below.
The third problem, PM, is a classic search problem, and is motivated, for example, by clients wishing to store
(and search) their email on the cloud. Cormode et al. [8] considered the PATTERN M ATCHING WITH W ILDCARDS
problem, where the pattern and text can contain wildcard symbols that match with any character, but for simplicity we
did not implement this additional functionality.
We chose the fourth problem, matrix multiplication, for several reasons. First was its practical importance. Second
was a desire to experiment on problems requiring super-linear time to solve (in contrast to F2 and F0 ): running on a
super-linear problem allowed us to demonstrate that our implementation as well as that of [8] saves the verifier time in
addition to space, and it also forced us to grapple with the memory-intensive nature of the GKR protocol (see Section
4). Third was its status as a benchmark enabling us to compare the implementations of [8] and [19]. Although there
are also efficient special-purpose protocols to verify matrix multiplication (see Freivald’s algorithm [16, Section 7.1],
as well as Chakrabarti et al. [6, Theorem 5.2]), it is still interesting to see how a general-purpose implementation
performs on this problem. Finally, matrix multiplication is an attractive primitive to have at one’s disposal when
verifying more complicated computations using the GKR protocol.
6.2.1
Description of circuits
We briefly review the circuits for our benchmark problems.
The circuit for F2 is by far the simplest (see Figure 4 for an illustration). This circuit simply computes the square
of each input wire using a layer of multiplication gates, and then sums the results using a single sum-gate of very large
fan-in. We remark that the GKR protocol typically assumes that all gates have fan-in two, but [8] explains how the
protocol can be modified to handle a single sum-gate of very large fan-in at the output.
The circuit for F0 exploits Fermat’s Little Theorem, which says that for prime p, ap−1 ≡ 1 mod p if and only if
a 6= 0. Thus, this circuit computes the p − 1’th power of each input wire (taking all non-zero inputs to 1, and leaving
all 0-inputs at 0), and sums the results via a single sum-gate of high fan-in.
10
Figure 4: The circuit for F2 .
The circuit for PM is similar to that for F0 : essentially, for each possible location of the pattern, it computes a
value that is 0 if the pattern is at the location, and non-zero otherwise. It then computes the (p − 1)th power of each
such value and sums the results (i.e., it uses the F0 circuit as a subroutine) to determine the number of locations where
the pattern does (not) appear in the input.
Our circuit for M AT M ULT uses similar ideas. We could run a separate instance of the GKR protocol to verify each
of the n2 entries in the output matrix AB and compare them to C, but this would be very expensive for both the client
and the server. Instead, we specify a suitable circuit with a single output gate, allowing us to run a single instance
of the protocol to verify the output. Our circuit computes the n2 entries in AB via naive matrix multiplication, and
subtracts the corresponding entry of C from each. It then computes the number of non-zero values using the F0 circuit
as a subroutine. The final output of the circuit is zero if and only if C = AB.
6.2.2
Scaling to large inputs
As described in Section 5, the memory-intensive nature of the GKR protocol made it challenging to scale to large
inputs, especially given the limited amount of device memory. Indeed, with the no-copying approach (where we
simply keep the entire circuit in device memory), we were only able to scale to inputs of size roughly 150, 000 for the
F0 problem, and to 32 × 32 matrices for the M AT M ULT problem on a machine with 1 GB of device memory. Using
the copy-once-per-layer approach, we were able to scale to inputs with over 2 million entries for the F0 problem, and
128 × 128 matrices for the M AT M ULT problem. By running on a NVIDIA Tesla C2070 GPU with 6 GBs of device
memory, we were able to push to 256 × 256 matrices for the M AT M ULT problem; the data from this experiment is
reported in Table 2.
6.2.3
Evaluation of previous implementations
To our knowledge, the only existing implementation for verifiable computation that can be directly compared to
that of Cormode et al. [8] is that of Setty et al. [19]. We therefore performed a brief comparison of the sequential
implementation of [8] with that of [19]. This provides important context in which to evaluate our results: our 40-120×
speedups compared to the sequential implementation of [8] would be less interesting if the sequential implementation
of [8] was slower than alternative methods. Prior to this paper, these implementations had never been run on the same
problems, so we picked a benchmark problem (matrix multiplication) evaluated in [19] and compared to the results
reported there.
We stress that our goal is not to provide a rigorous quantitative comparison of the two implementations. Indeed, we
only compare the implementation of [8] to the numbers reported in [19]; we never ran the implementations on the same
system, leaving this more rigorous comparison for future work. Moreover, both implementations may be amenable to
further optimization. Despite these caveats, the comparison between the two implementations seems clear. The results
are summarized in Table 1.
11
Implementation
Matrix Size
[8]
[19], Pepper
[19], Habanero
512 × 512
400 × 400
400 × 400
P Time
3.11 hours
8.1 years∗
17 days†
V Time
0.12 seconds
14 hours∗
2.1 minutes†
Total Communication
138.1 KB
Not Reported
17.1 GB†
Table 1: Comparison of the costs for the sequential implementations of [8] and [19]. Entries marked with ∗ indicate
that the costs given are total costs over 45,000 queries. Entries marked with † indicate that the costs are total costs
over 111 queries.
Problem
Input Size
(number of
entries)
Circuit Size
(number of
gates)
GPU P
Time (s)
Sequential
P Time (s)
Circuit
Evaluation
Time (s)
GPU V
Time (s)
Sequential
V Time (s)
Unverified
Algorithm
Time (s)
F2
F0
PM
M AT M ULT
8.4 million
2.1 million
524,288
65,536
25.2 million
255.8 million
76.0 million
42.3 million
3.7
128.5
38.9
39.6
424.6
8,268.0
1,893.1
1,658.0
0.1
4.2
1.2
0.9
0.035
0.009
0.004
0.003
3.600
0.826
0.124
0.045
0.028
0.005
0.006
0.080
Table 2: Prover runtimes in the GKR protocol for all four problems considered.
In Table 1, Pepper refers to an implementation in [19] which is actually proven secure against polynomial-time adversaries under cryptographic assumptions, while Habenero is an implementation in [19] which runs faster by allowing
for a very high soundness probability of 79 that a deviating prover can fool the verifier, and utilizing what the authors
themselves refer to as heuristics (not proven secure in [19], though the authors indicate this may be due to space constraints). In contrast, the soundness probability in the implementation of [8] is roughly 2150 (roughly proportional to
the reciprocal of the field size p = 261 − 1), and the protocol is unconditionally secure even against computationally
unbounded adversaries.
The implementation of [19] has very high set-up costs for both P and V, and therefore the costs of a single query
are very high. But this set-up cost can be amortized over many queries, and the most detailed experimental results
provided in [19] give the costs for batches of hundreds or thousands of queries. The costs reported in the second and
third rows of Table 1 are therefore the total costs of the implementation when run on a large number of queries.
When we run the implementation of [8] on a single 512 × 512 matrix, the server takes 3.11 hours, the client takes
0.12 seconds, and the total length of all messages transmitted between the two parties is 138.1 KB. In contrast, the
server in the heuristic implementation of [19], Habanero, requires 17 days amortized over 111 queries when run on
considerably smaller matrices (400 × 400). This translates to roughly 3.7 hours per query, but the cost of a single
query without batching is likely about two orders of magnitude higher. The client in Habanero requires 2.1 minutes to
process the same 111 queries, or a little over 1 second per query, while the total communication is 17.1 GBs, or about
157 MBs per query. Again, the per query costs will be roughly two orders of magnitude higher without the batching.
We conclude that, even under large batching the per-query time for the server of the sequential implementation
of [8] is competitive with the heuristic implementation of [19], while the per-query time for the verifier is about two
orders of magnitude smaller, and the per-query communication cost is between two and three orders of magnitude
smaller. Without the batching, the per-query time of [8] is roughly 100× smaller for the server and 1,000× smaller
for the client, and the communication cost is about 100,000× smaller.
Likewise, the implementation of [8] is over 5 orders of magnitude faster for the client than the non-heuristic
implementation Pepper, and four orders of magnitude faster for the server.
6.2.4
Evaluation of our GPU-based implementation
Figure 5 demonstrates the performance of our GPU-based implementation of the GKR protocol. Table 2 also gives a
succinct summary of our results, showing the costs for the largest instance of each problem we ran on. We consider
the main takeaways of our experiments to be the following.
12
105
F2 Prover (parallel)
102
101
100
10−1
10−2
10−3
10−4 4
2
26
28
210 212 214 216 218 220 222 224
Input Size
(a)
Computation Time (seconds)
105
F0 Prover (parallel)
103
102
101
100
10−1
10−2
10−3
10−4 4
2
26
28
210 212 214 216 218 220 222 224
Input Size
(b)
105
M AT M ULT Prover (sequential)
M AT M ULT Prover (parallel)
104
103
Computation Time (seconds)
103
102
101
100
10−1
10−2
10−3
10−4 4
2
26
105
F0 Prover (sequential)
104
Computation Time (seconds)
F2 Prover (sequential)
104
Computation Time (seconds)
Computation Time (seconds)
105
Input Size
(d)
103
102
101
100
10−1
10−2
10−3
10−4 4
2
26
28
210 212 214 216 218 220 222 224
Input Size
(c)
Verifier (sequential)
Verifier (parallel)
104
103
102
101
100
10−1
10−2
10−3
10−4 4
2
28 210 212 214 216 218 220 222 224
PM Prover (sequential)
PM Prover (parallel)
104
26
28 210 212 214 216 218 220 222 224
Input Size
(e)
Figure 5: Comparison of prover and verifier runtimes between the sequential implementation of the GKR protocol due
to [8] and our GPU-based implementation. Note that all plots are on a log-log scale. Plots (a), (b), (c), and (d) depict the
prover runtimes for F0 , F2 , PM, M AT M ULT respectively. Plot (e) depicts the verifier runtimes for the GKR protocol.
We include only one plot for the verifier, since its dominant cost in the GKR protocol is problem-independent.
Server-side speedup obtained by GPU computing. Compared to the sequential implementation of [8], our GPUbased server implementation ran close to 115× faster for the F2 circuit, about 60× faster for the F0 circuit, 45× faster
for PM, and about 40× faster for M AT M ULT (see Figure 5).
Notice that for the first three problems, we need to look at large inputs to see the asymptotic behavior of the
curve corresponding to the parallel prover’s runtime. Due to the log-log scale in Figure 5, the curves for both the
sequential and parallel implementations are asymptotically linear, and the 45-120× speedup obtained by our GPUbased implementation is manifested as an additive gap between the two curves. The explanation for this is simple:
there is considerable overhead relative to the total computation time in parallelizing the computation at small inputs,
but this overhead is more effectively amortized as the input size grows.
In contrast, notice that for M AT M ULT the slope of the curve for the parallel prover remains significantly smaller
than that of the sequential prover throughout the entire plot. This is because our GPU-based implementation ran out
of device memory well before the overhead in parallelizing the prover’s computation became negligible. We therefore
believe the speedup for M AT M ULT would be somewhat higher than the 40× speedup observed if we were able to run
on larger inputs.
Could a parallel verifiable program be faster than a sequential unverifiable one? The very first step of the prover’s
computation in the GKR protocol is to evaluate the circuit. In theory this can be done efficiently in parallel, by proceeding sequentially layer by layer and evaluating all gates at a given layer in parallel. However, in practice we observed
that the time it takes to copy the circuit to the device exceeds the time it takes to evaluate the circuit sequentially. This
observation suggests that on the current generation of GPUs, no GPU-based implementation of the prover could run
faster than a sequential unverifiable algorithm. This is because sequentially evaluating the circuit takes at least as long
as the unverifiable sequential algorithm, and copying the data to the GPU takes longer than sequentially evaluating the
13
circuit. This observation applies not just to the GKR protocol, but to any protocol that uses a circuit representation of
the computation (which is a standard technique in the theory literature [13, 18]). Nonetheless, we can certainly hope
to obtain a GPU-based implementation that is competitive with sequential unverifiable algorithms.
Server-side slowdown relative to unverifiable sequential algorithms. For F2 , the total slowdown for the prover
was roughly 130× (3.7 seconds compared to 0.028 seconds for the unverifiable algorithm, which simply iterates over
all entries of the frequency vector and computes the sum of the squares of each entry). We stress that it is likely
that we overestimate the slowdown resulting from our protocol, because we did not count the time it takes for the
unverifiable implementation to compute the number of occurrences of each item i, that is, to aggregate the stream into
its frequency vector representation (a1 , . . . , an ). Instead, we simply generated the vector of frequencies at random
(we did not count the generation time), and calculated the time to compute the sum of their squares. In practice, this
aggregation step may take much longer than the time required to compute the sum of the squared frequencies once the
stream is in aggregated form.
For F0 , our GPU-based server implementation ran roughly 25,000× slower than the obvious unverifiable algorithm
which simply counts the number of non-zero items in a vector. The larger slowdown compared to the F2 problem is
unsurprising. Since F0 is a less arithmetic problem than F2 , its circuit representation is much larger. Once again, it is
likely that we overestimate the slowdowns for this problem, as we did not count the time for an unverifiable algorithm
to aggregate the stream into its frequency-vector representation. Despite the substantial slow-down incurred for F0
compared to a naive unverifiable algorithm, it remains valuable as a primitive for use in heavier-duty computations
like PM and M AT M ULT.
For PM, the bulk of the circuit consists of a F0 sub-routine, and so the runtime of our GPU-based implementation
was similar to those for F0 . However, the sequential unverifiable algorithm for PM takes longer than that for F0 .
Thus, our GPU-based server implementation ran roughly 6,500× slower than the naive unverifiable algorithm, which
exhaustively searches all possible locations for occurrences of the pattern.
For M AT M ULT, our GPU-based server implementation ran roughly 500× slower than naive matrix-multiplication
for 256 × 256 matrices. Moreover, this number is likely inflated due to cache effects from which the naive unverifiable
algorithm benefited. That is, the naive unverifiable algorithm takes only 0.09 seconds for 256 × 256 matrices, but takes
7.1 seconds for 512 × 512 matrices, likely because the algorithm experiences very few cache misses on the smaller
matrix. We therefore expect the slowdown of our implementation to fall to under 100× if we were to scale to larger
matrices. Furthermore, the GKR protocol is capable of verifying matrix-multiplication over the finite field Fp rather
than over the integers at no additional cost. Naive matrix-multiplication over this field is between 2-3× slower than
matrix multiplication over the integers (even using the fast arithmetic operations available for this field). Thus, if our
goal was to work over this finite field rather than the integers, our slowdown would fall by another 2-3×. It is therefore
possible that our server-side slowdown may be less than 50× at larger inputs compared to naive matrix multiplication
over Fp .
Client-side speedup obtained by GPU computing. The bulk of V’s computation consists of evaluating a single
symbol in an error-corrected encoding of the input; this computation is independent of the circuit being verified. For
reasonably large inputs (see the row for F2 in Table 2), our GPU-based client implementation performed this computation over 100× faster than the sequential implementation of [8]. For smaller inputs the speedup was unsurprisingly
smaller due to increased overhead relative to total computation time. Still, we obtained a 15× speedup even for an
input of length 65,536 (256 × 256 matrix multiplication).
Client-side speedup relative to unverifiable sequential algorithms. Our matrix-multiplication results clearly demonstrate that for problems requiring super-linear time to solve, even the sequential implementation of [8] will save the
client time compared to doing the computation locally. Indeed, the runtime of the client is dominated by the cost of
evaluating a single symbol in an error-corrected encoding of the input, and this cost grows linearly with the input size.
Even for relatively small matrices of size 256 × 256, the client in the implementation of [8] saved time. For matrices
with tens of millions of entries, our results demonstrate that the client will still take just a few seconds, while performing the matrix multiplication computation would require orders of magnitude more time. Our results demonstrate that
GPU computing can be used to reduce the verifier’s computation time by another 100×.
14
103
Special-Purpose F2 Prover (sequential)
Computation Time (seconds)
Computation Time (seconds)
103
Special-Purpose F2 Prover (parallel)
102
101
100
10−1
10−2
10−3 17
2
218
219
220
221
222
Input Size
(a)
223
224
225
Special-Purpose F2 Verifier (sequential)
Special-Purpose F2 Verifier (parallel)
102
101
100
10−1
10−2
10−3 17
2
218
219
220
221
222
Input Size
(b)
223
224
225
Figure 6: Comparison of prover (a) and verifier (b) runtimes in the sequential and GPU-based implementations of the
special-purpose F2 protocol.√Note that all plots are on a log-log scale. Throughout, the verifier’s space usage and the
proof length are both set to n.
V space
(KB)
Proof length
(KB)
GPU P
Time (s)
Sequential P
Time (s)
GPU V
Time (s)
Sequential V
Time (s)
39.1
78.2
156.5
313.2
1953.1
78.1
39.1
19.5
9.8
0.78
2.901
1.872
1.154
0.909
0.357
43.773
43.544
37.254
36.554
20.658
0.019
0.010
0.010
0.008
0.007
0.858
0.639
0.577
0.552
0.551
Table 3: Prover and verifier runtimes for the special-purpose F2 protocol. All results are for fixed universe size n = 25
million, varying the tradeoff between proof length and the client’s space usage. This universe size corresponds to 190.7
MB of data.
6.3
Special-purpose protocols.
We implemented both the client and the server of the non-interactive F2 protocol of [6, 8] on the GPU. As described in
Section 2.3, this protocol is the fundamental building block for a host of non-interactive protocols achieving optimal
tradeoffs between the space usage of the client and the length of the proof. Figure 6 demonstrates the performance
of our GPU-based implementation of this protocol. Our GPU implementation obtained a 20-50× server-side speedup
relative to the sequential implementation of [8]. This speedup was only possible after transposing the data grid into
column-major order so as to achieve perfect memory coalescing, as described in Section 5.2.1.
The server-side speedups we observed depended on the desired tradeoff between proof length and space usage.
That is, the protocol partitions the universe [n] into a v × h grid where h is roughly the proof length and v is the
verifier’s space usage. The prover processes each row of the grid independently (many rows in parallel). When v is
large, each row requires a substantial amount of processing. In this case, the overhead of parallelization is effectively
amortized over the total computation time. If v is smaller, then the overhead is less effectively amortized and we see
less impressive speedups.
We note that Figure 6 depicts the prover runtime √
for both the sequential implementation of [8] and our GPUbased implementation with the parameters h = v = n. With these parameters, our GPU-based implementation
achieved roughly a 20× speedup relative to the sequential program. Table 3 shows the costs of the protocol for fixed
universe size n = 25 million as we vary the tradeoff between h and v. The data in this table shows that our parallel
implementation enjoys a 40-60× speedup relative to the sequential implementation
√ when v is substantially larger than
h. This indicates that we would see similar speedups even when h = v = n if we scaled to larger input sizes
n. Notice that universe size n = 25 million corresponds to over 190 MBs of data, while the verifier’s space usage
15
and the proof length are hundreds or thousands of times smaller in all our experiments. An unverifiable sequential
algorithm for computing the second frequency moment over this universe required 0.031 seconds; thus, our parallel
server implementation achieved a slowdown of 10-100× relative to an unverifiable algorithm.
In contrast, the verifier’s computation was much easier to parallelize, as its memory access pattern is highly regular.
Our GPU-based implementation obtained 40-70×
speedups relative to the sequential verifier of [8] across all input
√
lengths n, including when we set h = v = n.
7
Conclusions
This paper adds to a growing line of work focused on obtaining fully practical methods for verifiable computation.
Our primary contribution in this paper was in demonstrating the power of parallelization, and GPU computing in
particular, to obtain robust speedups in some of the most promising protocols in this area. We believe the additional
costs of obtaining correctness guarantees demonstrated in this paper would already be considered modest in many
correctness-critical applications. Moreover, it seems likely that future advances in interactive proof methodology will
also be amenable to parallelization. This is because the protocols we implement utilize a number of common primitives
(such as the sum-check protocol [15]) as subroutines, and these primitives are likely to appear in future protocols as
well.
Several avenues for future work suggest themselves. First, the GKR protocol is rather inefficient for the prover
when applied to computations which are non-arithmetic in nature, as the circuit representation of such a computation is necessarily large. Developing improved protocols for such problems (even special-purpose ones) would be
interesting. Prime candidates include many graph problems like minimum spanning tree and perfect matching. More
generally, a top priority is to further reduce the slowdown or the memory-intensity for the prover in general-purpose
protocols. Both these goals could be accomplished by developing an entirely new construction that avoids the circuit
representation of the computation; it is also possible that the the prover within the GKR construction can be further
optimized without fundamentally altering the protocol.
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[9] G. Cormode, J. Thaler, and K. Yi. Verifying computations with streaming interactive proofs. Proc. VLDB
Endowment, 5(1):25–36, 2011.
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[14] V. W. Lee, C. Kim, J. Chhugani, M. Deisher, D. Kim, A. D. Nguyen, N. Satish, M. Smelyanskiy, S. Chennupaty, P. Hammarlund, R. Singhal, and P. Dubey. Debunking the 100x gpu vs. cpu myth: an evaluation of
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Unreported Side Effects of Drugs Are Found Using
Internet Search Data, Study Finds
By JOHN MARKOFF
Published: March 6, 2013
184 Comments
Using data drawn from queries entered into Google, Microsoft and
Yahoo search engines, scientists at Microsoft, Stanford and Columbia
University have for the first time been able to detect evidence of
unreported prescription drug side effects before they were found by
the Food and Drug Administration’s warning system.
FACEBOOK
TWITTER
GOOGLE+
SAVE
E-MAIL
Multimedia
Using automated software tools to
SHARE
examine queries by six million
PRINT
Internet users taken from Web search
REPRINTS
logs in 2010, the researchers looked
for searches relating to an
antidepressant, paroxetine, and a
cholesterol lowering drug, pravastatin.
They were able to find evidence that
the combination of the two drugs caused high blood sugar.
Software and Side Effects
Connect With
Us on Social
Media
@nytimesscience on
Twitter.
Science Reporters
and Editors on Twitter
Like the science desk on Facebook.
Readers’ Comments
The study, which was reported in the Journal of the
American Medical Informatics Association on Wednesday,
is based on data-mining techniques similar to those
employed by services like Google Flu Trends, which has
been used to give early warning of the prevalence of the
sickness to the public.
The F.D.A. asks physicians to report side effects through a
system known as the Adverse Event Reporting System. But
its scope is limited by the fact that data is generated only
when a physician notices something and reports it.
Readers shared their thoughts
on this article.
The new approach is a refinement of work done by the
laboratory
of Russ B. Altman, the chairman of the Stanford
Read All Comments (184) »
bioengineering department. The group had explored
whether it was possible to automate the process of
discovering “drug-drug” interactions by using software to hunt through the data found in
F.D.A. reports.
The group reported in May 2011 that it was able to detect the interaction between
paroxetine and pravastatin in this way. Its research determined that the patient’s risk of
developing hyperglycemia was increased compared with taking either drug individually.
The new study was undertaken after Dr. Altman wondered whether there was a more
immediate and more accurate way to gain access to data similar to what the F.D.A. had
access to.
AUTOS
He turned to computer scientists at Microsoft, who created software for
scanning anonymized data collected from a software toolbar installed in Web browsers by
users who permitted their search histories to be collected. The scientists were able to
explore 82 million individual searches for drug, symptom and condition information.
The researchers first identified individual searches for the terms paroxetine and
pravastatin, as well as searches for both terms, in 2010. They then computed the
likelihood that users in each group would also search for hyperglycemia as well as roughly
80 of its symptoms — words or phrases like “high blood sugar” or “blurry vision.”
They determined that people who searched for both drugs during the 12-month period
were significantly more likely to search for terms related to hyperglycemia than were those
who searched for just one of the drugs. (About 10 percent, compared with 5 percent and 4
percent for just one drug.)
They also found that people who did the searches for symptoms relating to both drugs
were likely to do the searches in a short time period: 30 percent did the search on the same
day, 40 percent during the same week and 50 percent during the same month.
“You can imagine how this kind of combination would be very, very hard to study given all
the different drug pairs or combinations that are out there,” said Eric Horvitz, a managing
co-director of Microsoft Research’s laboratory in Redmond, Wash.
The researchers said they were surprised by the strength of the “signal” that they detected
in the searches and argued that it would be a valuable tool for the F.D.A. to add to its
current system for tracking adverse effects. “There is a potential public health benefit in
listening to such signals,” they wrote in the paper, “and integrating them with other
sources of information.”
The researchers said that they were now thinking about how to add new sources of
information, like behavioral data and information from social media sources. The
challenge, they noted, was to integrate new sources of data while protecting individual
privacy.
Currently the F.D.A. has financed the Sentinel Initiative, an effort begun in 2008 to assess
the risks of drugs already on the market. Eventually, that project plans to monitor drug
use by as many as 100 million people in the United States. The system will be based on
information collected by health care providers on a massive scale.
“I think there are tons of drug-drug interactions — that’s the bad news,” Dr. Altman said.
“The good news is we also have ways to evaluate the public health impact.
“This is why I’m excited about F.D.A. involvement here. They do have mechanisms and
ways to pick up the things that we find and triage them based on anticipated public health
impact.”
