Coalgebraic Weak Bisimulation from Recursive Equations over

Transcription

Coalgebraic Weak Bisimulation from
Recursive Equations over Monads
Sergey Goncharov, Dirk Pattinson
Oberseminar, 10. Dezember 2014
FAU Erlangen-Nürnberg, Informatik 8
A Ridiculously Simple Example
Interaction of a computer scientist (CS) and a coffee machine (CM):
CS = coin.coffee.pub.CS
CM = coin.(coffee.CM + tea.CM)
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FAU Erlangen-Nürnberg, Informatik
2
Interaction of a computer scientist (CS) and a coffee machine (CM):
CS = coin.coffee.pub.CS
CM = coin.(coffee.CM + tea.CM)
System S = (CS | CM)\coin\coffee satisfies the equation:
S = τ.τ.pub.S.
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System S = (CS | CM)\coin\coffee satisfies the equation:
S = τ.τ.pub.S.
We could formalize that the computer scientist is productive as
νγ. hτihτihpubiγ.
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We could formalize that the computer scientist is productive as
νγ. hτihτihpubiγ.
Or, better: νγ. hhpubiiγ where
hhaiiφ = hhiihhaiihhiiφ
hhiiφ = µγ. (φ ∨ hτi]γ)
[[a]]φ = [[ ]][[a]][[ ]]φ
[[ ]]φ = νγ. (φ ∧ [τ]γ).
However, this trick does not work with probabilistic systems.
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Strong Bisimulation, Coalgebraically
a
Given an LTS (X, −
→), R is a strong bisimulation (equivalence) if
x
R
y
x
and
a
y
a
x0
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R
x0
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6
a
x
R
y
a
x
a
x0
|
y
a
y0
R
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and
R
a
x0
|
R
y0
7
a
x
R
y
a
x
a
x0
and
y
a
y0
R
R
a
x0
R
y0
Coalgebraic approach:
• replace LTS with a coalgebra, i.e. a map f : X → FX with
FX = P(X × A)
• identify equivalences E on X with projections π : X → (X × X)/E
• then R is recovered a the kernel of some F-coalgebra morphism
g : Z → FZ. Thus we obtain kernel bisimulation
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Kernel Bisimulation
Kernel bisimulation is a robust device:
• works uniformly for a diversity of systems: automata, probabilistic,
stohastic, graded, etc.
• has a proof-theoretic characterization — coinduction
• has a modal characterization — coalgebraic modal logic
• has a relational characterization — Aczel-Mendler(-style) bisimulation
(under weak pullback preservation).
• yeilds final semantics, as the unique morphism to final coalgebra νF.
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Kernel Bisimulation
Kernel bisimulation is a robust device:
• works uniformly for a diversity of systems: automata, probabilistic,
stohastic, graded, etc.
• has a proof-theoretic characterization — coinduction
• has a modal characterization — coalgebraic modal logic
• has a relational characterization — Aczel-Mendler(-style) bisimulation
(under weak pullback preservation).
• yeilds final semantics, as the unique morphism to final coalgebra νF.
But, in concurrency strong bisimulation is only an auxiliary notion.
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Weak Bisimulation: Why?
Rule for paralell composition:
a
P−
→ P0
a
Q−
→ Q0
τ
P | Q −−→ P 0 | Q 0
Hence R is a weak bisimulation if
x
y
R
x
∗
τ
y
τ
x
τ∗
x0
•
a
R
R
y0
x
τ∗
y0
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R
y
τ∗
R
y0
|
y0
•
τ
x0
|
a
a
•
R
y
•
a
x0
R
τ∗
τ∗
R
x0
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Weak Bisimulation: Why?
Internal action
Rule for paralell composition:
a
P−
→ P0
a
Q−
→ Q0
τ
P | Q −−→ P 0 | Q 0
Hence R is a weak bisimulation if
x
y
R
x
∗
τ
y
τ
x
τ∗
x0
•
a
R
R
y0
x
τ∗
y0
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R
y
τ∗
R
y0
|
y0
•
τ
x0
|
a
a
•
R
y
•
a
x0
R
τ∗
τ∗
R
x0
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Delay Bisimulation
Alternatively, R is a delay bisimulation if
x
y
R
x
∗
τ
R
y0
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y
y0
|
y
•
τ∗
R
R
τ∗
R
x
|
x
R
a
x0
y
x0
•
a
τ∗
R
a
a
x0
R
y0
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Weak Transitions and Partial Observations