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
Six Provocations for Big Data
danah boyd
Microsoft Research
dmb@microsoft.com
Kate Crawford
University of New South Wales
k.crawford@unsw.edu.au
Technology is neither good nor bad; nor is it neutral...technology’s interaction with the
social ecology is such that technical developments frequently have environmental, social,
and human consequences that go far beyond the immediate purposes of the technical
devices and practices themselves.
Melvin Kranzberg (1986, p. 545)
We need to open a discourse – where there is no effective discourse now – about the
varying temporalities, spatialities and materialities that we might represent in our
databases, with a view to designing for maximum flexibility and allowing as possible for
an emergent polyphony and polychrony. Raw data is both an oxymoron and a bad idea; to
the contrary, data should be cooked with care.
Geoffrey Bowker (2005, p. 183-184)
The era of Big Data has begun. Computer scientists, physicists, economists,
mathematicians, political scientists, bio-informaticists, sociologists, and many others are
clamoring for access to the massive quantities of information produced by and about
people, things, and their interactions. Diverse groups argue about the potential benefits
and costs of analyzing information from Twitter, Google, Verizon, 23andMe, Facebook,
Wikipedia, and every space where large groups of people leave digital traces and deposit
data. Significant questions emerge. Will large-scale analysis of DNA help cure diseases?
Or will it usher in a new wave of medical inequality? Will data analytics help make
people’s access to information more efficient and effective? Or will it be used to track
protesters in the streets of major cities? Will it transform how we study human
communication and culture, or narrow the palette of research options and alter what
‘research’ means? Some or all of the above?
Big Data is, in many ways, a poor term. As Lev Manovich (2011) observes, it has been
used in the sciences to refer to data sets large enough to require supercomputers, although
now vast sets of data can be analyzed on desktop computers with standard software.
There is little doubt that the quantities of data now available are indeed large, but that’s
not the most relevant characteristic of this new data ecosystem. Big Data is notable not
because of its size, but because of its relationality to other data. Due to efforts to mine
1
Electronic copy available at: http://ssrn.com/abstract=1926431
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
and aggregate data, Big Data is fundamentally networked. Its value comes from the
patterns that can be derived by making connections between pieces of data, about an
individual, about individuals in relation to others, about groups of people, or simply about
the structure of information itself.
Furthermore, Big Data is important because it refers to an analytic phenomenon playing
out in academia and industry. Rather than suggesting a new term, we are using Big Data
here because of its popular salience and because it is the phenomenon around Big Data
that we want to address. Big Data tempts some researchers to believe that they can see
everything at a 30,000-foot view. It is the kind of data that encourages the practice of
apophenia: seeing patterns where none actually exist, simply because massive quantities
of data can offer connections that radiate in all directions. Due to this, it is crucial to
begin asking questions about the analytic assumptions, methodological frameworks, and
underlying biases embedded in the Big Data phenomenon.
While databases have been aggregating data for over a century, Big Data is no longer just
the domain of actuaries and scientists. New technologies have made it possible for a
wide range of people – including humanities and social science academics, marketers,
governmental organizations, educational institutions, and motivated individuals – to
produce, share, interact with, and organize data. Massive data sets that were once
obscure and distinct are being aggregated and made easily accessible. Data is
increasingly digital air: the oxygen we breathe and the carbon dioxide that we exhale. It
can be a source of both sustenance and pollution.
How we handle the emergence of an era of Big Data is critical: while it is taking place in
an environment of uncertainty and rapid change, current decisions will have considerable
impact in the future. With the increased automation of data collection and analysis – as
well as algorithms that can extract and inform us of massive patterns in human behavior –
it is necessary to ask which systems are driving these practices, and which are regulating
them. In Code, Lawrence Lessig (1999) argues that systems are regulated by four forces:
the market, the law, social norms, and architecture – or, in the case of technology, code.
When it comes to Big Data, these four forces are at work and, frequently, at odds. The
market sees Big Data as pure opportunity: marketers use it to target advertising, insurance
providers want to optimize their offerings, and Wall Street bankers use it to read better
readings on market temperament. Legislation has already been proposed to curb the
collection and retention of data, usually over concerns about privacy (for example, the Do
Not Track Online Act of 2011 in the United States). Features like personalization allow
rapid access to more relevant information, but they present difficult ethical questions and
fragment the public in problematic ways (Pariser 2011).
There are some significant and insightful studies currently being done that draw on Big
Data methodologies, particularly studies of practices in social network sites like
Facebook and Twitter. Yet, it is imperative that we begin asking critical questions about
what all this data means, who gets access to it, how it is deployed, and to what ends. With
Big Data come big responsibilities. In this essay, we are offering six provocations that we
hope can spark conversations about the issues of Big Data. Social and cultural researchers
2
Electronic copy available at: http://ssrn.com/abstract=1926431
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
have a stake in the computational culture of Big Data precisely because many of its
central questions are fundamental to our disciplines. Thus, we believe that it is time to
start critically interrogating this phenomenon, its assumptions, and its biases.
1. Automating Research Changes the Definition of Knowledge.
In the early decades of the 20th century, Henry Ford devised a manufacturing system of
mass production, using specialized machinery and standardized products.
Simultaneously, it became the dominant vision of technological progress. Fordism meant
automation and assembly lines, and for decades onward, this became the orthodoxy of
manufacturing: out with skilled craftspeople and slow work, in with a new machine-made
era (Baca 2004). But it was more than just a new set of tools. The 20th century was
marked by Fordism at a cellular level: it produced a new understanding of labor, the
human relationship to work, and society at large.
Big Data not only refers to very large data sets and the tools and procedures used to
manipulate and analyze them, but also to a computational turn in thought and research
(Burkholder 1992). Just as Ford changed the way we made cars – and then transformed
work itself – Big Data has emerged a system of knowledge that is already changing the
objects of knowledge, while also having the power to inform how we understand human
networks and community. ‘Change the instruments, and you will change the entire social
theory that goes with them,’ Latour reminds us (2009, p. 9).
We would argue that Bit Data creates a radical shift in how we think about research.
Commenting on computational social science, Lazer et al argue that it offers ‘the capacity
to collect and analyze data with an unprecedented breadth and depth and scale’ (2009, p.
722). But it is not just a matter of scale. Neither is enough to consider it in terms of
proximity, or what Moretti (2007) refers to as distant or close analysis of texts. Rather, it
is a profound change at the levels of epistemology and ethics. It reframes key questions
about the constitution of knowledge, the processes of research, how we should engage
with information, and the nature and the categorization of reality. Just as du Gay and
Pryke note that ‘accounting tools...do not simply aid the measurement of economic
activity, they shape the reality they measure’ (2002, pp. 12-13), so Big Data stakes out
new terrains of objects, methods of knowing, and definitions of social life.
Speaking in praise of what he terms ‘The Petabyte Age’, Chris Anderson, Editor-in-Chief
of Wired, writes:
This is a world where massive amounts of data and applied mathematics replace
every other tool that might be brought to bear. Out with every theory of human
behavior, from linguistics to sociology. Forget taxonomy, ontology, and
psychology. Who knows why people do what they do? The point is they do it, and
we can track and measure it with unprecedented fidelity. With enough data, the
numbers speak for themselves. (2008)
3
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
Do numbers speak for themselves? The answer, we think, is a resounding ‘no’.
Significantly, Anderson’s sweeping dismissal of all other theories and disciplines is a tell:
it reveals an arrogant undercurrent in many Big Data debates where all other forms of
analysis can be sidelined by production lines of numbers, privileged as having a direct
line to raw knowledge. Why people do things, write things, or make things is erased by
the sheer volume of numerical repetition and large patterns. This is not a space for
reflection or the older forms of intellectual craft. As David Berry (2011, p. 8) writes, Big
Data provides ‘destablising amounts of knowledge and information that lack the
regulating force of philosophy.’ Instead of philosophy – which Kant saw as the rational
basis for all institutions – ‘computationality might then be understood as an ontotheology,
creating a new ontological “epoch” as a new historical constellation of intelligibility’
(Berry 2011, p. 12).
We must ask difficult questions of Big Data’s models of intelligibility before they
crystallize into new orthodoxies. If we return to Ford, his innovation was using the
assembly line to break down interconnected, holistic tasks into simple, atomized,
mechanistic ones. He did this by designing specialized tools that strongly predetermined
and limited the action of the worker. Similarly, the specialized tools of Big Data also
have their own inbuilt limitations and restrictions. One is the issue of time. ‘Big Data is
about exactly right now, with no historical context that is predictive,’ observes Joi Ito, the
director of the MIT Media Lab (Bollier 2010, p. 19). For example, Twitter and Facebook
are examples of Big Data sources that offer very poor archiving and search functions,
where researchers are much more likely to focus on something in the present or
immediate past – tracking reactions to an election, TV finale or natural disaster – because
of the sheer difficulty or impossibility of accessing older data.
If we are observing the automation of particular kinds of research functions, then we
must consider the inbuilt flaws of the machine tools. It is not enough to simply ask, as
Anderson suggests ‘what can science learn from Google?’, but to ask how Google and
the other harvesters of Big Data might change the meaning of learning, and what new
possibilities and new limitations may come with these systems of knowing.
2. Claims to Objectivity and Accuracy are Misleading
‘Numbers, numbers, numbers,’ writes Latour (2010). ‘Sociology has been obsessed by
the goal of becoming a quantitative science.’ Yet sociology has never reached this goal,
in Latour’s view, because of where it draws the line between what is and is not
quantifiable knowledge in the social domain.
Big Data offers the humanistic disciplines a new way to claim the status of quantitative
science and objective method. It makes many more social spaces quantifiable. In reality,
working with Big Data is still subjective, and what it quantifies does not necessarily have
a closer claim on objective truth – particularly when considering messages from social
media sites. But there remains a mistaken belief that qualitative researchers are in the
business of interpreting stories and quantitative researchers are in the business
4
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
of producing facts. In this way, Big Data risks reinscribing established divisions in the
long running debates about scientific method.
The notion of objectivity has been a central question for the philosophy of science and
early debates about the scientific method (Durkheim 1895). Claims to objectivity suggest
an adherence to the sphere of objects, to things as they exist in and for themselves.
Subjectivity, on the other hand, is viewed with suspicion, colored as it is with various
forms of individual and social conditioning. The scientific method attempts to remove
itself from the subjective domain through the application of a dispassionate process
whereby hypotheses are proposed and tested, eventually resulting in improvements in
knowledge. Nonetheless, claims to objectivity are necessarily made by subjects and are
based on subjective observations and choices.
All researchers are interpreters of data. As Lisa Gitelman (2011) observes, data needs to
be imagined as data in the first instance, and this process of the imagination of data
entails an interpretative base: ‘every discipline and disciplinary institution has its own
norms and standards for the imagination of data.’ As computational scientists have
started engaging in acts of social science, there is a tendency to claim their work as the
business of facts and not interpretation. A model may be mathematically sound, an
experiment may seem valid, but as soon as a researcher seeks to understand what it
means, the process of interpretation has begun. The design decisions that determine what
will be measured also stem from interpretation.
For example, in the case of social media data, there is a ‘data cleaning’ process: making
decisions about what attributes and variables will be counted, and which will be ignored.
This process is inherently subjective. As Bollier explains,
As a large mass of raw information, Big Data is not self-explanatory. And yet the
specific methodologies for interpreting the data are open to all sorts of
philosophical debate. Can the data represent an ‘objective truth’ or is any
interpretation necessarily biased by some subjective filter or the way that data is
‘cleaned?’ (2010, p. 13)
In addition to this question, there is the issue of data errors. Large data sets from Internet
sources are often unreliable, prone to outages and losses, and these errors and gaps are
magnified when multiple data sets are used together. Social scientists have a long history
of asking critical questions about the collection of data and trying to account for any
biases in their data (Cain & Finch, 1981; Clifford & Marcus, 1986). This requires
understanding the properties and limits of a dataset, regardless of its size. A dataset may
have many millions of pieces of data, but this does not mean it is random or
representative. To make statistical claims about a dataset, we need to know where data is
coming from; it is similarly important to know and account for the weaknesses in that
data. Furthermore, researchers must be able to account for the biases in their
interpretation of the data. To do so requires recognizing that one’s identity and
perspective informs one’s analysis (Behar & Gordon, 1996).
5
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
Spectacular errors can emerge when researchers try to build social science findings into
technological systems. A classic example arose when Friendster chose to implement
Robin Dunbar’s (1998) work. Analyzing gossip practices in humans and grooming
habits in monkeys, Dunbar found that people could only actively maintain 150
relationships at any time and argued that this number represented the maximum size of a
person's personal network. Unfortunately, Friendster believed that people were
replicating their pre-existing personal networks on the site, so they inferred that no one
should have a friend list greater than 150. Thus, they capped the number of ‘Friends’
people could have on the system (boyd, 2006).
Interpretation is at the center of data analysis. Regardless of the size of a data set, it is
subject to limitation and bias. Without those biases and limitations being understood and
outlined, misinterpretation is the result. Big Data is at its most effective when researchers
take account of the complex methodological processes that underlie the analysis of social
data.
3. Bigger Data are Not Always Better Data
Social scientists have long argued that what makes their work rigorous is rooted in their
systematic approach to data collection and analysis (McClosky, 1985). Ethnographers
focus on reflexively accounting for bias in their interpretations. Experimentalists control
and standardize the design of their experiment. Survey researchers drill down on
sampling mechanisms and question bias. Quantitative researchers weigh up statistical
significance. These are but a few of the ways in which social scientists try to assess the
validity of each other’s work. Unfortunately, some who are embracing Big Data presume
the core methodological issues in the social sciences are no longer relevant. There is a
problematic underlying ethos that bigger is better, that quantity necessarily means
quality.
Twitter provides an example in the context of a statistical analysis. First, Twitter does not
represent ‘all people’, although many journalists and researchers refer to ‘people’ and
‘Twitter users’ as synonymous. Neither is the population using Twitter representative of
the global population. Nor can we assume that accounts and users are equivalent. Some
users have multiple accounts. Some accounts are used by multiple people. Some people
never establish an account, and simply access Twitter via the web. Some accounts are
‘bots’ that produce automated content without involving a person. Furthermore, the
notion of an ‘active’ account is problematic. While some users post content frequently
through Twitter, others participate as ‘listeners’ (Crawford 2009, p. 532). Twitter Inc. has
revealed that 40 percent of active users sign in just to listen (Twitter, 2011). The very
meanings of ‘user’ and ‘participation’ and ‘active’ need to be critically examined.
Due to uncertainties about what an account represents and what engagement looks like, it
is standing on precarious ground to sample Twitter accounts and make claims about
people and users. Twitter Inc. can make claims about all accounts or all tweets or a
random sample thereof as they have access to the central database. Even so, they cannot
6
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
easily account for lurkers, people who have multiple accounts or groups of people who
all access one account. Additionally, the central database is also prone to outages, and
tweets are frequently lost and deleted.
Twitter Inc. makes a fraction of its material available to the public through its APIs1. The
‘firehose’ theoretically contains all public tweets ever posted and explicitly excludes any
tweet that a user chose to make private or ‘protected.’ Yet, some publicly accessible
tweets are also missing from the firehose. Although a handful of companies and startups
have access to the firehose, very few researchers have this level of access. Most either
have access to a ‘gardenhose’ (roughly 10% of public tweets), a ‘spritzer’ (roughly 1% of
public tweets), or have used ‘white-listed’ accounts where they could use the APIs to get
access to different subsets of content from the public stream.2 It is not clear what tweets
are included in these different data streams or sampling them represents. It could be that
the API pulls a random sample of tweets or that it pulls the first few thousand tweets per
hour or that it only pulls tweets from a particular segment of the network graph. Given
uncertainty, it is difficult for researchers to make claims about the quality of the data that
they are analyzing. Is the data representative of all tweets? No, because it excludes
tweets from protected accounts.3 Is the data representative of all public tweets? Perhaps,
but not necessarily.
These are just a few of the unknowns that researchers face when they work with Twitter
data, yet these limitations are rarely acknowledged. Even those who provide a
mechanism for how they sample from the firehose or the gardenhose rarely reveal what
might be missing or how their algorithms or the architecture of Twitter’s system
introduces biases into the dataset. Some scholars simply focus on the raw number of
tweets: but big data and whole data are not the same. Without taking into account the
sample of a dataset, the size of the dataset is meaningless. For example, a researcher may
seek to understand the topical frequency of tweets, yet if Twitter removes all tweets that
contain problematic words or content – such as references to pornography – from the
stream, the topical frequency would be wholly inaccurate. Regardless of the number of
tweets, it is not a representative sample as the data is skewed from the beginning.
Twitter has become a popular source for mining Big Data, but working with Twitter data
has serious methodological challenges that are rarely addressed by those who embrace it.
When researchers approach a dataset, they need to understand – and publicly account for
– not only the limits of the dataset, but also the limits of which questions they can ask of
a dataset and what interpretations are appropriate.
1
API stands for application programming interface; this refers to a set of tools that developers can use to
access structured data.
2
Details of what Twitter provides can be found at https://dev.twitter.com/docs/streaming-api/methods
White-listed accounts were a common mechanism of acquiring access early on, but they are no longer
available.
3
The percentage of protected accounts is unknown. In a study of Twitter where they attempted to locate
both protected and public Twitter accounts, Meeder et al (2010) found that 8.4% of the accounts they
identified were protected.
7
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
This is especially true when researchers combine multiple large datasets. Jesper
Anderson, co-founder of open financial data store FreeRisk, explains that combining data
from multiple sources creates unique challenges: ‘Every one of those sources is errorprone…I think we are just magnifying that problem [when we combine multiple data
sets]’ (Bollier 2010, p. 13). This does not mean that combining data doesn’t have value –
studies like those by Alessandro Acquisti and Ralph Gross (2009), which reveal how
databases can be combined to reveal serious privacy violations are crucial. Yet, it is
imperative that such combinations are not without methodological rigor and
transparency.
Finally, in the era of the computational turn, it is increasingly important to recognize the
value of ‘small data’. Research insights can be found at any level, including at very
modest scales. In some cases, focusing just on a single individual can be extraordinarily
valuable. Take, for example, the work of Tiffany Veinot (2007), who followed one
worker - a vault inspector at a hydroelectric utility company - in order to understand the
information practices of blue-collar worker. In doing this unusual study, Veinot reframed
the definition of ‘information practices’ away from the usual focus on early-adopter,
white-collar workers, to spaces outside of the offices and urban context. Her work tells a
story that could not be discovered by farming millions of Facebook or Twitter accounts,
and contributes to the research field in a significant way, despite the smallest possible
participant count. The size of data being sampled should fit the research question being
asked: in some cases, small is best.
4. Not All Data Are Equivalent
Some researchers assume that analyses done with small data can be done better with Big
Data. This argument also presumes that data is interchangeable. Yet, taken out of
context, data lose meaning and value. Context matters. When two datasets can be
modeled in a similar way, this does not mean that they are equivalent or can be analyzed
in the same way. Consider, for example, the rise of interest in social network analysis
that has emerged alongside the rise of social network sites (boyd & Ellison 2007) and the
industry-driven obsession with the ‘social graph’. Countless researchers have flocked to
Twitter and Facebook and other social media to analyze the resultant social graphs,
making claims about social networks.
The study of social networks dates back to early sociology and anthropology (e.g.,
Radcliffe-Brown 1940), with the notion of a ‘social network’ emerging in 1954 (Barnes)
and the field of ‘social network analysis’ emerging shortly thereafter (Freeman 2006).
Since then, scholars from diverse disciplines have been trying to understand people’s
relationships to one another using diverse methodological and analytical approaches. As
researchers began interrogating the connections between people on public social media,
there was a surge of interest in social network analysis. Now, network analysts are
turning to study networks produced through mediated communication, geographical
movement, and other data traces.
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Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
However, the networks produced through social media and resulting from
communication traces are not necessarily interchangeable with other social network data.
Just because two people are physically co-present – which may be made visible to cell
towers or captured through photographs – does not mean that they know one another.
Furthermore, rather than indicating the presence of predictable objective patterns, social
network sites facilitate connectedness across structural boundaries and act as a dynamic
source of change: taking a snapshot, or even witnessing a set of traces over time does not
capture the complexity of all social relations. As Kilduff and Tsai (2003, p. 117) note,
‘network research tends to proceed from a naive ontology that takes as unproblematic the
objective existence and persistence of patterns, elementary parts and social systems.’ This
approach can yield a particular kind of result when analysis is conducted only at a fixed
point in time, but quickly unravels as soon as broader questions are asked (Meyer et al.
2005).
Historically speaking, when sociologists and anthropologists were the primary scholars
interested in social networks, data about people’s relationships was collected through
surveys, interviews, observations, and experiments. Using this data, social scientists
focused on describing one’s ‘personal networks’ – the set of relationships that individuals
develop and maintain (Fischer 1982). These connections were evaluated based on a series
of measures developed over time to identify personal connections. Big Data introduces
two new popular types of social networks derived from data traces: ‘articulated networks’
and ‘behavioral networks.’
Articulated networks are those that result from people specifying their contacts through a
mediating technology (boyd 2004). There are three common reasons in which people
articulate their connections: to have a list of contacts for personal use; to publicly display
their connections to others; and to filter content on social media. These articulated
networks take the form of email or cell phone address books, instant messaging buddy
lists, ‘Friends’ lists on social network sites, and ‘Follower’ lists on other social media
genres. The motivations that people have for adding someone to each of these lists vary
widely, but the result is that these lists can include friends, colleagues, acquaintances,
celebrities, friends-of-friends, public figures, and interesting strangers.
Behavioral networks are derived from communication patterns, cell coordinates, and
social media interactions (Meiss et al. 2008; Onnela et al. 2007). These might include
people who text message one another, those who are tagged in photos together on
Facebook, people who email one another, and people who are physically in the same
space, at least according to their cell phone.
Both behavioral and articulated networks have great value to researchers, but they are not
equivalent to personal networks. For example, although often contested, the concept of
‘tie strength’ is understood to indicate the importance of individual relationships
(Granovetter, 1973). When a person chooses to list someone as their ‘Top Friend’ on
MySpace, this may or may not be their closest friend; there are all sorts of social reasons
to not list one’s most intimate connections first (boyd, 2006). Likewise, when mobile
phones recognize that a worker spends more time with colleagues than their spouse, this
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Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
does not necessarily mean that they have stronger ties with their colleagues than their
spouse. Measuring tie strength through frequency or public articulation is a common
mistake: tie strength – and many of the theories built around it – is a subtle reckoning in
how people understand and value their relationships with other people.
Fascinating network analysis can be done with behavioral and articulated networks. But
there is a risk in an era of Big Data of treating every connection as equivalent to every
other connection, of assuming frequency of contact is equivalent to strength of
relationship, and of believing that an absence of connection indicates a relationship
should be made. Data is not generic. There is value to analyzing data abstractions, yet the
context remains critical.
5. Just Because it is Accessible Doesn’t Make it Ethical
In 2006, a Harvard-based research project started gathering the profiles of 1,700 collegebased Facebook users to study how their interests and friendships changed over time
(Lewis et al. 2008). This supposedly anonymous data was released to the world, allowing
other researchers to explore and analyze it. What other researchers quickly discovered
was that it was possible to de-anonymize parts of the dataset: compromising the privacy
of students, none of whom were aware their data was being collected (Zimmer 2008).
The case made headlines, and raised a difficult issue for scholars: what is the status of socalled ‘public’ data on social media sites? Can it simply be used, without requesting
permission? What constitutes best ethical practice for researchers? Privacy campaigners
already see this as a key battleground where better privacy protections are needed. The
difficulty is that privacy breaches are hard to make specific – is there damage done at the
time? What about twenty years hence? ‘Any data on human subjects inevitably raise
privacy issues, and the real risks of abuse of such data are difficult to quantify’ (Nature,
cited in Berry 2010).
Even when researchers try to be cautious about their procedures, they are not always
aware of the harm they might be causing in their research. For example, a group of
researchers noticed that there was a correlation between self-injury (‘cutting’) and
suicide. They prepared an educational intervention seeking to discourage people from
engaging in acts of self-injury, only to learn that their intervention prompted an increase
in suicide attempts. For some, self-injury was a safety valve that kept the desire to
attempt suicide at bay. They immediately ceased their intervention (Emmens & Phippen
2010).
Institutional Review Boards (IRBs) – and other research ethics committees – emerged in
the 1970s to oversee research on human subjects. While unquestionably problematic in
implementation (Schrag, 2010), the goal of IRBs is to provide a framework for evaluating
the ethics of a particular line of research inquiry and to make certain that checks and
balances are put into place to protect subjects. Practices like ‘informed consent’ and
protecting the privacy of informants are intended to empower participants in light of
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Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
earlier abuses in the medical and social sciences (Blass, 2004; Reverby, 2009). Although
IRBs cannot always predict the harm of a particular study – and, all too often, prevent
researchers from doing research on grounds other than ethics – their value is in prompting
scholars to think critically about the ethics of their research.
With Big Data emerging as a research field, little is understood about the ethical
implications of the research being done. Should someone be included as a part of a large
aggregate of data? What if someone’s ‘public’ blog post is taken out of context and
analyzed in a way that the author never imagined? What does it mean for someone to be
spotlighted or to be analyzed without knowing it? Who is responsible for making certain
that individuals and communities are not hurt by the research process? What does consent
look like?
It may be unreasonable to ask researchers to obtain consent from every person who posts
a tweet, but it is unethical for researchers to justify their actions as ethical simply because
the data is accessible. Just because content is publicly accessible doesn’t mean that it was
meant to be consumed by just anyone (boyd & Marwick, 2011). There are serious issues
involved in the ethics of online data collection and analysis (Ess, 2002). The process of
evaluating the research ethics cannot be ignored simply because the data is seemingly
accessible. Researchers must keep asking themselves – and their colleagues – about the
ethics of their data collection, analysis, and publication.
In order to act in an ethical manner, it is important that scholars reflect on the importance
of accountability. In the case of Big Data, this means both accountability to the field of
research, and accountability to the research subjects. Academic researchers are held to
specific professional standards when working with human participants in order to protect
their rights and well-being. However, many ethics boards do not understand the processes
of mining and anonymizing Big Data, let alone the errors that can cause data to become
personally identifiable. Accountability to the field and to human subjects required
rigorous thinking about the ramifications of Big Data, rather than assuming that ethics
boards will necessarily do the work of ensuring people are protected. Accountability here
is used as a broader concept that privacy, as Troshynski et al. (2008) have outlined,
where the concept of accountability can apply even when conventional expectations of
privacy aren’t in question. Instead, accountability is a multi-directional relationship: there
may be accountability to superiors, to colleagues, to participants and to the public
(Dourish & Bell 2011).
There are significant questions of truth, control and power in Big Data studies:
researchers have the tools and the access, while social media users as a whole do not.
Their data was created in highly context-sensitive spaces, and it is entirely possible that
some social media users would not give permission for their data to be used elsewhere.
Many are not aware of the multiplicity of agents and algorithms currently gathering and
storing their data for future use. Researchers are rarely in a user’s imagined audience,
neither are users necessarily aware of all the multiple uses, profits and other gains that
come from information they have posted. Data may be public (or semi-public) but this
does not simplistically equate with full permission being given for all uses. There is a
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Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
considerable difference between being in public and being public, which is rarely
acknowledged by Big Data researchers.
6. Limited Access to Big Data Creates New Digital Divides
In an essay on Big Data, Scott Golder (2010) quotes sociologist George Homans
(1974): ‘The methods of social science are dear in time and money and getting dearer
every day.’ Historically speaking, collecting data has been hard, time consuming, and
resource intensive. Much of the enthusiasm surrounding Big Data stems from the
perception that it offers easy access to massive amounts of data.
But who gets access? For what purposes? In what contexts? And with what constraints?
While the explosion of research using data sets from social media sources would suggest
that access is straightforward, it is anything but. As Lev Manovich (2011) points out,
‘only social media companies have access to really large social data - especially
transactional data. An anthropologist working for Facebook or a sociologist working for
Google will have access to data that the rest of the scholarly community will not.’ Some
companies restrict access to their data entirely; other sell the privilege of access for a high
fee; and others offer small data sets to university-based researchers. This produces
considerable unevenness in the system: those with money – or those inside the company
– can produce a different type of research than those outside. Those without access can
neither reproduce nor evaluate the methodological claims of those who have privileged
access.
It is also important to recognize that the class of the Big Data rich is reinforced through
the university system: top-tier, well-resourced universities will be able to buy access to
data, and students from the top universities are the ones most likely to be invited to work
within large social media companies. Those from the periphery are less likely to get those
invitations and develop their skills. The result is that the divisions between those who
went to the top universities and the rest will widen significantly.
In addition to questions of access, there are questions of skills. Wrangling APIs, scraping
and analyzing big swathes of data is a skill set generally restricted to those with a
computational background. When computational skills are positioned as the most
valuable, questions emerge over who is advantaged and who is disadvantaged in such a
context. This, in its own way, sets up new hierarchies around ‘who can read the
numbers’, rather than recognizing that computer scientists and social scientists both have
valuable perspectives to offer. Significantly, this is also a gendered division. Most
researchers who have computational skills at the present moment are male and, as
feminist historians and philosophers of science have demonstrated, who is asking the
questions determines which questions are asked (Forsythe 2001; Harding 1989). There
are complex questions about what kinds of research skills are valued in the future and
how those skills are taught. How can students be educated so that they are equally
comfortable with algorithms and data analysis as well as with social analysis and theory?
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Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
Finally, the difficulty and expense of gaining access to Big Data produces a restricted
culture of research findings. Large data companies have no responsibility to make their
data available, and they have total control over who gets to see it. Big Data researchers
with access to proprietary data sets are less likely to choose questions that are contentious
to a social media company, for example, if they think it may result in their access being
cut. The chilling effects on the kinds of research questions that can be asked - in public or
private - are something we all need to consider when assessing the future of Big Data.
The current ecosystem around Big Data creates a new kind of digital divide: the Big Data
rich and the Big Data poor. Some company researchers have even gone so far as to
suggest that academics shouldn’t bother studying social media - as in-house people can
do it so much better.4 Such explicit efforts to demarcate research ‘insiders’ and
‘outsiders’ – while by no means new – undermine the utopian rhetoric of those who
evangelize about the values of Big Data. ‘Effective democratisation can always be
measured by this essential criterion,’ Derrida claimed, ‘the participation in and access to
the archive, its constitution, and its interpretation’ (1996, p. 4). Whenever inequalities are
explicitly written into the system, they produce class-based structures. Manovich writes
of three classes of people in the realm of Big Data: ‘those who create data (both
consciously and by leaving digital footprints), those who have the means to collect it, and
those who have expertise to analyze it’ (2011). We know that the last group is the
smallest, and the most privileged: they are also the ones who get to determine the rules
about how Big Data will be used, and who gets to participate. While institutional
inequalities may be a forgone conclusion in academia, they should nevertheless be
examined and questioned. They produce a bias in the data and the types of research that
emerge.
By arguing that the Big Data phenomenon is implicated in some much broader historical
and philosophical shifts is not to suggest it is solely accountable; the academy is by no
means the sole driver behind the computational turn. There is a deep government and
industrial drive toward gathering and extracting maximal value from data, be it
information that will lead to more targeted advertising, product design, traffic planning or
criminal policing. But we do think there are serious and wide-ranging implications for the
operationalization of Big Data, and what it will mean for future research agendas. As
Lucy Suchman (2011) observes, via Levi Strauss, ‘we are our tools.’ We should consider
how they participate in shaping the world with us as we use them. The era of Big Data
has only just begun, but it is already important that we start questioning the assumptions,
values, and biases of this new wave of research. As scholars who are invested in the
production of knowledge, such interrogations are an essential component of what we do.
4
During his keynote talk at the International Conference on Weblogs and Social Media (ICWSM) in
Barcelona on July 19, 2011, Jimmy Lin – a researcher at Twitter – discouraged researchers from pursuing
lines of inquiry that internal Twitter researchers could do better given their preferential access to Twitter
data.
13
Paper to be presented at Oxford Internet Institute’s “A Decade in Internet Time: Symposium
on the Dynamics of the Internet and Society” on September 21, 2011.
Acknowledgements
We wish to thank Heather Casteel for her help in preparing this article. We are also
deeply grateful to Eytan Adar, Tarleton Gillespie, and Christian Sandvig for inspiring
conversations, suggestions, and feedback.
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17
By Cade Metz
02.25.13
1:30 PM
Scott Yara, the co-founder of Greenplum, a company that seeks to reinvent data analysis under the
aegis of tech giant EMC. Photo: EMC
Jeff Hammerbacher says that Facebook tried them all. And none of them did what the web giant
needed them to do.
Hammerbacher is the Harvard-trained mathematician Facebook hired in 2006. His job was to
harness all the digital data generated by Mark Zuckerberg’s social network — to make sense of what
people were doing on the service and find new ways of improving the thing. But as the service
expanded to tens of millions of people, Hammerbacher remembers, it was generating more data
than the company could possibly analyze with the software at hand: a good old-fashioned Oracle
database.
At the time, a long line of startups were offering a new breed of database designed to store and
analyze much larger amounts of data. Greenplum. Vertica. Netezza. Hammerbacher and Facebook
tested them all. But they weren’t suited to the task either.
In the end, Facebook turned to a little-known open source software platform that had only just
gotten off the ground at Yahoo. It was called Hadoop, and it was built to harness the power of
thousands of ordinary computer servers. Unlike the Greenplums and the Verticas, Hammerbacher
says, Hadoop could store and process the ever-expanding sea of data generated by what was quickly
becoming the world’s most popular social network.
Over the next few years, Hadoop reinvented data analysis not only at Facebook and Yahoo but so
many other web services. And then an army of commercial software vendors started selling the thing
to the rest of the world. Soon, even the likes of Oracle and Greenplum were hawking Hadoop. These
companies still treated Hadoop as an adjunct to the traditional database — as a tool suited only to
certain types of data analysis. But now, that’s changing too.
On Monday, Greenplum — now owned by tech giant EMC — revealed that it has spent the last two
years building a new Hadoop platform that it believes will leave the traditional database behind.
Known as Pivotal HD, this tool can store the massive amounts of information Hadoop was created to
store, but it’s designed to ask questions of this data significantly faster than you can with the existing
open source platform.
“We think we’re one the verge of a major shift where businesses are looking at a set of canonical
applications that can’t be easily run on existing data fabrics and relational databases,” says Paul
Martiz, the former Microsoft exec who now oversees Greenplum. Businesses need a new data fabric,
Maritz says, and the starting point for that fabric is Hadoop.
That’s a somewhat surprising statement from a company whose original business was built around a
relational database — software that stores data in neat rows and columns. But Greenplum and EMC
are just acknowledging what Jeff Hammerbacher and Facebook learned so many years ago: Hadoop
— for all its early faults — is so well suited to storing and processing the massive amounts of data
facing the modern business.
What’s more, Greenplum is revamping Hadoop to operate more like a relational database, letting
you rapidly ask questions of data using the structured query language, or SQL, which has been a
staple of the database world for decades. “When we were acquired [by EMC], we really believed that
the two worlds were going to fuse together,” says Greenplum co-founder Scott Yara. “What was
going to be exciting is if you cold take the massively parallel query processing technology in a
database system [like Greenplum] and basically fuse it with the Hadoop platform.”
The trouble with Hadoop has always been that it takes so much time to analyze data. It was a “batch
system.” Using a framework called Hadoop MapReduce, you had the freedom to build all sorts of
complex programs that crunch enormous amounts of data, but when you gave it a task, you could
wait hours — or even days — for a response.
With its new system Greenplum has worked to change that. A team led by former Microsoft database
designer Florian Waas has designed a new “query engine” that can more quickly run SQL queries on
data stored across a massive cluster of systems using the Hadoop File System, or HDFS. Open
source tools such as Hive have long provided ways of running SQL queries on Hadoop data, but this
too was a batch system that needed a fair amount of time to complete queries.
This query engine will make its debut later this year as part of Pivotal HD. Greenplum is now a key
component of an EMC subsidiary called The Pivotal Initiative, which seeks to bring several new age
web technologies and techniques to the average business.
This time, Greenplum is in lock-step with Jeff Hammerbacher. After leaving Facebook,
Hammerbacher helped found a Hadoop startup known as a Cloudera, and late last year, he unveiled
a system called Impala, which also seeks to run real-time queries atop Hadoop. But according to
Waas and Yara, Pivotal HD is significantly faster than Impala and the many other tools that run SQL
queries atop Hadoop. Yara claims that it’s at least 100 times faster than Impala.
The caveat, says Waas, is that if a server crashes when Pivotal HD is running a query, you’re forced
to restart the query. This is a little different from what people have come to expect when running
jobs at Hadoop, which was specifically designed to keep running across a large cluster of servers
even as individual machines started to fail — as they inevitably do.
“The query extensions of Pivotal HD behave slightly differently in that they require a restart of the
query when a machine is lost,” he says. “An individual query needs to be restarted but the integrity,
accessibility and functionality of the system is guaranteed to continue. We consider this a small price
to pay for several orders of magnitude performance enhancement as we do not materialize any
results during processing.”
The traditional database will always have its place. Even Greenplum will continue to offer its original
data warehouse tool, which was based on the open source PostgreSQL database. But the company’s
new query engine is yet another sign that Hadoop will continue to reinvent the way businesses
crunch their data. Not just web giants. But any business.
Update: This story has been updated with additional comment from Florian Waas and to clarify
how Pivotal HD deals with hardware failures.
March 10, 2013
By STEVE LOHR
Trading stocks, targeting ads, steering political campaigns, arranging dates, besting people on
“Jeopardy” and even choosing bra sizes: computer algorithms are doing all this work and more.
But increasingly, behind the curtain there is a decidedly retro helper — a human being.
Although algorithms are growing ever more powerful, fast and precise, the computers themselves
are literal-minded, and context and nuance often elude them. Capable as these machines are, they
are not always up to deciphering the ambiguity of human language and the mystery of reasoning.
Yet these days they are being asked to be more humanlike in what they figure out.
“For all their brilliance, computers can be thick as a brick,” said Tom M. Mitchell, a computer
scientist at Carnegie Mellon University.
And so, while programming experts still write the step-by-step instructions of computer code,
additional people are needed to make more subtle contributions as the work the computers do has
become more involved. People evaluate, edit or correct an algorithm’s work. Or they assemble
online databases of knowledge and check and verify them — creating, essentially, a crib sheet the
computer can call on for a quick answer. Humans can interpret and tweak information in ways that
are understandable to both computers and other humans.
Question-answering technologies like Apple’s Siri and I.B.M.’s Watson rely particularly on the
emerging machine-man collaboration. Algorithms alone are not enough.
Twitter uses a far-flung army of contract workers, whom it calls judges, to interpret the meaning
and context of search terms that suddenly spike in frequency on the service.
For example, when Mitt Romney talked of cutting government money for public broadcasting in a
presidential debate last fall and mentioned Big Bird, messages with that phrase surged. Human
judges recognized instantly that “Big Bird,” in that context and at that moment, was mainly a
political comment, not a reference to “Sesame Street,” and that politics-related messages should
pop up when someone searched for “Big Bird.” People can understand such references more
accurately and quickly than software can, and their judgments are fed immediately into Twitter’s
search algorithm.
“Humans are core to this system,” two Twitter engineers wrote in a blog post in January.
Even at Google, where algorithms and engineers reign supreme in the company’s business and
culture, the human contribution to search results is increasing. Google uses human helpers in two
ways. Several months ago, it began presenting summaries of information on the right side of a
search page when a user typed in the name of a well-known person or place, like “Barack Obama”
or “New York City.” These summaries draw from databases of knowledge like Wikipedia, the C.I.A.
World Factbook and Freebase, whose parent company, Metaweb, Google acquired in 2010. These
databases are edited by humans.
When Google’s algorithm detects a search term for which this distilled information is available, the
search engine is trained to go fetch it rather than merely present links to Web pages.
“There has been a shift in our thinking,” said Scott Huffman, an engineering director in charge of
search quality at Google. “A part of our resources are now more human curated.”
Other human helpers, known as evaluators or raters, help Google develop tweaks to its search
algorithm, a powerhouse of automation, fielding 100 billion queries a month. “Our engineers
evolve the algorithm, and humans help us see if a suggested change is really an improvement,” Mr.
Huffman said.
Katherine Young, 23, is a Google rater — a contract worker and a college student in Macon, Ga. She
is shown an ambiguous search query like “what does king hold,” presented with two sets of Google
search results and asked to rate their relevance, accuracy and quality. The current search result for
that imprecise phrase starts with links to Web pages saying that kings typically hold ceremonial
scepters, a reasonable inference.
Her judgments, Ms. Young said, are “not completely black and white; some of it is subjective.” She
added, “You try to put yourself in the shoes of the person who typed in the query.”
I.B.M.’s Watson, the powerful question-answering computer that defeated “Jeopardy” champions
two years ago, is in training these days to help doctors make diagnoses. But it, too, is turning to
humans for help.
To prepare for its role in assisting doctors, Watson is being fed medical texts, scientific papers and
digital patient records stripped of personal identifying information. Instead of answering
questions, however, Watson is asking them of clinicians at the Cleveland Clinic and medical school
students. They are giving answers and correcting the computer’s mistakes, using a “Teach Watson”
feature.
Watson, for example, might come across this question in a medical text: “What neurological
condition contraindicates the use of bupropion?” The software may have bupropion, an
antidepressant, in its database, but stumble on “contraindicates.” A human helper will confirm
that the word means “do not use,” and Watson returns to its data trove to reason that the
neurological condition is seizure disorder.
“We’re using medical experts to help Watson learn, make it smarter going forward,” said Eric
Brown, a scientist on I.B.M.’s Watson team.
Ben Taylor, 25, is a product manager at FindTheBest, a fast-growing start-up in Santa Barbara,
Calif. The company calls itself a “comparison engine” for finding and comparing more than 100
topics and products, from universities to nursing homes, smartphones to dog breeds. Its Web site
went up in 2010, and the company now has 60 full-time employees.
Mr. Taylor helps design and edit the site’s education pages. He is not an engineer, but an English
major who has become a self-taught expert in the arcane data found in Education Department
studies and elsewhere. His research methods include talking to and e-mailing educators. He is an
information sleuth.
On FindTheBest, more than 8,500 colleges can be searched quickly according to geography,
programs and tuition costs, among other criteria. Go to the page for a university, and a wealth of
information appears in summaries, charts and graphics — down to the gender and race
breakdowns of the student body and faculty.
Mr. Taylor and his team write the summaries and design the initial charts and graphs. From
hundreds of data points on college costs, for example, they select the ones most relevant to college
students and their parents. But much of their information is prepared in templates and tagged with
code a computer can read. So the process has become more automated, with Mr. Taylor and others
essentially giving “go fetch” commands that the computer algorithm obeys.
The algorithms are getting better. But they cannot do it alone.
“You need judgment, and to be able to intuitively recognize the smaller sets of data that are most
important,” Mr. Taylor said. “To do that, you need some level of human involvement.”
How Systems Fail
How Complex Systems Fail
(Being a Short Treatise on the Nature of Failure; How Failure is Evaluated; How Failure is
Attributed to Proximate Cause; and the Resulting New Understanding of Patient Safety)
Richard I. Cook, MD
Cognitive technologies Laboratory
University of Chicago
1) Complex systems are intrinsically hazardous systems.
All of the interesting systems (e.g. transportation, healthcare, power generation) are
inherently and unavoidably hazardous by the own nature. The frequency of hazard
exposure can sometimes be changed but the processes involved in the system are
themselves intrinsically and irreducibly hazardous. It is the presence of these hazards
that drives the creation of defenses against hazard that characterize these systems.
2) Complex systems are heavily and successfully defended against failure.
The high consequences of failure lead over time to the construction of multiple layers of
defense against failure. These defenses include obvious technical components (e.g.
backup systems, ‘safety’ features of equipment) and human components (e.g. training,
knowledge) but also a variety of organizational, institutional, and regulatory defenses
(e.g. policies and procedures, certification, work rules, team training). The effect of these
measures is to provide a series of shields that normally divert operations away from
accidents.
3) Catastrophe requires multiple failures – single point failures are not enough..
The array of defenses works. System operations are generally successful. Overt
catastrophic failure occurs when small, apparently innocuous failures join to create
opportunity for a systemic accident. Each of these small failures is necessary to cause
catastrophe but only the combination is sufficient to permit failure. Put another way,
there are many more failure opportunities than overt system accidents. Most initial
failure trajectories are blocked by designed system safety components. Trajectories that
reach the operational level are mostly blocked, usually by practitioners.
4) Complex systems contain changing mixtures of failures latent within them.