a
a
Milner’s weak transition construction: From (X, −
→) switch to (X, =
⇒)
where
a
=
⇒
τ
=
⇒
is
is
τ ∗
a
τ ∗
(a 6= τ)
→
− −
→→
−
τ ∗
→
−
a
a
Weak bisimulation for (X, −
→) is strong bisimulation for (X, =
⇒).
a
τ
The observable =
⇒ is constructed out of partial observables →
− and
a
−
→. This can be understood a coalgebra B → 2 × BA, equivelently
o:B→2
(acceptance)
−/− : B × A → B
(evolution)
We call B an observation pattern.
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Observation Patterns
Any observation pattern is a subcoalgebra of the final coalgebra
∼ P(A∗)
νγ. (2 × γA) =
E.g. for weak bisimulation B = {0/ , τ∗, τ∗aτ∗},
o(s) = 1 ⇐⇒ s = τ∗
0/ /a = 0/
τ∗aτ∗/τ = τ∗aτ∗
/
τ∗ / a = 0
/ (a 6= b)
τ∗aτ∗/b = 0
τ∗/τ = τ∗
τ∗aτ∗/a = τ∗
Analogously,
• B = {0/ , τ∗, τ∗a} for delay bisimulation
• B = {0/ , {}, {a}} for strong bisimulation
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Probabilistic Weak Bisimulation: The Problem
Milner’s construction fails at probabilistic systems.
τ(0.5)
a(0.5)
b(0.5)
s2
s1
a(0.5)
b(1.0)
s4
s3
s5
a
• The probability of weak transition s1 =
⇒ s2 is not
0.5 + 0.52 + · · · = 1.
• R = ∆ ∪ {hs2, s4i, hs3, s5i} is a weak bisimulation.
Moral: point-to-set transitions are irreducible to point-to-point ones.
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Probabilistic Weak Bisimulation: The Solution
Let P∗(x, Λ, S) be the total probability of the move from x ∈ X to
S ⊆ X via Λ ⊆ A∗.
Definition [BaierHermanns97]. R ⊆ X × X is a probabilistic
bisimulation equivalence if xRy implies
P∗(x, τ∗aτ∗, S) = P∗(y, τ∗aτ∗, S)
(a 6= τ)
P∗(x, τ∗, S) = P∗(y, τ∗, S)
for any R-equivalence class S.
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How to Calculate Total Probabilities?
Approach from [BaierHermanns97]:
• Consider a set-algebra of path-cones generated by finite paths.
• Complete it to a σ-algebra.
• Extend probabilistic measure using a Caratheodory-style theorem.
Our approach: Solve recursive system
P∗(x, Λ, S) = 1
∗
P (x, Λ, S) =
X
(x ∈ S and ∈ Λ)
P(x, a, y) · P∗(y, Λ/a, S)
a,y
Equivalently,
X
1 if x ∈ S and ∈ Λ
P (x, Λ, S) =
t
P(x, a, y) · P∗(y, Λ/a, S)
0 otherwise
a,y
∗
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From Total Probabilities to Total Multiplicities
a
a
For graded systems, P −
→ Q + Q is not the same as P −
→ Q.
a(n)
Hence, one uses weighted transitions P −−→ Q.
Following the ideas from [CorradiniNicolaEtAl99] “total multiplicities”
M∗(x, Λ, S) must mesure the degree of nondeterminism.
Surprisingly, the same formula is suitable for M∗:
X
if x ∈ S and ∈ Λ
t
M(x, a, y) · M∗(y, Λ/a, S)
otherwise
a,y
1
M ( x, Λ , S ) =
0
∗
τ
E.g.
s1
a
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s2
|
M∗(s1, τ∗aτ∗, {s1, s2}) = 1
M∗(s2, τ∗bτ∗, {s2, s3}) = ∞
b
s3
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The General Case
LTS, graded and probabilistic systems are coalgebras of type
f : X → T(X × A)
where T is a completely ordered monad
(≈ T is a monad + TX is a ω-cpo with ⊥).
Main Definition:
fhB(x)(b) =
η(h(x))
⊥
if o(b)
otherwise
⊕
do hy, ai ← f (x); fhB(y)(b/a)
Then R ⊆ X × X is a B-⊕-bisimulation equivalence on f if R ⊆ ker fπB
where π is the projection X → R/X.
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The General Case
continuous operation T2 → T
LTS, graded and probabilistic systems are coalgebras of type
observation pattern
f : X → T(X × A)
where T is a completely ordered monad
(≈ T is a monad + TX is a ω-cpo with ⊥).
Main Definition:
B
fh (x)(b) =
η(h(x))
⊥
if o(b)
otherwise
⊕
do hy, ai ← f (x); fhB(y)(b/a)
Then R ⊆ X × X is a B-⊕-bisimulation equivalence on f if R ⊆ ker fπB
where π is the projection X → R/X.
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Probability and Nondeterminism
The mixture of probability and nondeterminsm is modelled by
(simple) Segala systems.
The corresponding monad is a monad of convex sets of valuations
[Jacobs08,Brengos13].
Our definition unravels as follows (and agrees with [Segala94]):
τ
x=
⇒ δx
a
x=
⇒ζ
iff
∃ξ ∈ f (x). ζ ∈