The complexity of these systems makes it impossible for them to run without multiple
flaws being present. Because these are individually insufficient to cause failure they are
regarded as minor factors during operations. Eradication of all latent failures is limited
primarily by economic cost but also because it is difficult before the fact to see how such
failures might contribute to an accident. The failures change constantly because of
changing technology, work organization, and efforts to eradicate failures.
5) Complex systems run in degraded mode.
A corollary to the preceding point is that complex systems run as broken systems. The
system continues to function because it contains so many redundancies and because
people can make it function, despite the presence of many flaws. After accident reviews
nearly always note that the system has a history of prior ‘proto-accidents’ that nearly
generated catastrophe. Arguments that these degraded conditions should have been
recognized before the overt accident are usually predicated on naïve notions of system
performance. System operations are dynamic, with components (organizational, human,
technical) failing and being replaced continuously.
Copyright © 1998, 1999, 2000 by R.I.Cook, MD, for CtL
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6) Catastrophe is always just around the corner.
Complex systems possess potential for catastrophic failure. Human practitioners are
nearly always in close physical and temporal proximity to these potential failures –
disaster can occur at any time and in nearly any place. The potential for catastrophic
outcome is a hallmark of complex systems. It is impossible to eliminate the potential for
such catastrophic failure; the potential for such failure is always present by the system’s
own nature.
7) Post-accident attribution accident to a ‘root cause’ is fundamentally wrong.
Because overt failure requires multiple faults, there is no isolated ‘cause’ of an accident.
There are multiple contributors to accidents. Each of these is necessary insufficient in
itself to create an accident. Only jointly are these causes sufficient to create an accident.
Indeed, it is the linking of these causes together that creates the circumstances required
for the accident. Thus, no isolation of the ‘root cause’ of an accident is possible. The
evaluations based on such reasoning as ‘root cause’ do not reflect a technical
understanding of the nature of failure but rather the social, cultural need to blame
specific, localized forces or events for outcomes.1
8) Hindsight biases post-accident assessments of human performance.
Knowledge of the outcome makes it seem that events leading to the outcome should have
appeared more salient to practitioners at the time than was actually the case. This means
that ex post facto accident analysis of human performance is inaccurate. The outcome
knowledge poisons the ability of after-accident observers to recreate the view of
practitioners before the accident of those same factors. It seems that practitioners “should
have known” that the factors would “inevitably” lead to an accident.2 Hindsight bias
remains the primary obstacle to accident investigation, especially when expert human performance
is involved.
9) Human operators have dual roles: as producers & as defenders against failure.
The system practitioners operate the system in order to produce its desired product and
also work to forestall accidents. This dynamic quality of system operation, the balancing
of demands for production against the possibility of incipient failure is unavoidable.
Outsiders rarely acknowledge the duality of this role. In non-accident filled times, the
production role is emphasized. After accidents, the defense against failure role is
emphasized. At either time, the outsider’s view misapprehends the operator’s constant,
simultaneous engagement with both roles.
10) All practitioner actions are gambles.
After accidents, the overt failure often appears to have been inevitable and the
practitioner’s actions as blunders or deliberate willful disregard of certain impending
failure. But all practitioner actions are actually gambles, that is, acts that take place in the
face of uncertain outcomes. The degree of uncertainty may change from moment to
moment. That practitioner actions are gambles appears clear after accidents; in general,
Anthropological field research provides the clearest demonstration of the social construction of the notion
of ‘cause’ (cf. Goldman L (1993), The Culture of Coincidence: accident and absolute liability in Huli, New York:
Clarendon Press; and also Tasca L (1990), The Social Construction of Human Error, Unpublished doctoral
dissertation, Department of Sociology, State University of New York at Stonybrook.
1
This is not a feature of medical judgements or technical ones, but rather of all human cognition about past
events and their causes.
2
Copyright © 1998, 1999, 2000 by R.I.Cook, MD, for CtL
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How Systems Fail
post hoc analysis regards these gambles as poor ones. But the converse: that successful
outcomes are also the result of gambles; is not widely appreciated.
11) Actions at the sharp end resolve all ambiguity.
Organizations are ambiguous, often intentionally, about the relationship between
production targets, efficient use of resources, economy and costs of operations, and
acceptable risks of low and high consequence accidents. All ambiguity is resolved by
actions of practitioners at the sharp end of the system. After an accident, practitioner
actions may be regarded as ‘errors’ or ‘violations’ but these evaluations are heavily
biased by hindsight and ignore the other driving forces, especially production pressure.
12) Human practitioners are the adaptable element of complex systems.
Practitioners and first line management actively adapt the system to maximize
production and minimize accidents. These adaptations often occur on a moment by
moment basis. Some of these adaptations include: (1) Restructuring the system in order
to reduce exposure of vulnerable parts to failure. (2) Concentrating critical resources in
areas of expected high demand. (3) Providing pathways for retreat or recovery from
expected and unexpected faults. (4) Establishing means for early detection of changed
system performance in order to allow graceful cutbacks in production or other means of
increasing resiliency.
13) Human expertise in complex systems is constantly changing
Complex systems require substantial human expertise in their operation and
management. This expertise changes in character as technology changes but it also
changes because of the need to replace experts who leave. In every case, training and
refinement of skill and expertise is one part of the function of the system itself. At any
moment, therefore, a given complex system will contain practitioners and trainees with
varying degrees of expertise. Critical issues related to expertise arise from (1) the need to
use scarce expertise as a resource for the most difficult or demanding production needs
and (2) the need to develop expertise for future use.
14) Change introduces new forms of failure.
The low rate of overt accidents in reliable systems may encourage changes, especially the
use of new technology, to decrease the number of low consequence but high frequency
failures. These changes maybe actually create opportunities for new, low frequency but
high consequence failures. When new technologies are used to eliminate well
understood system failures or to gain high precision performance they often introduce
new pathways to large scale, catastrophic failures. Not uncommonly, these new, rare
catastrophes have even greater impact than those eliminated by the new technology.
These new forms of failure are difficult to see before the fact; attention is paid mostly to
the putative beneficial characteristics of the changes. Because these new, high
consequence accidents occur at a low rate, multiple system changes may occur before an
accident, making it hard to see the contribution of technology to the failure.
15) Views of ‘cause’ limit the effectiveness of defenses against future events.
Post-accident remedies for “human error” are usually predicated on obstructing activities
that can “cause” accidents. These end-of-the-chain measures do little to reduce the
likelihood of further accidents. In fact that likelihood of an identical accident is already
extraordinarily low because the pattern of latent failures changes constantly. Instead of
increasing safety, post-accident remedies usually increase the coupling and complexity of
Copyright © 1998, 1999, 2000 by R.I.Cook, MD, for CtL
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How Systems Fail
the system. This increases the potential number of latent failures and also makes the
detection and blocking of accident trajectories more difficult.
16) Safety is a characteristic of systems and not of their components
Safety is an emergent property of systems; it does not reside in a person, device or
department of an organization or system. Safety cannot be purchased or manufactured; it
is not a feature that is separate from the other components of the system. This means that
safety cannot be manipulated like a feedstock or raw material. The state of safety in any
system is always dynamic; continuous systemic change insures that hazard and its
management are constantly changing.
17) People continuously create safety.
Failure free operations are the result of activities of people who work to keep the system
within the boundaries of tolerable performance. These activities are, for the most part,
part of normal operations and superficially straightforward. But because system
operations are never trouble free, human practitioner adaptations to changing conditions
actually create safety from moment to moment. These adaptations often amount to just
the selection of a well-rehearsed routine from a store of available responses; sometimes,
however, the adaptations are novel combinations or de novo creations of new approaches.
18) Failure free operations require experience with failure.
Recognizing hazard and successfully manipulating system operations to remain inside
the tolerable performance boundaries requires intimate contact with failure. More robust
system performance is likely to arise in systems where operators can discern the “edge of
the envelope”. This is where system performance begins to deteriorate, becomes difficult
to predict, or cannot be readily recovered. In intrinsically hazardous systems, operators
are expected to encounter and appreciate hazards in ways that lead to overall
performance that is desirable. Improved safety depends on providing operators with
calibrated views of the hazards. It also depends on providing calibration about how their
actions move system performance towards or away from the edge of the envelope.
Other materials:
Cook, Render, Woods (2000). Gaps in the continuity of care and progress on patient
safety. British Medical Journal 320: 791-4.
Cook (1999). A Brief Look at the New Look in error, safety, and failure of complex
systems. (Chicago: CtL).
Woods & Cook (1999). Perspectives on Human Error: Hindsight Biases and Local
Rationality. In Durso, Nickerson, et al., eds., Handbook of Applied Cognition. (New
York: Wiley) pp. 141-171.
Woods & Cook (1998). Characteristics of Patient Safety: Five Principles that Underlie
Productive Work. (Chicago: CtL)
Cook & Woods (1994), “Operating at the Sharp End: The Complexity of Human Error,”
in MS Bogner, ed., Human Error in Medicine, Hillsdale, NJ; pp. 255-310.
Copyright © 1998, 1999, 2000 by R.I.Cook, MD, for CtL
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How Systems Fail
Woods, Johannesen, Cook, & Sarter (1994), Behind Human Error: Cognition, Computers and
Hindsight, Wright Patterson AFB: CSERIAC.
Cook, Woods, & Miller (1998), A Tale of Two Stories: Contrasting Views of Patient Safety,
Chicago, IL: NPSF, (available as PDF file on the NPSF web site at www.npsf.org).
Copyright © 1998, 1999, 2000 by R.I.Cook, MD, for CtL
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OP-ED COLUMNIST
What Data Can’t Do
By DAVID BROOKS
Published: February 18, 2013
254 Comments
Not long ago, I was at a dinner with the chief executive of a large
bank. He had just had to decide whether to pull out of Italy, given the
weak economy and the prospect of a future euro crisis.
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The C.E.O. had his economists project
out a series of downside scenarios and
calculate what they would mean for
his company. But, in the end, he made
his decision on the basis of values.
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His bank had been in Italy for
decades. He didn’t want Italians to
think of the company as a fair-weather
friend. He didn’t want people inside
the company thinking they would cut
and run when times got hard. He decided to stay in Italy
and ride out any potential crisis, even with the short-term
costs.
He wasn’t oblivious to data in making this decision, but
ultimately, he was guided by a different way of thinking.
And, of course, he was right to be. Commerce depends on
trust. Trust is reciprocity coated by emotion. People and
companies that behave well in tough times earn affection
and self-respect that is extremely valuable, even if it is hard
to capture in data.
I tell this story because it hints at the strengths and
limitations of data analysis. The big novelty of this historic
moment is that our lives are now mediated through
data-collecting computers. In this world, data can be used
to make sense of mind-bogglingly complex situations. Data
can help compensate for our overconfidence in our own
intuitions and can help reduce the extent to which our
desires distort our perceptions.
But there are many things big data does poorly. Let’s note a
few in rapid-fire fashion:
Data struggles with the social. Your brain is pretty bad at
math (quick, what’s the square root of 437), but it’s
excellent at social cognition. People are really good at mirroring each other’s emotional
states, at detecting uncooperative behavior and at assigning value to things through
emotion.
AUTOS
Computer-driven data analysis, on the other hand, excels at measuring the quantity of
social interactions but not the quality. Network scientists can map your interactions with
the six co-workers you see during 76 percent of your days, but they can’t capture your
devotion to the childhood friends you see twice a year, let alone Dante’s love for Beatrice,
whom he met twice.
Therefore, when making decisions about social relationships, it’s foolish to swap the
amazing machine in your skull for the crude machine on your desk.
Data struggles with context. Human decisions are not discrete events. They are embedded
in sequences and contexts. The human brain has evolved to account for this reality. People
are really good at telling stories that weave together multiple causes and multiple contexts.
Data analysis is pretty bad at narrative and emergent thinking, and it cannot match the
explanatory suppleness of even a mediocre novel.
Data creates bigger haystacks. This is a point Nassim Taleb, the author of “Antifragile,”
has made. As we acquire more data, we have the ability to find many, many more
statistically significant correlations. Most of these correlations are spurious and deceive us
when we’re trying to understand a situation. Falsity grows exponentially the more data we
collect. The haystack gets bigger, but the needle we are looking for is still buried deep
inside.
One of the features of the era of big data is the number of “significant” findings that don’t
replicate the expansion, as Nate Silver would say, of noise to signal.
Big data has trouble with big problems. If you are trying to figure out which e-mail
produces the most campaign contributions, you can do a randomized control experiment.
But let’s say you are trying to stimulate an economy in a recession. You don’t have an
alternate society to use as a control group. For example, we’ve had huge debates over the
best economic stimulus, with mountains of data, and as far as I know not a single major
player in this debate has been persuaded by data to switch sides.
Data favors memes over masterpieces. Data analysis can detect when large numbers of
people take an instant liking to some cultural product. But many important (and
profitable) products are hated initially because they are unfamiliar.
Data obscures values. I recently saw an academic book with the excellent title, “ ‘Raw
Data’ Is an Oxymoron.” One of the points was that data is never raw; it’s always structured
according to somebody’s predispositions and values. The end result looks disinterested,
but, in reality, there are value choices all the way through, from construction to
interpretation.
This is not to argue that big data isn’t a great tool. It’s just that, like any tool, it’s good at
some things and not at others. As the Yale professor Edward Tufte has said, “The world is
much more interesting than any one discipline.”
By Natalie Wolchover, Simons Science News
02.06.13
9:30 AM
In Cuernavaca, Mexico, a “spy” network makes the decentralized bus system more efficient. As a
consequence, the departure times of buses exhibit a ubiquitous pattern known as “universality.”
(Photo: Marco de Leija)
In 1999, while sitting at a bus stop in Cuernavaca, Mexico, a Czech physicist named Petr Šeba
noticed young men handing slips of paper to the bus drivers in exchange for cash. It wasn’t
organized crime, he learned, but another shadow trade: Each driver paid a “spy” to record when the
bus ahead of his had departed the stop. If it had left recently, he would slow down, letting passengers
accumulate at the next stop. If it had departed long ago, he sped up to keep other buses from passing
him. This system maximized profits for the drivers. And it gave Šeba an idea.
“We felt here some kind of similarity with quantum chaotic systems,” explained Šeba’s co-author,
Milan Krbálek, in an email.
Original story reprinted with permission from Simons Science News, an editorially independent
division of SimonsFoundation.org whose mission is to enhance public understanding of science by
covering research developments and trends in mathematics and the computational, physical and
life sciences.
After several failed attempts to talk to the spies himself, Šeba asked his student to explain to them
that he wasn’t a tax collector, or a criminal — he was simply a “crazy” scientist willing to trade
tequila for their data. The men handed over their used papers. When the researchers plotted
thousands of bus departure times on a computer, their suspicions were confirmed: The interaction
between drivers caused the spacing between departures to exhibit a distinctive pattern previously
observed in quantum physics experiments.
“I was thinking that something like this could come out, but I was really surprised that it comes
exactly,” Šeba said.
Subatomic particles have little to do with decentralized bus systems. But in the years since the odd
coupling was discovered, the same pattern has turned up in other unrelated settings. Scientists now
believe the widespread phenomenon, known as “universality,” stems from an underlying connection
to mathematics, and it is helping them to model complex systems from the internet to Earth’s
climate.
The red pattern exhibits a precise balance of randomness and regularity known as “universality,”
which has been observed in the spectra of many complex, correlated systems. In this spectrum, a
mathematical formula called the “correlation function” gives the exact probability of finding two
lines spaced a given distance apart. (Illustration: Simons Science News)
The pattern was first discovered in nature in the 1950s in the energy spectrum of the uranium
nucleus, a behemoth with hundreds of moving parts that quivers and stretches in infinitely many
ways, producing an endless sequence of energy levels. In 1972, the number theorist Hugh
Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely
related to the distribution of prime numbers. In 2000, Krbálek and Šeba reported it in the
Cuernavaca bus system. And in recent years it has shown up in spectral measurements of composite
materials, such as sea ice and human bones, and in signal dynamics of the Erdös–Rényi model, a
simplified version of the internet named for Paul Erdös and Alfréd Rényi.
Each of these systems has a spectrum — a sequence like a bar code representing data such as energy
levels, zeta zeros, bus departure times or signal speeds. In all the spectra, the same distinctive
pattern appears: The data seem haphazardly distributed, and yet neighboring lines repel one
another, lending a degree of regularity to their spacing. This fine balance between chaos and order,
which is defined by a precise formula, also appears in a purely mathematical setting: It defines the
spacing between the eigenvalues, or solutions, of a vast matrix filled with random numbers.
“Why so many physical systems behave like random matrices is still a mystery,” said Horng-Tzer
Yau, a mathematician at Harvard University. “But in the past three years, we have made a very
important step in our understanding.”
By investigating the “universality” phenomenon in random matrices, researchers have developed a
better sense of why it arises elsewhere — and how it can be used. In a flurry of recent papers, Yau
and other mathematicians have characterized many new types of random matrices, which can
conform to a variety of numerical distributions and symmetry rules. For example, the numbers
filling a matrix’s rows and columns might be chosen from a bell curve of possible values, or they
might simply be 1s and –1s. The top right and bottom left halves of the matrix might be mirror
images of one another, or not. Time and again, regardless of their specific characteristics, the
random matrices are found to exhibit that same chaotic yet regular pattern in the distribution of
their eigenvalues. That’s why mathematicians call the phenomenon “universality.”
“It seems to be a law of nature,” said Van Vu, a mathematician at Yale University who, with Terence
Tao of the University of California, Los Angeles, has proven universality for a broad class of random
matrices.
Universality is thought to arise when a system is very complex, consisting of many parts that
strongly interact with each other to generate a spectrum. The pattern emerges in the spectrum of a
random matrix, for example, because the matrix elements all enter into the calculation of that
spectrum. But random matrices are merely “toy systems” that are of interest because they can be
rigorously studied, while also being rich enough to model real-world systems, Vu said. Universality
is much more widespread. Wigner’s hypothesis (named after Eugene Wigner, the physicist who
discovered universality in atomic spectra) asserts that all complex, correlated systems exhibit
universality, from a crystal lattice to the internet.
The more complex a system is, the more robust its universality should be, said László Erdös of the
University of Munich, one of Yau’s collaborators. “This is because we believe that universality is the
typical behavior.”
Mathematicians are using random matrix models to study and predict some of the internet’s
properties, such as the size of typical computer clusters. (Illustration: Matt Britt)
In many simple systems, individual components can assert too great an influence on the outcome of
the system, changing the spectral pattern. With larger systems, no single component dominates. “It’s
like if you have a room with a lot of people and they decide to do something, the personality of one
person isn’t that important,” Vu said.
Whenever a system exhibits universality, the behavior acts as a signature certifying that the system
is complex and correlated enough to be treated like a random matrix. “This means you can use a
random matrix to model it,” Vu said. “You can compute other parameters of the matrix model and
use them to predict that the system may behave like the parameters you computed.”
This technique is enabling scientists to understand the structure and evolution of the internet.
Certain properties of this vast computer network, such as the typical size of a cluster of computers,
can be closely estimated by measurable properties of the corresponding random matrix. “People are
very interested in clusters and their locations, partially motivated by practical purposes such as
advertising,” Vu said.
A similar technique may lead to improvements in climate change models. Scientists have found that
the presence of universality in features similar to the energy spectrum of a material indicates that its
components are highly connected, and that it will therefore conduct fluids, electricity or heat.
Conversely, the absence of universality may show that a material is sparse and acts as an insulator.
In new work presented in January at the Joint Mathematics Meetings in San Diego, Ken Golden, a
mathematician at the University of Utah, and his student, Ben Murphy, used this distinction to
predict heat transfer and fluid flow in sea ice, both at the microscopic level and through patchworks
of Arctic melt ponds spanning thousands of kilometers.
When Arctic melt ponds are sufficiently connected, as pictured here, they exhibit a property called
universality that researchers believe is common to all complex, correlated systems. (Photo: Don
Perovich)
The spectral measure of a mosaic of melt ponds, taken from a helicopter, or a similar measurement
taken of a sample of sea ice in an ice core, instantly exposes the state of either system. “Fluid flow
through sea ice governs or mediates very important processes that you need to understand in order
to understand the climate system,” Golden said. “The transitions in the eigenvalue statistics presents
a brand new, mathematically rigorous approach to incorporating sea ice into climate models.”
The same trick may also eventually provide an easy test for osteoporosis. Golden, Murphy and their
colleagues have found that the spectrum of a dense, healthy bone exhibits universality, while that of
a porous, osteoporotic bone does not.
“We’re dealing with systems where the ‘particles’ can be on the millimeter or even on the kilometer
scale,” Murphy said, referring to the systems’ component parts. “It’s amazing that the same
underlying mathematics describes both.”
The reason a real-world system would exhibit the same spectral behavior as a random matrix may be
easiest to understand in the case of the nucleus of a heavy atom. All quantum systems, including
atoms, are governed by the rules of mathematics, and specifically by those of matrices. “That’s what
quantum mechanics is all about,” said Freeman Dyson, a retired mathematical physicist who helped
develop random matrix theory in the 1960s and 1970s while at Princeton’s Institute for Advanced
Study. “Every quantum system is governed by a matrix representing the total energy of the system,
and the eigenvalues of the matrix are the energy levels of the quantum system.”
The matrices behind simple atoms, such as hydrogen or helium, can be worked out exactly, yielding
eigenvalues that correspond with stunning precision to the measured energy levels of the atoms. But
the matrices corresponding to more complex quantum systems, such as a uranium nucleus, quickly
grow too thorny to grasp. According to Dyson, this is why such nuclei can be compared to random
matrices. Many of the interactions inside uranium — the elements of its unknown matrix — are so
complex that they become washed out, like a mélange of sounds blending into noise. Consequently,
the unknown matrix that governs the nucleus behaves like a matrix filled with random numbers, and
so its spectrum exhibits universality.
Scientists have yet to develop an intuitive understanding of why this particular random-yet-regular
pattern, and not some other pattern, emerges for complex systems. “We only know it from
calculations,” Vu said. Another mystery is what it has to do with the Riemann zeta function, whose
spectrum of zeros exhibits universality. The zeros of the zeta function are closely tied to the
distribution of the prime numbers — the irreducible integers out of which all others are constructed.
Mathematicians have long wondered at the haphazard way in which the primes are sprinkled along
the number line from one to infinity, and universality offers a clue. Some think there may be a
matrix underlying the Riemann zeta function that is complex and correlated enough to exhibit
universality. Discovering such a matrix would have “big implications” for finally understanding the
distribution of the primes, said Paul Bourgade, a mathematician at Harvard.
Or perhaps the explanation lies deeper still. “It may happen that it is not a matrix that lies at the core
of both Wigner’s universality and the zeta function, but some other, yet undiscovered, mathematical
structure,” Erdös said. “Wigner matrices and zeta functions may then just be different
representations of this structure.”
Many mathematicians are searching for the answer, with no guarantee that there is one. “Nobody
imagined that the buses in Cuernavaca would turn out to be an example of this. Nobody imagined
that the zeroes of the zeta function would be another example,” Dyson said. “The beauty of science is
it’s completely unpredictable, and so everything useful comes out of surprises.”
Questioning the Foundations
FQXi's 2012 Essay Contest Winners!
FQXI ARTICLE
March 12, 2013
Ideas inspired by microscopic physics and magnetism could one day help predict the
spread of disease, financial markets, and the fates of Facebook friendships.
by Graeme Stemp Morlock
November 6, 2012
Your grade ten math teacher probably
wrote this several times on your tests:
SIMPLIFY. And, for much of science, that’s
part of the work: SIMPLIFY. The universe
can be broken down into smaller and
smaller chunks in an attempt to find its
most basic level and functions. But what
do you do when that doesn’t work?
Complex systems that defy reduction are
all around us, from the elaborate workings
of an ant colony—which could never be
predicted from the physiology of a single
ant—to fluctuations in the financial system
that can send ripples around the globe.
When broken into their constituent pieces,
examined and put back together, such
systems do not behave as expected. The
sum of the parts does not equal the whole.
Surprisingly, the best way to analyze
these decidedly large-scale systems may
be by exploiting techniques first developed
not in biology or economics, but in
microscopic physics. Raissa D’Souza, a
complexity scientist at UC Davis and an
external professor at the Santa Fe
Institute, is applying lessons learned from
studying how physical systems go through
phase transitions—becoming magnetized,
for instance—to try to predict when
Raissa D’Souza
everyday networks will go through
Univers ty of California, Davis.
potentially catastrophic changes. Her work
has implications for the spread of disease,
sustaining the energy infrastructure, the financial health of countries, and for the way we connect with our
friends in online communities.
While completing her PhD in statistical physics at MIT in the 1990s, D’Souza became fascinated with
complex systems and the behavior patterns that emerge from them. Since she did not know of anyone who
specialized in the subject, she went to the library and searched the entire Boston area for someone who did,
before finding Norm Margolus, who it turned out was handily also at MIT and with whom she studied pattern
formation and computing in natural systems. D’Souza’s background in statistical physics introduced her to
the prototypical phase transition. It considers a collection of atoms, each with a magnetic moment, that
could either line-up with each other—so that the overall system becomes magnetized—or remain in a
disordered mess. There is a tension in this case: on the one hand, the atoms want to line-up, lowering the
system’s energy; on the other hand, the laws of thermodynamics tell us that systems prefer to move to a
state of increasing disorder, mathematically expressed as having a higher entropy. It was first discovered
experimentally that the outcome depends on temperature. At high temperatures entropy rules, the atoms
remain disordered and the system does not become magnetized. But below some critical temperature, the
system undergoes a phase transition and the atoms align.
That sounds simple enough, but the phase transitions that change a system’s behaviour so profoundly are
often unpredictable, especially if you only study the system in its smallest components. How for instance,
could you predict what the critical temperature would be, in theory, if you only focus your attention down
onto one isolated atom? Instead, you’ve got to see the big picture. And sometimes that picture is very big.
The Power of Networking
Taking a step back, D’Souza sees everything as being interconnected. Her job is to work out when linked
objects or entities will go through profound phase transitions, which could lead to a negative (or sometimes
positive) outcome. For instance, the United States power grid was once a collection of small isolated grids,
made of a few powerplants run by some municipality or corporation. Then, local grids were connected to
create state-wide and regional grids that remained largely independent. Distinct regions were then
interconnected to allow power transfer in emergency situations. But, with deregulation, those
interconnections now transfer massive amounts of power bought and sold on power auction markets each
day. As D’Souza points out, this interdependence has changed networks in ways that were originally never
intended, leading to unforeseen bad consequences. The U.S. power grid has grown to a point where it has
seemingly encountered a phase transition and now occasionally suffers from large cascading blackouts,
where a problem in one area can drag many areas down. Worse, a failure in one network can actually drag
down many different networks. So a failure in the power grid can cause a failure in the telecommunications
grid which causes a failure in the transportation grid and the impact keeps rippling through time and space.
Speaking at FQXi’s Setting Time Aright meeting, D’Souza discussed the conception of time that emerges
from considering examples of interconnected networks in terms of complexity theory:
"In the last 3-4 years, I’ve been working to take the ideas from single networks, such as the structure of the
network and the dynamics happening on top of the network substrate, and extending it to this bigger
context where we have networks of networks," says D’Souza. "I’ve been trying to understand what it does
to emergent properties like phase transitions and what it means for the vulnerability of these systems. So, if
there is a small ripple in one layer can it go into other layers and how long does it take?"
Understanding how networks interconnect and evolve has huge implications for public health, for instance.
D’Souza cites the example of pandemics, where infection rates have changed drastically over time based on
advancements in our transportation networks. In the Middle Ages, the bubonic plague took years to spread
across Europe, for example; by contrast the Spanish flu pandemic of 1918 killed over 20 million people
across the globe, taking only a matter of weeks or months to spread. But now, with the arrival of mass air
travel, it only takes hours for SARS, swine flu or bird flu to reach new countries.
Pinpointing the critical point of a phase transition is not easy in the world of networked networks, but part of
D’Souza’s work has been to find a generalised model, or set of equations that will apply to many different
examples, not just the power grid or the transportation network. In February 2012, D’Souza and colleagues
published a paper in the Proceedings of the National Academies of Sciences (PNAS) in which they analysed
such a universal model and predicted where the optimal level of connection and interdependence would
be—and that, ultimately, too much connectivity would be detrimental.
There are drawbacks to basing your mathematical analyses on equations inspired by mathematical physics
that are usually used to analyse the collective behavior of atoms and molecules, however. Such statistical
26
equations usually work by considering a collection of around 10 atoms (that’s 10 followed by 26 zeros,
Avogradro’s number). By contrast, even the biggest real-world networks today only get up to about a billion
9
(10 ), which makes it difficult to take theoretical predictions from the equations and apply them directly to
real-world networks. Nonetheless, independent network scientists aiming to forecast financial crises have
found intriguing evidence that backs D’Souza’s theoretical predictions about interdependence and phase
transitions.
Financial Contagion
Soon after D’Souza’s PNAS paper appeared, Stefano Battiston, at ETH Zurich, and colleagues published an
independent study in the Journal of Economic Dynamics and Control that investigated the dominant view in
finance that diversification is good. The idea is that it is better to spread your money around, so that even if
one investment goes bad, you will still win overall. However, Battiston’s group found that diversification may
not be the best strategy. In fact, they calculated that it could actually spread "financial contagion."
What Battiston’s group realized was
that a naturally occurring financial
mechanism known as trend
reinforcement was enough to push
a single failure through the entire
system. Trend reinforcement works
through a rating body that
evaluates an entity’s performance.
In the financial case that Battiston’s
group evaluated, when the market
was disappointed by a company’s
returns, they downgraded that
company, which resulted in
additional selling, which caused the
company to underperform and
disappoint the rating body again.
This negative cycle and penalization
altered the probability of the
company doing well and magnified
the initial shock. Furthermore, they
found that if the shock became big
enough, then a phase transition
would occur, as D’Souza
hypothesizes, allowing the shock to
travel through the entire system.
"There are some benefits in
Dangerous Liaisons?
Networks of networks share the good and the bad.
diversification and connections of
Credit: aleksandarvelasevic
course," says Battiston, "but there
are serious threats that come from
connecting everything in a single
system that behaves in synchrony because recovering from a complete collapse is obviously very costly, not
just economically but socially as well."
Extending their tentacles beyond the financial world, networks can also help expose the way politicians and
nation-states act. Zeev Maoz, a political scientist at UC Davis and a distinguished fellow at the
Interdisciplinary Center in Herzliya, Israel, has found that geopolitical networks have significant spillover to
other networks, for instance security and trade. Importantly, Maoz has also shown that nations are not
connected equally; often smaller states are connected through more central players. So, you get a situation
where there are a few major players each with a large cadre of states connected on their periphery, and this
can be destabilizing.
"The uneven structure is a cause of potential instability because if everyone is connected to a few major
partners and the major powers experience a shock then everyone suffers," says Maoz. Unfortunately, there
aren’t any levers that can help mitigate a shock like that because of the nature of connectivity, he explains.
Take for instance Greece, which is dependent on Germany and France and the United States. If a shock
because of the recession hits the big players, then Greece suffers more than Germany, France, or the USA,
because Greece is dependent on them and does not have trading partners of its own.
Complex Conceptions of Time
All these studies converge on one conclusion: complex systems are, fittingly, even more complex than first
thought. Complexity theorists have long accepted that you cannot just look at components and understand
the whole system—their discipline is based on that assumption, after all. But now complexity scientists have
learned that you cannot even look at a single system and understand it without the context of all the other
systems it interacts with. "So we’re at the point where we can begin to analyze systems of systems," says
D’Souza, each evolving on its own timescale, with feedbacks between these systems. Take, for instance,
online social networks that evolve much faster than say social norms or transportation networks. Users of
Facebook or Twitter typically develop a web of "friends" or "followers" that extends well beyond the number
of people they would have time to physically meet up with and interact with face-to-face, says D’Souza:
"How do we characterize time in these disparate systems?"
At first sight, the ability of online social networks to bring
people around the world closer together and shrink the
Social networks could breakdown if time that it takes to interact may seem like an
unambiguously positive thing. But even social networks
they got so dense you couldn’t
are vulnerable to phase transitions, so D’Souza urges
distinguish meaningful information caution: At some point that connectivity might backfire
and potentially cause the network to collapse. "Maybe we
from noise anymore.
will find that Facebook becomes a great big mush and isn’t
- Raissa D’Souza
interesting anymore because there is no way to
differentiate who is a true friend and who is someone that
you used to know 20 years ago and you’re just
overwhelmed with information," D’Souza says. "That could be one way that a network like Facebook could
fail. It could break down if it got so dense that you couldn’t distinguish meaningful information from noise
anymore."
And your number of Facebook friends is only going to increase, according to D’Souza. In fact, she believes
that to be almost a rule in thinking about networks of networks. "I firmly believe networks become more
interdependent in time," says D’Souza. "We see the global economy becoming more interdependent. We see
Facebook making everyone more interconnected. We’re relying increasingly on technologies like the Internet
and communications networks, for instance, the smart-grid, a cyber-physical system. All these networks
that used to operate more independently are now becoming more interconnected, and to me that is really a
signature of time."
1
Optimization of Lyapunov Invariants in Verification of
Software Systems
Mardavij Roozbehani, Member, IEEE, Alexandre Megretski , Member, IEEE, and Eric Feron Member, IEEE
Abstract
The paper proposes a control-theoretic framework for verification of numerical software systems, and puts
forward software verification as an important application of control and systems theory. The idea is to transfer
arXiv:1108.0170v1 [cs.SY] 31 Jul 2011
Lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications. These include but are not
limited to absence of overflow, absence of division-by-zero, termination in finite time, presence of dead-code, and
certain user-specified assertions. Central to this framework are Lyapunov invariants. These are properly constructed
functions of the program variables, and satisfy certain properties—resembling those of Lyapunov functions—along
the execution trace. The search for the invariants can be formulated as a convex optimization problem. If the
associated optimization problem is feasible, the result is a certificate for the specification.
Index Terms
Software Verification, Lyapunov Invariants, Convex Optimization.
I. I NTRODUCTION
S
OFTWARE in safety-critical systems implement complex algorithms and feedback laws that control
the interaction of physical devices with their environments. Examples of such systems are abundant
in aerospace, automotive, and medical applications. The range of theoretical and practical issues that arise
in analysis, design, and implementation of safety-critical software systems is extensive, see, e.g., [26],
[37] , and [22]. While safety-critical software must satisfy various resource allocation, timing, scheduling,
and fault tolerance constraints, the foremost requirement is that it must be free of run-time errors.
A. Overview of Existing Methods
1) Formal Methods: Formal verification methods are model-based techniques [44], [41], [36] for
proving or disproving that a mathematical model of a software (or hardware) satisfies a given specification,
Mardavij Roozbehani and Alexandre Megretski are with the Laboratory for Information and Decision Systems (LIDS), Massachusetts
Institute of Technology, Cambridge, MA. E-mails: {mardavij,ameg}@mit.edu. Eric Feron is professor of aerospace software engineering at
the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA. E-mail: feron@gatech.edu.
2
i.e., a mathematical expression of a desired behavior. The approach adopted in this paper too, falls under
this category. Herein, we briefly review model checking and abstract interpretation.
a) Model Checking: In model checking [14] the system is modeled as a finite state transition system
and the specifications are expressed in some form of logic formulae, e.g., temporal or propositional logic.
The verification problem then reduces to a graph search, and symbolic algorithms are used to perform
an exhaustive exploration of all possible states. Model checking has proven to be a powerful technique
for verification of circuits [13], security and communication protocols [33], [38] and stochastic processes
[3]. Nevertheless, when the program has non-integer variables, or when the state space is continuous,
model checking is not directly applicable. In such cases, combinations of various abstraction techniques
and model checking have been proposed [2], [17], [54]; scalability, however, remains a challenge.
b) Abstract Interpretation: is a theory for formal approximation of the operational semantics of
computer programs in a systematic way [15]. Construction of abstract models involves abstraction of
domains—typically in the form of a combination of sign, interval, polyhedral, and congruence abstractions
of sets of data—and functions. A system of fixed-point equations is then generated by symbolic forward/backward executions of the abstract model. An iterative equation solving procedure, e.g., Newton’s
method, is used for solving the nonlinear system of equations, the solution of which results in an inductive
invariant assertion, which is then used for checking the specifications. In practice, to guarantee finite
convergence of the iterates, narrowing (outer approximation) operators are used to estimate the solution,
followed by widening (inner approximation) to improve the estimate [16]. This compromise can be a
source of conservatism in analysis. Nevertheless, these methods have been used in practice for verification
of limited properties of embedded software of commercial aircraft [7].
Alternative formal methods can be found in the computer science literature mostly under deductive
verification [32], type inference [45], and data flow analysis [23]. These methods share extensive similarities in that a notion of program abstraction and symbolic execution or constraint propagation is present
in all of them. Further details and discussions of the methodologies can be found in [16], and [41].
3
2) System Theoretic Methods: While software analysis has been the subject of an extensive body of
research in computer science, treatment of the topic in the control systems community has been less
systematic. The relevant results in the systems and control literature can be found in the field of hybrid
systems [11]. Much of the available techniques for safety verification of hybrid systems are explicitly
or implicitly based on computation of the reachable sets, either exactly or approximately. These include
but are not limited to techniques based on quantifier elimination [29], ellipsoidal calculus [27], and
mathematical programming [5]. Alternative approaches aim at establishing properties of hybrid systems
through barrier certificates [46], numerical computation of Lyapunov functions [10], [24], or by combined
use of bisimulation mechanisms and Lyapunov techniques [20], [28], [54], [2].
Inspired by the concept of Lyapunov functions in stability analysis of nonlinear dynamical systems
[25], in this paper we propose Lyapunov invariants for analysis of computer programs. While Lyapunov
functions and similar concepts have been used in verification of stability or temporal properties of system
level descriptions of hybrid systems [48], [10], [24], to the best of our knowledge, this paper is the first
to present a systematic framework based on Lyapunov invariance and convex optimization for verification
of a broad range of code-level specifications for computer programs. Accordingly, it is in the systematic
integration of new ideas and some well-known tools within a unified software analysis framework that we
see the main contribution of our work, and not in carrying through the proofs of the underlying theorems
and propositions. The introduction and development of such framework provides an opportunity for the
field of control to systematically address a problem of great practical significance and interest to both
computer science and engineering communities. The framework can be summarized as follows:
1) Dynamical system interpretation and modeling (Section II). We introduce generic dynamical system
representations of programs, along with specific modeling languages which include Mixed-Integer
Linear Models (MILM), Graph Models, and MIL-over-Graph Hybrid Models (MIL-GHM).
2) Lyapunov invariants as behavior certificates for computer programs (Section III). Analogous to a
Lyapunov function, a Lyapunov invariant is a real-valued function of the program variables, and
satisfies a difference inequality along the trace of the program. It is shown that such functions can
4
be formulated for verification of various specifications.
3) A computational procedure for finding the Lyapunov invariants (Section IV). The procedure is
standard and constitutes these steps: (i) Restricting the search space to a linear subspace. (ii) Using
convex relaxation techniques to formulate the search problem as a convex optimization problem,
e.g., a linear program [6], semidefinite program [9], [55], or a SOS program [42]. (iii) Using convex
optimization software for numerical computation of the certificates.
II. DYNAMICAL S YSTEM I NTERPRETATION AND M ODELING OF C OMPUTER P ROGRAMS
We interpret computer programs as discrete-time dynamical systems and introduce generic models that
formalize this interpretation. We then introduce MILMs, Graph Models, and MIL-GHMs as structured
cases of the generic models. The specific modeling languages are used for computational purposes.
A. Generic Models
1) Concrete Representation of Computer Programs: We will consider generic models defined by a
finite state space set X with selected subsets X0 ⊆ X of initial states, and X∞ ⊂ X of terminal states,
and by a set-valued state transition function f : X 7→ 2X , such that f (x) ⊆ X∞ , ∀x ∈ X∞ . We denote
such dynamical systems by S(X, f, X0 , X∞ ).
Definition 1: The dynamical system S(X, f, X0 , X∞ ) is a C-representation of a computer program
P, if the set of all sequences that can be generated by P is equal to the set of all sequences X =
(x(0), x(1), . . . , x(t), . . . ) of elements from X, satisfying
x (0) ∈ X0 ⊆ X,
x (t + 1) ∈ f (x (t))
∀t ∈ Z+
(1)
The uncertainty in x(0) allows for dependence of the program on different initial conditions, and the
uncertainty in f models dependence on parameters, as well as the ability to respond to real-time inputs.
Example 1: Integer Division (adopted from [44]): The functionality of Program 1 is to compute the
result of the integer division of dd (dividend) by dr (divisor). A C-representation of the program is
displayed alongside. Note that if dd ≥ 0, and dr ≤ 0, then the program never exits the “while” loop and
the value of q keeps increasing, eventually leading to either an overflow or an erroneous answer. The
program terminates if dd and dr are positive.
5
int IntegerDivision ( int dd, int dr )
{int q = {0}; int r = {dd};
while (r >= dr)
{ q = q + 1;
r = r − dr; }
return r; }
Z = Z∩ [−32768, 32767]
X = Z4
X0 = {(dd, dr, q, r) ∈ X | q = 0, r = dd}
X∞ = {(dd, dr, q, r) ∈ X | r < dr}
(
(dd, dr, q + 1, r − dr),
f : (dd, dr, q, r) 7→
(dd, dr, q, r),
(dd, dr, q, r) ∈ X\X∞
(dd, dr, q, r) ∈ X∞
Program 1: The Integer Division Program (left) and its Dynamical System Model (right)
2) Abstract Representation of Computer Programs: In a C-representation, the elements of the state
space X belong to a finite subset of the set of rational numbers that can be represented by a fixed
number of bits in a specific arithmetic framework, e.g., fixed-point or floating-point arithmetic. When
the elements of X are non-integers, due to the quantization effects, the set-valued map f often defines
very complicated dependencies between the elements of X, even for simple programs involving only
elementary arithmetic operations. An abstract model over-approximates the behavior set in the interest of
tractability. The drawbacks are conservatism of the analysis and (potentially) undecidability. Nevertheless,
abstractions in the form of formal over-approximations make it possible to formulate computationally
tractable, sufficient conditions for a verification problem that would otherwise be intractable.
Definition 2: Given a program P and its C-representation S(X, f, X0 , X∞ ), we say that S(X, f , X 0 , X ∞ )
is an A-representation, i.e., an abstraction of P, if X ⊆ X, X0 ⊆ X 0 , and f (x) ⊆ f (x) for all x ∈ X,
and the following condition holds:
X ∞ ∩ X ⊆ X∞ .
(2)
Thus, every trajectory of the actual program is also a trajectory of the abstract model. The definition
of X ∞ is slightly more subtle. For proving Finite-Time Termination (FTT), we need to be able to infer
that if all the trajectories of S eventually enter X ∞ , then all trajectories of S will eventually enter X∞ .
It is tempting to require that X ∞ ⊆ X∞ , however, this may not be possible as X∞ is often a discrete set,
while X ∞ is dense in the domain of real numbers. The definition of X ∞ as in (2) resolves this issue.
6
Construction of S(X, f , X 0 , X ∞ ) from S(X, f, X0 , X∞ ) involves abstraction of each of the elements
X, f, X0 , X∞ in a way that is consistent with Definition 2. Abstraction of the state space X often involves
replacing the domain of floats or integers or a combination of these by the domain of real numbers.
Abstraction of X0 or X∞ often involves a combination of domain abstractions and abstraction of functions
that define these sets. Semialgebraic set-valued abstractions of some commonly-used nonlinearities are
presented in Appendix I. Interested readers may refer to [49] for more examples including abstractions
of fixed-point and floating point operations.
B. Specific Models of Computer Programs
Specific modeling languages are particularly useful for automating the proof process in a computational
framework. Here, three specific modeling languages are proposed: Mixed-Integer Linear Models (MILM),
Graph Models, and Mixed-Integer Linear over Graph Hybrid Models (MIL-GHM).
1) Mixed-Integer Linear Model (MILM): Proposing MILMs for software modeling and analysis is
motivated by the observation that by imposing linear equality constraints on boolean and continuous
variables over a quasi-hypercube, one can obtain a relatively compact representation of arbitrary piecewise
affine functions defined over compact polytopic subsets of Euclidean spaces (Proposition 1). The earliest
reference to the statement of universality of MILMs appears to be [39], in which a constructive proof is
given for the one-dimensional case. A constructive proof for the general case is given in [49].
Proposition 1: Universality of Mixed-Integer Linear Models. Let f : X 7→ Rn be a piecewise affine
map with a closed graph, defined on a compact state space X ⊆ [−1, 1]n , consisting of a finite union of
compact polytopes. That is:
f (x) ∈ 2Ai x + 2Bi
subject to
x ∈ Xi , i ∈ Z (1, N )
where, each Xi is a compact polytopic set. Then, f can be specified precisely, by imposing linear
equality constraints on a finite number of binary and continuous variables ranging over compact intervals.
7
Specifically, there exist matrices F and H, such that the following two sets are equal:
G1 = {(x, f (x)) | x ∈ X}
T
T
G2 = {(x, y) | F [ x w v 1 ] = y, H[ x w v 1 ] = 0, (w, v) ∈ [−1, 1]nw × {−1, 1}nv }
Mixed Logical Dynamical Systems (MLDS) with similar structure were considered in [4] for analysis
of a class of hybrid systems. The main contribution here is in the application of the model to software
analysis. A MIL model of a computer program is defined via the following elements:
1) The state space X ⊂ [−1, 1]n .
2) Letting ne = n + nw + nv + 1, the state transition function f : X 7→ 2X is defined by two matrices
F, and H of dimensions n-by-ne and nH -by-ne respectively, according to:
T
T
nw
nv
f (x) ∈ F [ x w v 1 ] | H[ x w v 1 ] = 0, (w, v) ∈ [−1, 1] × {−1, 1}
.
(3)
3) The set of initial conditions is defined via either of the following:
a) If X0 is finite with a small cardinality, then it can be conveniently specified by its elements.
We will see in Section IV that per each element of X0 , one constraint needs to be included
in the set of constraints of the optimization problem associated with the verification task.
b) If X0 is not finite, or |X0 | is too large, an abstraction of X0 can be specified by a matrix
H0 ∈ RnH0 ×ne which defines a union of compact polytopes in the following way:
T
X0 = {x ∈ X | H0 [ x w v 1 ] = 0, (w, v) ∈ [−1, 1]nw × {−1, 1}nv }.
(4)
4) The set of terminal states X∞ is defined by
T
X∞ = {x ∈ X | H[ x w v 1 ] 6= 0, ∀w ∈ [−1, 1]nw , ∀v ∈ {−1, 1}nv }.
(5)
Therefore, S(X, f, X0 , X∞ ) is well defined. A compact description of a MILM of a program is either
of the form S (F, H, H0 , n, nw , nv ) , or of the form S (F, H, X0 , n, nw , nv ). The MILMs can represent a
broad range of computer programs of interest in control applications, including but not limited to control
programs of gain scheduled linear systems in embedded applications. In addition, generalization of the
model to programs with piecewise affine dynamics subject to quadratic constraints is straightforward.
Example 2: A MILM of an abstraction of the IntegerDivision program (Program 1: Section II-A), with
8
all the integer variables replaced with real variables, is given by S (F, H, H0 , 4, 3, 0) , where
H0 =
H=
F =





1
0 0 0 0 0 0
1
0 0 −1 0 0 0 0
0
2 0 −2 1 0 0 1



1 0 0 0 0 0
0 1
0 0 0 0 0  
  0
 0
 ,  0 −2 0

0 0 1 0 1, 


 0 −2 0
0
0 1 0 0 0 0
0 0 1 0 1
−2
0 0
0 0 0 1 1
0 −1 0 1 0 0 0
−2
0 0
0 0 0 1 1
0


0 

1/M 
0
Here, M is a scaling parameter used for bringing all the variables within the interval [−1, 1] .
2) Graph Model: Practical considerations such as universality and strong resemblance to the natural
flow of computer code render graph models an attractive and convenient model for software analysis.
Before we proceed, for convenience, we introduce the following notation: Pr (i, x) denotes the projection
operator defined as Pr (i, x) = x, for all i ∈ Z∪ {o
n} , and all x ∈ Rn .
A graph model is defined on a directed graph G (N , E) with the following elements:
1) A set of nodes N = {∅} ∪ {1, . . . , m} ∪ {o
n} . These can be thought of as line numbers or code
locations. Nodes ∅ and o
n are starting and terminal nodes, respectively. The only possible transition
from node o
n is the identity transition to node o
n.
2) A set of edges E = {(i, j, k) | i ∈ N , j ∈ O (i)} , where the outgoing set O (i) is the set of all
nodes to which transition from node i is possible in one step. Definition of the incoming set I (i)
is analogous. The third element in the triplet (i, j, k) is the index for the kth edge between i and
j, and Aji = {k | (i, j, k) ∈ E} .
3) A set of program variables xl ∈ Ω ⊆ R, l ∈ Z (1, n) . Given N and n, the state space of a graph
model is X = N × Ωn . The state x
e = (i, x) of a graph model has therefore, two components: The
discrete component i ∈ N , and the continuous component x ∈ Ωn ⊆ Rn .
k
k
k
4) A set of transition labels T ji assigned to every edge (i, j, k) ∈ E, where T ji maps x to the set T ji x =
k
{Tjik (x, w, v) | (x, w, v) ∈ Sji
}, where (w, v) ∈ [−1, 1]nw × {−1, 1}nv , and Tjik : Rn+nw +nv 7→ Rn
k
k
k
is a polynomial function and Sji
is a semialgebraic set. If T ji is a deterministic map, we drop Sji
k
and define T ji ≡ Tjik (x).
5) A set of passport labels Πkji assigned to all edges (i, j, k) ∈ E, where Πkji is a semialgebraic set. A
state transition along edge (i, j, k) is possible if and only if x ∈ Πkji .
9
6) A set of semialgebraic invariant sets Xi ⊆ Ωn , i ∈ N are assigned to every node on the graph,
such that Pr (i, x) ∈ Xi . Equivalently, a state x
e = (i, x) satisfying x ∈ X\Xi is unreachable.
Therefore, a graph model is a well-defined specific case of the generic model S(X, f, X0 , X∞ ), with
X = N × Ωn , X0 = {∅} × X∅ , X∞ = {o
n} × Xno and f : X 7→ 2X defined as:
o
n
k
f (e
x) ≡ f (i, x) = (j, T ji x) | j ∈ O (i) , x ∈ Πkji ∩ Xi .
(6)
Conceptually similar models have been reported in [44] for software verification, and in [1], [12]
for modeling and verification of hybrid systems. Interested readers may consult [49] for further details
regarding treatment of graph models with time-varying state-dependent transitions labels which arise in
modeling operations with arrays.
Remarks
– The invariant set of node ∅ contains all the available information about the initial conditions of the
program variables: Pr (∅, x) ∈ X∅ .
– Multiple edges between nodes enable modeling of logical ”or” or ”xor” type conditional transitions.
This allows for modeling systems with nondeterministic discrete transitions.
k
– The transition label T ji may represent a simple update rule which depends on the real-time input.
T
For instance, if T = Ax + Bw, and S = Rn × [−1, 1] , then x 7→ {Ax + Bw | w ∈ [−1, 1]} .
k
In other cases, T ji may represent an abstraction of a nonlinearity. For instance, the assignment
T
x 7→ sin (x) can be abstracted by x 7→ {T (x, w) | (x, w) ∈ S} (see Eqn. (46) in Appendix I).
Before we proceed, we introduce the following notation: Given a semialgebraic set Π, and a polynomial
function τ : Rn 7→ Rn , we denote by Π (τ ) , the set: Π(τ ) = {x | τ (x) ∈ Π} .
a) Construction of Simple Invariant Sets: Simple invariant sets can be included in the model if they
are readily available or easily computable. Even trivial invariants can simplify the analysis and improve
the chances of finding stronger invariants via numerical optimization.
– Simple invariant sets may be provided by the programmer. These can be trivial sets representing
simple algebraic relations between variables, or they can be more complicated relationships that
reflect the programmer’s knowledge about the functionality and behavior of the program.
10
– Invariant Propagation: Assuming that Tijk are deterministic and invertible, the set
−1 S
Xi =
Πkij Tijk
(7)
j∈I(i), k∈Aij
is an invariant set for node i. Furthermore, if the invariant sets Xj are strict subsets of Ωn for all
j ∈ I (i) , then (7) can be improved. Specifically, the set
S
k
k −1
k −1
Xi =
Πij Tij
∩ Xj Tij
(8)
j∈I(i), k∈Aij
is an invariant set for node i. Note that it is sufficient that the restriction of Tijk to the lower
dimensional spaces in the domains of Πkij and Xj be invertible.
– Preserving Equality Constraints: Simple assignments of the form Tijk : xl 7→ f (ym ) result in invariant
sets of the form Xi = {x | xl − f (ym ) = 0} at node i, provided that Tijk does not simultaneously
update ym . Formally, let Tijk be such that (Tijk x)l −xl is non-zero for at most one element ˆl ∈ Z (1, n) ,
and that (Tijk x)l̂ is independent of xl̂ . Then, the following set is an invariant set at node i :
Xi =
S
x | Tijk − I x = 0
j∈I(i), k∈Aij
3) Mixed-Integer Linear over Graph Hybrid Model (MIL-GHM): The MIL-GHMs are graph models
in which the effects of several lines and/or functions of code are compactly represented via a MILM. As
a result, the graphs in such models have edges (possibly self-edges) that are labeled with matrices F and
H corresponding to a MILM as the transition and passport labels. Such models combine the flexibility
provided by graph models and the compactness of MILMs. An example is presented in Section V.
C. Specifications
The specification that can be verified in our framework can generically be described as unreachability
and finite-time termination.
Definition 3: A Program P ≡ S(X, f, X0 , X∞ ) is said to satisfy the unreachability property with
respect to a subset X− ⊂ X, if for every trajectory X ≡ x (·) of (1), and every t ∈ Z+ , x(t) does
not belong to X− . A program P ≡ S(X, f, X0 , X∞ ) is said to terminate in finite time if every solution
X = x (·) of (1) satisfies x(t) ∈ X∞ for some t ∈ Z+ .
Several critical specifications associated with runtime errors are special cases of unreachability.
11
1) Overflow: Absence of overflow can be characterized as a special case of unreachability by defining:
X− = x ∈ X | α−1 x∞ > 1, α = diag {αi }
where αi > 0 is the overflow limit for variable i.
2) Out-of-Bounds Array Indexing: An out-of-bounds array indexing error occurs when a variable
exceeding the length of an array, references an element of the array. Assuming that xl is the corresponding
integer index and L is the array length, one must verify that xl does not exceed L at location i, where
referencing occurs. This can be accomplished by defining X− = {(i, x) ∈ X | |xl | > L} over a graph
model and proving that X− is unreachable. This is also similar to “assertion checking” defined next.
3) Program Assertions: An assertion is a mathematical expression whose validity at a specific location
in the code must be verified. It usually indicates the programmer’s expectation from the behavior of the
program. We consider assertions that are in the form of semialgebraic set memberships. Using graph
models, this is done as follows:
at location i : assert x ∈ Ai ⇒ define X− = {(i, x) ∈ X | x ∈ X\Ai } ,
at location i : assert x ∈
/ Ai ⇒ define X− = {(i, x) ∈ X | x ∈ Ai } .
In particular, safety assertions for division-by-zero or taking the square root (or logarithm) of positive
variables are standard and must be automatically included in numerical programs (cf. Sec. III-A, Table I).
4) Program Invariants: A program invariant is a property that holds throughout the execution of the
program. The property indicates that the variables reside in a semialgebraic subset XI ⊂ X. Essentially,
any method that is used for verifying unreachability of a subset X− ⊂ X, can be applied for verifying
invariance of XI by defining X− = X\XI , and vice versa.
D. Implications of the Abstractions
For mathematical correctness, we must show that if an A-representation of a program satisfies the
unreachability and FTT specifications, then so does the C-representation, i.e., the actual program. This is
established in the following proposition. The proof is omitted for brevity but can be found in [49].
Proposition 2: Let S(X, f , X 0 , X ∞ ) be an A-representation of program P with C-representation
S(X, f, X0 , X∞ ). Let X− ⊂ X and X − ⊂ X be such that X− ⊆ X − . Assume that the unreachability
12
property w.r.t. X − has been verified for S. Then, P satisfies the unreachability property w.r.t. X− .
Moreover, if the FTT property holds for S, then P terminates in finite time.
Since we are not concerned with undecidability issues, and in light of Proposition 2, we will not
differentiate between abstract or concrete representations in the remainder of this paper.
III. LYAPUNOV I NVARIANTS AS B EHAVIOR C ERTIFICATES
Analogous to a Lyapunov function, a Lyapunov invariant is a real-valued function of the program
variables satisfying a difference inequality along the execution trace.
Definition 4: A (θ, µ)-Lyapunov invariant for S(X, f, X0 , X∞ ) is a function V : X 7→ R such that
V (x+ ) − θV (x) ≤ −µ
∀x ∈ X, x+ ∈ f (x) : x ∈
/ X∞ .
(9)
where (θ, µ) ∈ [0, ∞) × [0, ∞). Thus, a Lyapunov invariant satisfies the difference inequality (9) along
the trajectories of S until they reach a terminal state X∞ .
It follows from Definition 4 that a Lyapunov invariant is not necessarily nonnegative, or bounded from
below, and in general it need not be monotonically decreasing. While the zero level set of V defines an
invariant set in the sense that V (xk ) ≤ 0 implies V (xk+l ) ≤ 0, for all l ≥ 0, monotonicity depends on
θ and the initial condition. For instance, if V (x0 ) ≤ 0, ∀x0 ∈ X0 , then (9) implies that V (x) ≤ 0 along
the trajectories of S, however, V (x) may not be monotonic if θ < 1, though it will be monotonic for
θ ≥ 1. Furthermore, the level sets of a Lyapunov invariant need not be bounded closed curves.
Proposition 3 (to follow) formalizes the interpretation of Definition 4 for the specific modeling languages. Natural Lyapunov invariants for graph models are functions of the form
V (e
x) ≡ V (i, x) = σi (x) ,
i ∈ N,
(10)
which assign a polynomial Lyapunov function to every node i ∈ N on the graph G (N , E) .
Proposition 3: Let S (F, H, X0 , n, nw , nv ) and properly labeled graph G (N , E) be the MIL and graph
models for a computer program P. The function V : [−1, 1]n 7→ R is a (θ, µ)-Lyapunov invariant for P
if it satisfies:
V (F xe ) − θV (x) ≤ −µ,
∀ (x, xe ) ∈ [−1, 1]n × Ξ,
13
where
T
Ξ = {(x, w, v, 1) | H[ x w v 1 ] = 0, (w, v) ∈ [−1, 1]nw × {−1, 1}nv }.
The function V : N ×Rn 7→ R, satisfying (10) is a (θ, µ)-Lyapunov invariant for P if
k
σj (x+ ) − θσi (x) ≤ −µ, ∀ (i, j, k) ∈ E, (x, x+ ) ∈ (Xi ∩ Πkji ) × T ji x.
(11)
Note that a generalization of (9) allows for θ and µ to depend on the state x, although simultaneous
search for θ (x) and V (x) leads to non-convex conditions, unless the dependence of θ on x is fixed
a-priori. We allow for dependence of θ on the discrete component of the state in the following way:
k
k
σi (x) ≤ −µji , ∀ (i, j, k) ∈ E, (x, x+ ) ∈ (Xi ∩ Πkji ) × T ji x
σj (x+ ) − θji
(12)
A. Behavior Certificates
1) Finite-Time Termination (FTT) Certificates: The following proposition is applicable to FTT analysis
of both finite and infinite state models.
Proposition 4: Finite-Time Termination. Consider a program P, and its dynamical system model
S(X, f, X0 , X∞ ). If there exists a (θ, µ)-Lyapunov invariant V : X 7→ R, uniformly bounded on X\X∞ ,
satisfying (9) and the following conditions
V (x) ≤ −η ≤ 0,
where kV k∞ =
∀x ∈ X0
(13)
µ + (θ − 1) kV k∞ > 0
(14)
max (µ, η) > 0
(15)
sup V (x) < ∞, then P terminates in finite time, and an upper-bound on the number
x∈X\X∞
of iterations is given by

log (µ + (θ − 1) kV k∞ ) − log (µ)



, θ 6= 1, µ > 0


log θ



log (kV k∞ ) − log (η)
Tu =
, θ 6= 1, µ = 0


log θ





 kV k /µ
, θ=1
∞
Proof: The proof is presented in Appendix II.
(16)
14
When the state-space X is finite, or when the Lyapunov invariant V is only a function of a subset of
the variables that assume values in a finite set, e.g., integer counters, it follows from Proposition 4 that V
being a (θ, µ)-Lyapunov invariant for any θ ≥ 1 and µ > 0 is sufficient for certifying FTT, and uniform
boundedness of V need not be established a-priori.
Example 3: Consider the IntegerDivision program presented in Example 1. The function V : X 7→ R,
defined according to V : (dd, dr, q, r) 7→ r is a (1, dr)-Lyapunov invariant for IntegerDivision: at every
step, V decreases by dr > 0. Since X is finite, the program IntegerDivision terminates in finite time. This,
however, only proves absence of infinite loops. The program could terminate with an overflow.
2) Separating Manifolds and Certificates of Boundedness: Let V be a Lyapunov invariant satisfying
def
(9) with θ = 1. The level sets of V, defined by Lr (V ) = {x ∈ X : V (x) < r}, are invariant with respect
to (1) in the sense that x(t + 1) ∈ Lr (V ) whenever x(t) ∈ Lr (V ). However, for r = 0, the level sets
Lr (V ) remain invariant with respect to (1) for any nonnegative θ. This is an important property with
the implication that θ = 1 (i.e., monotonicity) is not necessary for establishing a separating manifold
between the reachable set and the unsafe regions of the state space (cf. Theorem 1).
Theorem 1: Lyapunov Invariants as Separating Manifolds. Let V denote the set of all (θ, µ)Lyapunov invariants satisfying (9) for program P ≡ S(X, f, X0 , X∞ ). Let I be the identity map, and for
h ∈ {f, I} define
h−1 (X− ) = {x ∈ X|h (x) ∩ X− 6= ∅} .
A subset X− ⊂ X, where X− ∩ X0 = ∅ can never be reached along the trajectories of P, if there exists
V ∈ V satisfying
sup V (x) <
x∈X0
inf
V (x)
x∈h−1 (X− )
(17)
and either θ = 1, or one of the following two conditions hold:
(I) θ < 1
and
(II) θ > 1
and
inf
x∈h−1 (X− )
V (x) > 0.
sup V (x) ≤ 0.
x∈X0
(18)
(19)
15
Proof: The proof is presented in Appendix II.
The following corollary is based on Theorem 1 and Proposition 4 and presents computationally
implementable criteria for simultaneously establishing FTT and absence of overflow.
Corollary 1: Overflow and FTT Analysis Consider a program P, and its dynamical system model
S(X, f, X0 , X∞ ). Let α > 0 be a diagonal matrix specifying the overflow limit, and let X− = {x ∈
X | kα−1 xk∞ > 1}. Let q ∈ N∪{∞} , h ∈ {f, I} , and let the function V : X 7→ R be a (θ, µ)-Lyapunov
invariant for S satisfying
V (x) ≤ 0
∀x ∈ X0 .
n
o
−1
∀x ∈ X.
V (x) ≥ sup α h (x) q − 1
(20)
(21)
Then, an overflow runtime error will not occur during any execution of P. In addition, if µ > 0 and
µ + θ > 1, then, P terminates in at most Tu iterations where Tu = µ−1 if θ = 1, and for θ 6= 1 we have:
Tu =
where kV k∞ =
log (µ + (θ − 1) kV k∞ ) − log µ
log (µ + θ − 1) − log µ
≤
log θ
log θ
sup
(22)
|V (x)| .
x∈X\{X− ∪X∞ }
Proof: The proof is presented in Appendix II.
Application of Corollary 1 with h = f typically leads to much less conservative results compared with
h = I, though the computational costs are also higher. See [49] for remarks on variations of Corollary 1
to trade off conservativeness and computational complexity.
a) General Unreachability and FTT Analysis over Graph Models: The results presented so far in
this section (Theorem 1, Corollary 1, and Proposition 4) are readily applicable to MILMs. These results
will be applied in Section IV to formulate the verification problem as a convex optimization problem.
Herein, we present an adaptation of these results to analysis of graph models.
Definition 5: A cycle Cm on a graph G (N , E) is an ordered list of m triplets (n1 , n2 , k1 ) , (n2 , n3 , k2 ) , ...,
(nm , nm+1 , km ) , where nm+1 = n1 , and (nj , nj+1 , kj ) ∈ E, ∀j ∈ Z (1, m) . A simple cycle is a cycle
with no strict sub-cycles.
16
Corollary 2: Unreachability and FTT Analysis of Graph Models. Consider a program P and its
graph model G (N , E) . Let V (i, x) = σi (x) be a Lyapunov invariant for G (N , E) , satisfying (12) and
σ∅ (x) ≤ 0,
∀x ∈ X∅
(23)
and either of the following two conditions:
(I) : σi (x) > 0,
(II) : σi (x) > 0,
∀x ∈ Xi ∩ Xi− , i ∈ N \ {∅}
(24)
n ko
∀x ∈ Xj ∩ T −1 (Xi− ) , i ∈ N \ {∅} , j ∈ I (i) , T ∈ T ij
(25)
where
T −1 (Xi− ) = {x ∈ Xi |T (x) ∩ Xi− 6= ∅}
Then, P satisfies the unreachability property w.r.t. the collection of sets Xi− , i ∈ N \ {∅} . In addition,
if for every simple cycle C ∈ G, we have:
(θ (C) − 1) kσ (C)k∞ + µ (C) > 0, and µ (C) > 0, and kσ (C)k∞ < ∞,
(26)
where
θ (C) =
k
θij
,
Q
(i,j,k)∈C
µ (C) = max µkij ,
(i,j,k)∈C
kσ (C)k∞ = max
(i,.,.)∈C
|σi (x)|
sup
(27)
x∈Xi \Xi−
then P terminates in at most Tu iterations where
Tu =
X
C∈G:θ(C)6=1
log ((θ (C) − 1) kσ (C)k∞ + µ (C)) − log µ (C)
+
log θ (C)
X
C∈G:θ(C)=1
kσ (C)k∞
.
µ (C)
Proof: The proof is presented in Appendix II.
For verification against an overflow violation specified by a diagonal matrix α > 0, Corollary 2 is
applied with X− = {x ∈ Rn | kα−1 xk∞ > 1}. Hence, (24) becomes σi (x) ≥ p (x) (kα−1 xkq − 1),
∀x ∈ Xi , i ∈ N \ {∅} , where p (x) > 0. User-specified assertions, as well as many other standard safety
specifications such as absence of division-by-zero can be verified using Corollary 2 (See Table I).
– Identification of Dead Code: Suppose that we wish to verify that a discrete location i ∈ N \ {∅} in
a graph model G (N , E) is unreachable. If a function satisfying the criteria of Corollary 2 with Xi− = Rn
can be found, then location i can never be reached. Condition (24) then becomes σi (x) ≥ 0, ∀x ∈ Rn .
17
TABLE I
A PPLICATION OF C OROLLARY 2 TO THE VERIFICATION OF VARIOUS SAFETY SPECIFICATIONS .
apply Corollary 2 with:
At location i:
assert x ∈ Xa
⇒
Xi− := {x ∈ Rn | x ∈ Rn \Xa }
At location i:
assert x ∈
/ Xa
⇒
Xi− := {x ∈ Rn | x ∈ Xa }
At location i:
⇒
Xi− := {x ∈ Rn | xo = 0}
At location i:
(expr.)/xo
√
2k x
o
⇒
Xi− := {x ∈ Rn | xo < 0}
At location i:
log (xo )
⇒
Xi− := {x ∈ Rn | xo ≤ 0}
At location i:
dead code
⇒
Xi− := Rn
Example 4: Consider the following program
void ComputeTurnRate (void)
L0 : {double x = {0}; double y = {∗PtrToY};
L1 :
while (1)
L2 : {
y = ∗PtrToY;
L3 :
x = (5 ∗ sin(y) + 1)/3;
L4 :
if x > −1 {
L5 :
L6 :
L7 :
L8 :
x = x + 1.0472;
TurnRate = y/x; }
else {
TurnRate = 100 ∗ y/3.1416 }}
Program 3.
Graph of an abstraction of Program 3
Note that x can be zero right after the assignment x = (5 sin(y) + 1)/3. However, at location L6, x
cannot be zero and division-by-zero will not occur. The graph model of an abstraction of Program 3
is shown next to the program and is defined by the following elements: T65 : x 7→ x + 1.0472, and
T41 : x 7→ [−4/3, 2] . The rest of the transition labels are identity. The only non-universal passport labels
are Π54 and Π84 as shown in the figure. Define
σ6 (x) = −x2 − 100x + 1, σ5 (x) = −(x + 1309/1250)2 − 100x − 2543/25
σ0 (x) = σ1 (x) = σ4 (x) = σ8 (x) = −x2 + 2x − 3.
It can be verified that V (x) = σi (x) is a (θ, 1)-Lyapunov invariant for Program 3 with variable rates:
18
θ65 = 1, and θij = 0 ∀ (i, j) 6= (6, 5). Since
−2 = sup σ0 (x) < inf σ6 (x) = 1
x∈X0
x∈X−
the state (6, x = 0) cannot be reached. Hence, a division by zero will never occur. We will show in the
next section how to find such functions in general.
IV. C OMPUTATION OF LYAPUNOV I NVARIANTS
It is well known that the main difficulty in using Lyapunov functions in system analysis is finding
them. Naturally, using Lyapunov invariants in software analysis inherits the same difficulties. However,
the recent advances in hardware and software technology, e.g., semi-definite programming [18], [53],
and linear programming software [19] present an opportunity for new approaches to software verification
based on numerical optimization.
A. Preliminaries
1) Convex Parameterization of Lyapunov Invariants: The chances of finding a Lyapunov invariant are
increased when (9) is only required on a subset of X\X∞ . For instance, for θ ≤ 1, it is tempting to
replace (9) with
V (x+ ) − θV (x) ≤ −µ, ∀x ∈ X\X∞ : V (x) < 1, x+ ∈ f (x)
(28)
In this formulation V is not required to satisfy (9) for those states which cannot be reached from X0 .
However, the set of all functions V : X 7→ R satisfying (28) is not convex and finding a solution for
(28) is typically much harder than (9). Such non-convex formulations are not considered in this paper.
The first step in the search for a function V : X 7→ R satisfying (9) is selecting a finite-dimensional
linear parameterization of a candidate function V :
n
X
V (x) = Vτ (x) =
τk Vk (x) ,
τ = (τk )nk=1 , τk ∈ R,
(29)
k=1
where Vk : X 7→ R are fixed basis functions. Next, for every τ = (τk )N
k=1 let
φ(τ ) =
max
x∈X\X∞ , x+ ∈f (x)
Vτ (x+ ) − θVτ (x),
(assuming for simplicity that the maximum does exist). Since φ (·) is a maximum of a family of linear
19
functions, φ (·) is a convex function. If minimizing φ (·) over the unit disk yields a negative minimum,
the optimal τ ∗ defines a valid Lyapunov invariant Vτ ∗ (x). Otherwise, no linear combination (29) yields
a valid solution for (9).
The success and efficiency of the proposed approach depend on computability of φ (·) and its subgradients. While φ (·) is convex, the same does not necessarily hold for Vτ (x+ ) − θVτ (x). In fact, if X\X∞
is non-convex, which is often the case even for very simple programs, computation of φ (·) becomes a
non-convex optimization problem even if Vτ (x+ ) − Vτ (x) is a nice (e.g. linear or concave and smooth)
function of x. To get around this hurdle, we propose using convex relaxation techniques which essentially
lead to computation of a convex upper bound for φ (τ ).
2) Convex Relaxation Techniques: Such techniques constitute a broad class of techniques for constructing finite-dimensional, convex approximations for difficult non-convex optimization problems. Some of
the results most relevant to the software verification framework presented in this paper can be found
in [31] for SDP relaxation of binary integer programs, [34] and [40] for SDP relaxation of quadratic
programs, [56] for S-Procedure in robustness analysis, and [43],[42] for sum-of-squares relaxation in
polynomial non-negativity verification. We provide a brief overview of the latter two techniques.
a) The S-Procedure : The S-Procedure is commonly used for construction of Lyapunov functions
for nonlinear dynamical systems. Let functions φi : X 7→ R, i ∈ Z (0, m) , and ψj : X 7→ R, j ∈ Z (1, n)
be given, and suppose that we are concerned with evaluating the following assertions:
(I): φ0 (x) > 0, ∀x ∈ {x ∈ X | φi (x) ≥ 0, ψj (x) = 0, i ∈ Z (1, m) , j ∈ Z (1, n)}
+
(II): ∃τi ∈ R , ∃µj ∈ R, such that φ0 (x) >
m
X
i=1
τi φi (x) +
n
X
µj ψj (x) .
(30)
(31)
j=1
The implication (II) → (I) is trivial. The process of replacing assertion (I) by its relaxed version (II) is
called the S-Procedure. Note that condition (II) is convex in decision variables τi and µj . The implication
(I) → (II) is generally not true and the S-Procedure is called lossless for special cases where (I) and (II)
are equivalent. A well-known such case is when m = 1, n = 0, and φ0 , φ1 are quadratic functionals. A
comprehensive discussion of the S-Procedure as well as available results on its losslessness can be found
20
in [21]. Other variations of S-Procedure with non-strict inequalities exist as well.
b) Sum-of-Squares (SOS) Relaxation : The SOS relaxation technique can be interpreted as the
generalized version of the S-Procedure and is concerned with verification of the following assertion:
fj (x) ≥ 0, ∀j ∈ J,
gk (x) 6= 0, ∀k ∈ K,
hl (x) = 0, ∀l ∈ L ⇒ −f0 (x) ≥ 0,
(32)
where fj , gk , hl are polynomial functions. It is easy to see that the problem is equivalent to verification
of emptiness of a semialgebraic set, a necessary and sufficient condition for which is given by the Positivstellensatz Theorem [8]. In practice, sufficient conditions in the form of nonnegativity of polynomials
are formulated. The non-negativity conditions are in turn relaxed to SOS conditions. Let Σ [y1 , . . . , ym ]
denote the set of SOS polynomials in m variables y1 , ..., ym , i.e. the set of polynomials that can be
represented as p =
t
P
p2i , pi ∈ Pm , where Pm is the polynomial ring of m variables with real coefficients.
i=1
Then, a sufficient condition for (32) is that there exist SOS polynomials τ0 , τi , τij ∈ Σ [x] and polynomials
ρl , such that
τ0 +
X
i
τi fi +
X
i,j
τij fi fj +
X
l
ρl hl + (
Y
gk )2 = 0
Matlab toolboxes SOSTOOLS [47], or YALMIP [30] automate the process of converting an SOS problem
to an SDP, which is subsequently solved by available software packages such as LMILAB [18], or SeDumi
[53]. Interested readers are referred to [42], [35], [43], [47] for more details.
B. Optimization of Lyapunov Invariants for Mixed-Integer Linear Models
Natural Lyapunov invariant candidates for MILMs are quadratic and affine functionals.
1) Quadratic Invariants: The linear parameterization of the space of quadratic functionals mapping
Rn to R is given by:
(
Vx2
=
n
V : R 7→ R | V (x) =
x
1
T
P
x
1
)
n+1
,P ∈S
,
(33)
where Sn is the set of n-by-n symmetric matrices. We have the following lemma.
Lemma 1: Consider a program P and its MILM S (F, H, X0 , n, nw , nv ) . The program admits a quadratic
(θ, µ)-Lyapunov invariant V ∈ Vx2 , if there exists a matrix Y ∈ Rne ×nH , ne = n + nw + nv + 1, a
w
diagonal matrix Dv ∈ Dnv , a positive semidefinite diagonal matrix Dxw ∈ Dn+n
, and a symmetric
+
21
matrix P ∈ Sn+1 , satisfying the following LMIs:
LT1 P L1 − θLT2 P L2 He (Y H) + LT3 Dxw L3 + LT4 Dv L4 − (λ + µ) LT5 L5
λ = Trace Dxw + Trace Dv
where



 In
F 
L1 = 
 , L2 = 
01×(ne −1)
L5


T

0(n+nw )×nv
0n×(ne −n) 
 In+nw


 , L4 =  Inv
 , L3 = 
0(nv +1)×(n+nw )
1
0
1×nv
T

T

 0(ne −1)×1 

 , L5 = 
1
Proof: The proof is presented in Appendix II
The following theorem summarizes our results for verification of absence of overflow and/or FTT for
MILMs. The result follows from Lemma 1 and Corollary 1 with q = 2, h = f, though the theorem is
presented without a detailed proof.
Theorem 2: Optimization-Based MILM Verification. Let α : 0 ≺ α In be a diagonal positive
definite matrix specifying the overflow limit. An overflow runtime error does not occur during any
execution of P if there exist matrices Yi ∈ Rne ×nH , diagonal matrices Div ∈ Dnv , positive semidefinite
w
diagonal matrices Dixw ∈ Dn+n
, and a symmetric matrix P ∈ Sn+1 satisfying the following LMIs:
+
[ x0 1 ]P [ x0 1 ]T ≤ 0,
∀x0 ∈ X0
LT1 P L1 − θLT2 P L2 He (Y1 H) + LT3 D1xw L3 + LT4 D1v L4 − (λ1 + µ) LT5 L5
LT1 ΛL1 − LT2 P L2 He (Y2 H) + LT3 D2xw L3 + LT4 D2v L4 − λ2 LT5 L5
(34)
(35)
(36)
where Λ = diag {α−2 , −1} , λi = Trace Dixw + Trace Div , and 0 Dixw , i = 1, 2. In addition, if
µ + θ > 1, then P terminates in a most Tu steps where Tu is given in (22).
2) Affine Invariants: Affine Lyapunov invariants can often establish strong properties, e.g., boundedness, for variables with simple uncoupled dynamics (e.g. counters) at a low computational cost. For
variables with more complicated dynamics, affine invariants may simply establish sign-invariance (e.g.,
xi ≥ 0) or more generally, upper or lower bounds on some linear combination of certain variables. As
we will observe in Section V, establishing these simple behavioral properties is important as they can
be recursively added to the model (e.g., the matrix H in a MILM, or the invariant sets Xi in a graph
22
model) to improve the chances of success in proving stronger properties via higher order invariants. The
linear parameterization of the subspace of linear functionals mapping Rn to R, is given by:
n
Vx1 = V : Rn 7→ R | V (x) = K T [x
o
1]T , K ∈ Rn+1 .
(37)
It is possible to search for the affine invariants via semidefinite programming or linear programming.
Proposition 5: SDP Characterization of Linear Invariants: There exists a (θ, µ)-Lyapunov invariant
V ∈ Vx1 for a program P ≡ S (F, H, X0 , n, nw , nv ) , if there exists a matrix Y ∈ Rne ×nH , a diagonal
(n+nw )×(n+nw )
matrix Dv ∈ Dnv , a positive semidefinite diagonal matrix Dxw ∈ D+
, and a matrix K ∈ Rn+1
satisfying the following LMI:
He(LT1 KL5 − θLT5 K T L2 ) ≺ He(Y H) + LT3 Dxw L3 + LT4 Dv L4 − (λ + µ) LT5 L5
(38)
where λ = Trace Dxw + Trace Dv and 0 Dxw .
Proposition 6: LP Characterization of Linear Invariants: There exists a (θ, µ)-Lyapunov invariant
for a program P ≡ S (F, H, X0 , n, nw , nv ) in the class Vx1 , if there exists a matrix Y ∈ R1×nH , and
nonnegative matrices Dv , Dv ∈ R1×nv , Dxw , Dxw ∈ R1×(n+nw ) , and a matrix K ∈ Rn+1 satisfying:
K T L1 − θK T L2 − Y H − (Dxw − Dxw )L3 − (Dv − Dv )L4 − (D1 + µ) L5 = 0
(39a)
D1 + Dv + Dv 1r + Dxw + Dxw 1n+nw ≤ 0
(39b)
Dv , Dv , Dxw , Dxw ≥ 0
(39c)
where D1 is either 0 or −1. As a special case of (39), a subset of all the affine invariants is characterized
by the set of all solutions of the following system of linear equations:
K T L1 − θK T L2 + L5 = 0
(40)
Remark 1: When the objective is to establish properties of the form Kx ≥ a for a fixed K, (e.g., when
establishing sign-invariance for certain variables), matrix K in (38)−(40) is fixed and thus one can make
θ a decision variable subject to θ ≥ 0. Exploiting this convexity is extremely helpful for successfully
establishing such properties.
23
The advantage of using semidefinite programming is that efficient SDP relaxations for treatment of
binary variables exists, though the computational cost is typically higher than the LP-based approach.
In contrast, linear programming relaxations of the binary constraints are more involved than the corresponding SDP relaxations. Two extreme remedies can be readily considered. The first is to relax the
binary constraints and treat the variables as continuous variables vi ∈ [−1, 1] . The second is to consider
each of the 2nv different possibilities (one for each vertex of {−1, 1}nv ) separately. This approach can be
useful if nv is small, and is otherwise impractical. More sophisticated schemes can be developed based
on hierarchical linear programming relaxations of binary integer programs [52].
C. Optimization of Lyapunov Invariants for Graph Models
A linear parameterization of the subspace of polynomial functionals with total degree less than or equal
to d is given by:
n+d
= V : R 7→ R | V (x) = K Z (x) , K ∈ R , N =
(41)
d
where Z (x) is a vector of length n+d
, consisting of all monomials of degree less than or equal to d in n
d
Vxd
n
T
N
variables x1 , ..., xn . A linear parametrization of Lyapunov invariants for graph models is defined according
d(i)
to (10), where for every i ∈ N , we have σi (·) ∈ Vx , where d (i) is a selected degree bound for σi (·) .
Depending on the dynamics of the model, the degree bounds d (i) , and the convex relaxation technique,
the corresponding optimization problem will become a linear, semidefinite, or SOS optimization problem.
1) Node-wise Polynomial Invariants: We present generic conditions for verification over graph models
using SOS programming. Although LMI conditions for verification of linear graph models using quadratic
invariants and the S-Procedure for relaxation of non-convex constraints can be formulated, we do not
present them here due to space limitations. Such formulations are presented in the extended report [49],
along with executable Matlab code in [57]. The following theorem follows from Corollary 2.
Theorem 3: Optimization-Based Graph Model Verification. Consider a program P, and its graph
d(i)
model G (N , E) . Let V : Ωn 7→ R, be given by (10), where σi (·) ∈ Vx . Then, the functions σi (·) ,
24
i ∈ N define a Lyapunov invariant for P, if for all (i, j, k) ∈ E we have:
k
−σj (Tjik (x, w)) + θji
σi (x) − µkji ∈ Σ [x, w] subject to (x, w) ∈
k
Xi ∩ Πkji × [−1, 1]nw ∩ Sji
(42)
Furthermore, P satisfies the unreachability property w.r.t. the collection of sets Xi− , i ∈ N \ {∅} , if there
exist εi ∈ (0, ∞) , i ∈ N \ {∅} , such that
−σ∅ (x) ∈ Σ [x] subject to x ∈ X∅
(43)
σi (x) − εi ∈ Σ [x] subject to x ∈ Xi ∩ Xi− , i ∈ N \ {∅}
(44)
As discussed in Section IV-A2b, the SOS relaxation techniques can be applied for formulating the search
problem for functions σi satisfying (42)–(44) as a convex optimization problem. For instance, if
k
Xi ∩ Πkji × [−1, 1]nw ∩ Sji
= {(x, w) | fp (x, w) ≥ 0, hl (x, w) = 0} ,
then, (42) can be formulated as an SOS optimization problem of the following form:
k
−σj (Tjik (x, w)) + θji
σi (x) − µkji −
X
p
τ p fp −
X
τpq fp fq −
X
p,q
ρl hl ∈ Σ [x, w] , s.t. τp , τpq ∈ Σ [x, w] .
l
Software packages such as SOSTOOLS [47] or YALMIP [30] can then be used for formulating the SOS
optimization problems as semidefinite programs.
V. C ASE S TUDY
In this section we apply the framework to the analysis of Program 4 displayed below.
/ ∗ EuclideanDivision.c ∗ /
F0 :
int IntegerDivision ( int dd, int dr )
F1 :
{int q = {0}; int r = {dd};
F2 :
while (r >= dr) {
F3 :
q = q + 1;
F4 :
r = r − dr;
Fo
n:
return r; }
L0 :
int main ( int X, int Y ) {
L1 :
int rem = {0};
L2 :
while (Y > 0) {
L3 :
rem = IntegerDivision (X , Y);
L4 :
X = Y;
L5 :
Y = rem;
Lo
n:
return X; }}
Program 4: Euclidean Division and its Graph Model
25
Program 4 takes two positive integers X ∈ [1, M] and Y ∈ [1, M] as the input and returns their greatest
common divisor by implementing the Euclidean Division algorithm. Note that the M AIN function in
Program 4 uses the I NTEGER D IVISION program (Program 1).
A. Global Analysis
A global model can be constructed by embedding the dynamics of the I NTEGER D IVISION program
within the dynamics of M AIN. A labeled graph model is shown alongside the text of the program. This
model has a state space X = N × [−M, M]7 , where N is the set of nodes as shown in the graph, and the
global state x = [X, Y, rem, dd, dr, q, r] is an element of the hypercube [−M, M]7 . A reduced graph
model can be obtained by combining the effects of consecutive transitions and relabeling the reduced
graph model accordingly. While analysis of the full graph model is possible, working with a reduced
model is computationally advantageous. Furthermore, mapping the properties of the reduced graph model
to the original model is algorithmic. Interested readers may consult [51] for further elaboration on this
topic. For the graph model of Program 4, a reduced model can be obtained by first eliminating nodes
Fno, L4 , L5 , L3 , F0 , F1 , F3 , F4 , and L1 , (Figure 1 Left) and composing the transition and passport labels.
Node L2 can be eliminated as well to obtain a further reduced model with only three nodes: F2 , L0 , Lno.
(Figure 1 Right). This is the model that we will analyze. The passport and transition labels associated
with the reduced model are as follows:
Fig. 1.
Two reduced models of the graph model of Program 4.
26
1
2
T F2F2 : x 7→ [X, Y, rem, dd, dr, q + 1, r − dr]
T F2F2 : x 7→ [Y, r, r, Y, r, 0, Y]
T L0F2 : x 7→ [X, Y, 0, X, Y, 0, X]
T F2Lo
n : x 7→ [Y, r, r, dd, dr, q, r]
Π2F2F2 : {x | 1 ≤ r ≤ dr − 1}
Π1F2F2 : {x | r ≥ dr}
ΠF2Lo
n : {x | r ≤ dr − 1, r ≤ 0}
Finally, the invariant sets that can be readily included in the graph model (cf. Section II-B2a) are:
XL0 = {x | M ≥ X, M ≥ Y, X ≥ 1, Y ≥ 1} , XF2 = {x | dd = X, dr = Y} , XLo
n = {x | Y ≤ 0} .
We are interested in generating certificates of termination and absence of overflow. First, by recursively
searching for linear invariants we are able to establish simple lower bounds on all variables in just two
rounds (the properties established in the each round are added to the model and the next round of search
begins). For instance, the property X ≥ 1 is established only after Y ≥ 1 is established. These results,
which were obtained by applying the first part of Theorem 3 (equations (42)-(43) only) with linear
functionals are summarized in Table II.
TABLE II
Property
Proven in Round
σF2 (x) =
1
θF2F2
, µ1F2F2
2
θF2F2
, µ2F2F2
q≥0
Y≥1
dr ≥ 1
rem ≥ 0
dd ≥ 1
X≥1
r≥0
I
I
I
I
II
II
II
−q
1−Y
1 − dr
−rem
1 − dd
1−X
−r
(1, 1)
(1, 0)
(1, 0)
(1, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
We then add these properties to the node invariant sets to obtain stronger invariants that certify FTT and
boundedness of all variables in [−M, M]. By applying Theorem 3 and SOS programming using YALMIP
[30], the following invariants are found1 (after post-processing, rounding the coefficients, and reverifying):
σ1F2 (x) = 0.4 (Y − M) (2 + M − r)
σ2F2 (x) = (q × Y + r)2 − M2
σ3F2 (x) = (q + r)2 − M2
σ4F2 (x) = 0.1 (Y − M + 5Y × M + Y2 − 6M2 )
σ5F2 (x) = Y + r − 2M + Y × M − M2
σ6F2 (x) = r × Y + Y − M2 − M
The properties proven by these invariants are summarized in the Table III. The specifications that the
1
Different choices of polynomial degrees for the Lyapunov invariant function and the multipliers, as well as different choices for θ, µ
and different rounding schemes lead to different invariants. Note that rounding is not essential.
27
program terminates and that x ∈ [−M, M]7 for all initial conditions X, Y∈ [1, M] , could not be established
in one shot, at least when trying polynomials of degree d ≤ 4. For instance, σ1F2 certifies boundedness
of all the variables except q, while σ2F2 and σ3F2 which certify boundedness of all variables including q
do not certify FTT. Furthermore, boundedness of some of the variables is established in round II, relying
on boundedness properties proven in round I. Given σ (x) ≤ 0 (which is found in round I), second
round verification can be done by searching for a strictly positive polynomial p (x) and a nonnegative
polynomial q (x) ≥ 0 satisfying:
2
q (x) σ (x) − p (x) ( T x i − M2 ) ≥ 0,
1
2
T ∈ {T F2F2 , T F2F2 }
(45)
where the inequality (45) is further subject to boundedness properties established in round I, as well as
the usual passport conditions and basic invariant set conditions.
TABLE III
Invariant σF2 (x) =
σ1F2 (x)
σ2F2 (x) , σ3F2 (x)
σ4F2 (x)
σ5F2 (x) , σ6F2 (x)
1
θF2F2
, µ1F2F2
(1, 0)
(1, 0)
(1, 0)
(1, 1)
2
θF2F2
, µ2F2F2
(1, 0.8)
(0, 0)
(1, 0.7)
(1, 1)
Y, X, r, dr, rem, dd
q, Y, dr, rem
Y, X, r, dr, rem, dd
Y, dr, rem
Round I: x2i ≤ M2 for xi =
Round II: x2i ≤ M2 for xi =
Certificate for FTT
X, r, dd
NO
NO
X, r, dd
NO
YES, Tu = 2M2
In conclusion, σ2F2 (x) or σ3F2 (x) in conjunction with σ5F2 (x) or σ6F2 (x) prove finite-time termination
of the algorithm, as well as boundedness of all variables within [−M, M] for all initial conditions X, Y ∈
[1, M] , for any M ≥ 1. The provable bound on the number of iterations certified by σ5F2 (x) and σ6F2 (x)
is Tu = 2M2 (Corollary 2). If we settle for more conservative specifications, e.g., x ∈ [−kM, kM]7 for
all initial conditions X, Y ∈ [1, M] and sufficiently large k, then it is possible to prove the properties in
one shot. We show this in the next section.
B. MIL-GH Model
For comparison, we also constructed the MIL-GH model associated with the reduced graph in Figure 1.
The corresponding matrices are omitted for brevity, but details of the model along with executable Matlab
28
verification codes can be found in [57]. The verification theorem used in this analysis is an extension of
Theorem 2 to analysis of MIL-GHM for specific numerical values of M, though it is certainly possible to
perform this modeling and analysis exercise for parametric bounded values of M. The analysis using the
MIL-GHM is in general more conservative than SOS optimization over the graph model presented earlier.
This can be attributed to the type of relaxations proposed (similar to those used in Lemma 1) for analysis
of MILMs and MIL-GHMs. The benefits are simplified analysis at a typically much less computational
cost. The certificate obtained in this way is a single quadratic function (for each numerical value of
M), establishing a bound γ (M) satisfying γ (M) ≥ X2 + Y2 + rem2 + dd2 + dr2 + q2 + r2
1/2
. Table
IV summarizes the results of this analysis which were performed using both Sedumi 1 3 and LMILAB
solvers.
TABLE IV
102
103
104
105
106
Solver: LMILAB [18]: γ (M)
5.99M
6.34M
6.43M
6.49M
7.05M
Solver: SEDUMI [53]: γ (M)
6.00M
6.34M
6.44M
6.49M
NAN
1, 10−3
1, 10−3
1, 10−3
1, 10−3
1, 10−3
1, 10−3
1, 10−3
1, 10−3
M
−3
1
θF2F2
, µ1F2F2
1, 10
2
θF2F2
, µ2F2F2
1, 10−3
Upperbound on iterations
Tu = 2e4
Tu = 8e4
Tu = 8e5
Tu = 1.5e7
Tu = 3e9
C. Modular Analysis
The preceding results were obtained by analysis of a global model which was constructed by embedding
the internal dynamics of the program’s functions within the global dynamics of the Main function. In
contrast, the idea in modular analysis is to model software as the interconnection of the program’s
”building blocks” or ”modules”, i.e., functions that interact via a set of global variables. The dynamics
of the functions are then abstracted via Input/Output behavioral models, typically constituting equality
and/or inequality constraints relating the input and output variables. In our framework, the invariant sets
of the terminal nodes of the modules (e.g., the set Xno associated with the terminal node Fno in Program
4) provide such I/O model. Thus, richer characterization of the invariant sets of the terminal nodes of the
modules are desirable. Correctness of each sub-module must be established separately, while correctness
29
of the entire program will be established by verifying the unreachability and termination properties w.r.t.
the global variables, as well as verifying that a terminal global state will be reached in finite-time.
This way, the program variables that are private to each function are abstracted away from the global
dynamics. This approach has the potential to greatly simplify the analysis and improve the scalability
of the proposed framework as analysis of large size computer programs is undertaken. In this section,
we apply the framework to modular analysis of Program 4. Detailed analysis of the advantages in terms
of improving scalability, and the limitations in terms of conservatism the analysis is an important and
interesting direction of future research.
The first step is to establish correctness of the I NTEGER D IVISION module, for which we obtain
σ7F2 (dd, dr, q, r) = (q + r)2 − M2
The function σ7F2 is a (1, 0)-invariant proving boundedness of the state variables of I NTEGER D IVISION.
Subject to boundedness, we obtain the function
σ8F2 (dd, dr, q, r) = 2r − 11q − 6Z
which is a (1, 1)-invariant proving termination of I NTEGER D IVISION . The invariant set of node Fno can
thus be characterized by
Xno = (dd, dr, q, r) ∈ [0, M]4 | r ≤ dr − 1
The next step is construction of a global model. Given Xno, the assignment at L3:
L3 : rem = IntegerDivision (X , Y)
can be abstracted by
rem = W, s.t. W ∈ [0, M] , W ≤ Y − 1,
allowing for construction of a global model with variables X, Y, and rem, and an external state-dependent
input W ∈ [0, M] , satisfying W ≤ Y − 1. Finally, the last step is analysis of the global model. We obtain
the function σ9L2 (X, Y, rem) = Y ×M−M2 , which is (1, 1)-invariant proving both FTT and boundedness
of all variables within [M, M] .
30
VI. C ONCLUDING R EMARKS
We took a systems-theoretic approach to software analysis, and presented a framework based on convex
optimization of Lyapunov invariants for verification of a range of important specifications for software
systems, including finite-time termination and absence of run-time errors such as overflow, out-of-bounds
array indexing, division-by-zero, and user-defined program assertions. The verification problem is reduced
to solving a numerical optimization problem, which when feasible, results in a certificate for the desired
specification. The novelty of the framework, and consequently, the main contributions of this paper are in
the systematic transfer of Lyapunov functions and the associated computational techniques from control
systems to software analysis. The presented work can be extended in several directions. These include
understanding the limitations of modular analysis of programs, perturbation analysis of the Lyapunov
certificates to quantify robustness with respect to round-off errors, extension to systems with software in
closed loop with hardware, and adaptation of the framework to specific classes of software.
A PPENDIX I
Semialgebraic Set-Valued Abstractions of Commonly-Used Nonlinearities:
– Trigonometric Functions:
Abstraction of trigonometric functions can be obtained by approximation of the Taylor series expansion followed by representation of the absolute error by a static bounded uncertainty. For instance, an
abstraction of the sin (·) function can be constructed as follows:
Abstraction of sin (x)
x ∈ [− π2 , π2 ]
x ∈ [−π, π]
sin (x) ∈ {x + aw | w ∈ [−1, 1]}
a = 0.571
a = 3.142
sin (x) ∈ {x − 16 x3 + aw | w ∈ [−1, 1]}
a = 0.076
a = 2.027
Abstraction of cos (·) is similar. It is also possible to obtain piecewise linear abstractions by first
approximating the function by a piece-wise linear (PWL) function and then representing the absolute
error by a bounded uncertainty. Section II-B (Proposition 1) establishes universality of representation of
generic PWL functions via binary and continuous variables and an algorithmic construction can be found
in [49]. For instance, if x ∈ [0, π/2] then a piecewise linear approximation with absolute error less than
31
0.06 can be constructed in the following way:
S = (x, v, w) |x = 0.2 [(1 + v) (1 + w2 ) + (1 − v) (3 + w2 )] ,(w, v) ∈ [−1, 1]2 × {−1, 1}
sin (x) ∈ {T xE | xE ∈ S} , T : xE 7→ 0.45 (1 + v) x + (1 − v) (0.2x + 0.2) + 0.06w1
(46a)
(46b)
– The Sign Function (sgn) and the Absolute Value Function (abs):
The sign function (sgn(x) = 1I[0,∞) (x) − 1I(−∞,0) (x)) may appear explicitly or as an interpretation of
if-then-else blocks in computer programs (see [49] for more details). A particular abstraction of sgn (·)
is as follows: sgn(x) ∈ {v | xv ≥ 0, v ∈ {−1, 1}}. Note that sgn (0) is equal to 1, while the abstraction
is multi-valued at zero: sgn (0) ∈ {−1, 1} . The absolute value function can be represented (precisely)
over [−1, 1] in the following way:
abs (x) = {xv | x = 0.5 (v + w) , (w, v) ∈ [−1, 1] × {−1, 1}}
More on the systematic construction of function abstractions including those related to floating-point,
fixed-point, or modulo arithmetic can be found in the report [49].
A PPENDIX II
of Proposition 4: Note that (13)−(15) imply that V is negative-definite along the trajectories of S,
except possibly for V (x (0)) which can be zero when η = 0. Let X be any solution of S. Since V is
uniformly bounded on X, we have:
− kV k∞ ≤ V (x (t)) < 0, ∀x (t) ∈ X , t > 1.
Now, assume that there exists a sequence X ≡ (x(0), x(1), . . . , x(t), . . . ) of elements from X satisfying
(1), but not reaching a terminal state in finite time. That is, x (t) ∈
/ X∞ , ∀t ∈ Z+ . Then, it can be
verified that if t > Tu , where Tu is given by (16), we must have: V (x (t)) < − kV k∞ , which contradicts
boundedness of V.
of Theorem 1: Assume that S has a solution X =(x (0) , ..., x (t− ) , ...) , where x (0) ∈ X0 and
x (t− ) ∈ X− . Let
γh =
inf
x∈h−1 (X− )
V (x)
32
First, we claim that γh ≤ max {V (x (t− )), V (x (t− − 1))} . If h = I, we have x (t− ) ∈ h−1 (X− ) and
γh ≤ V (x (t− )). If h = f, we have x (t− − 1) ∈ h−1 (X− ) and γh ≤ V (x (t− − 1)), hence the claim.
Now, consider the θ = 1 case. Since V is monotonically decreasing along solutions of S, we must have:
γh =
inf
V (x) ≤ max {V (x (t− )), V (x (t− − 1))} ≤ V (x (0)) ≤ sup V (x)
x∈h−1 (X− )
(47)
x∈X0
which contradicts (17). Note that if µ > 0 and h = I, then (47) holds as a strict inequality and we can
replace (17) with its non-strict version. Next, consider case (I) , for which, V need not be monotonic
along the trajectories. Partition X0 into two subsets X 0 and X 0 such that X0 = X 0 ∪ X 0 and
V (x) ≤ 0 ∀x ∈ X 0 ,
and
V (x) > 0 ∀x ∈ X 0
Now, assume that S has a solution X = (x (0) , ..., x (t− ) , ...) , where x (0) ∈ X 0 and x (t− ) ∈ X− . Since
V (x (0)) > 0 and θ < 1, we have V (x (t)) < V (x (0)) ,
γh =
inf
∀t > 0. Therefore,
V (x) ≤ max {V (x (t− )), V (x (t− − 1))} ≤ V (x (0)) ≤ sup V (x)
x∈h−1 (X− )
x∈X0
which contradicts (17). Next, assume that S has a solution X = (x (0) , ..., x (t− ) , ...) , where x (0) ∈ X 0
and x (t− ) ∈ X− . In this case, regardless of the value of θ, we must have V (x (t)) ≤ 0, ∀t, implying that
γh ≤ 0, and hence, contradicting (18). Note that if h = I and either µ > 0, or θ > 0, then (18) can be
replaced with its non-strict version. Finally, consider case (II). Due to (19), V is strictly monotonically
decreasing along the solutions of S. The rest of the argument is similar to the θ = 1 case.
of Corollary 1: It follows from (21) and the definition of X− that:
n
o
V (x) ≥ sup α−1 h (x)q − 1 ≥ sup α−1 h (x)∞ − 1 > 0,
∀x ∈ X.
(48)
It then follows from (48) and (20) that:
inf
V (x) > 0 ≥ sup V (x)
x∈h−1 (X− )
x∈X0
Hence, the first statement of the Corollary follows from Theorem 1. The upperbound on the number of
33
iterations follows from Proposition 4 and the fact that supx∈X\{X− ∪X∞ } |V (x)| ≤ 1.
of Corollary 2: The unreachability property follows directly from Theorem 1. The finite time
termination property holds because it follows from (12), (23) and (26) along with Proposition 4, that
the maximum number of iterations around every simple cycle C is finite. The upperbound on the number
of iterations is the sum of the maximum number of iterations over every simple cycle.
of Lemma 1: Define xe = (x, w, v, 1)T , where x ∈ [−1, 1]n , w ∈ [−1, 1]nw , v ∈ {−1, 1}nv . Recall
that (x, 1)T = L2 xe , and that for all xe satisfying Hxe = 0, there holds: (x+ , 1) = (F xe , 1) = L1 xe . It
follows from Proposition 3 that (9) holds if:
xTe LT1 P L1 xe − θxTe LT2 P L2 xe ≤ −µ, s.t. Hxe = 0, L3 xe ∈ [−1, 1]n+nw , L4 xe ∈ {−1, 1}nv .
(49)
Recall from the S-Procedure ((30) and (31)) that the assertion σ (y) ≤ 0, ∀y ∈ [−1, 1]n holds if there exists
nonnegative constants τi ≥ 0, i = 1, ..., n, such that σ (y) ≤
P
τi (yi2 − 1) = y T τ y − Trace (τ ) , where
τ = diag (τi ) 0. Similarly, the assertion σ (y) ≤ 0, ∀y ∈ {−1, 1}n holds if there exists a diagonal
matrix µ such that σ (y) ≤
P
µi (yi2 − 1) = y T µy − Trace (µ) . Applying these relaxations to (49), we
obtain sufficient conditions for (49) to hold:
xTe LT1 P L1 xe −θxTe LT2 P L2 xe ≤ xTe Y H + H T Y T xe +xTe LT3 Dxw L3 xe +xTe LT4 Dv L4 xe −µ−Trace(Dxw +Dv )
Together with 0 Dxw , the above condition is equivalent to the LMIs in Lemma 1.
34
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