X

y∈ X


b
a
ξ(y, a) · θy + ξ(y, τ) · θy ∀y. y =
⇒ θby

τ
where =
⇒ ∈ X × B × [0, ∞)X.
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Killing τ with a Dagger?
Recall that a complete Elgot monad is a monad with an operator ---†:
f ∈ Hom(A, T(B + A))
f † ∈ Hom(A, TB))
7→
(satisfying suitable laws)
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7→
Roughly, this allows as to solve excursive equations like
f (x) = do z ← toss; case z of inl ? 7→ p; inr ? 7→ f (q),
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7→
equivalently,
f (x) = p ⊕ f (q),
with algebraic ⊕.
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7→
equivalently,
f (x) = p ⊕ f (q),
with algebraic ⊕. However, t is generally not algebraic, e.g. for
probabilistic systems:
do x ← (p t q); r 6= (do x ← p; r ) t (do x ← q; r ).
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Algebraic Operations
An operation ⊕ : T2 → T is algebraic if it distributes over sequential
composition
do x ← p ⊕ q; r = (do x ← p; r ) ⊕ (do x ← q; r ).
Examples: set union is algebraic for LTS and Segala systems.
Nonexample: join for probabilistic and weighted systems.
algebraicity ⇒ continuity
Theorem: If ⊕ is algebraic then E is a B-bisimulation equivalence for
f iff E is a strong bisimulation for fidB.
Bottom line: Weak transition construction works for LTS and Segala
system, but not for probabilistic and graded systems!
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A (Somewhat) Alternative Approach
The approach from [Tomasz Brengos, Marino Miculan, Marco Peressotti,
Behavioural equivalences for coalgebras with unobservable moves]:
∼ X × Aτ )
• Start from f : X → T(FX + X) (e.g. with FX + X =
where F distributes over T via some δ : FT → TF;
∼ 1);
• Equip T(FX + X) with a monad structure (assuming that T 0/ =
• Solve recursive equation g = h t g · f in the Kleisli category of
M = T(F + Id) where h : X → Y is the weak bisimulation in question
(assuming that T is ωCpo-enriched and has t).
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∼ A+1
Aτ =
∼ X × Aτ )
∼ 1);
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∼ A+1
Aτ =
∼ X × Aτ )
∼ 1);
Pro: Slick main definition, higher generality.
Con: Geared to the idea of saturation (i.e. aggregation of transitions
into τ∗aτ∗-sequences) — not observation; e.g. not suitable for delay
bisimulation.
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Further Generalization
• Let f : X → TFX be a coalgebra (e.g. FX = X × A);
• Let G a functor distributing over T (e.g. GX = X × B);
• Let, additionally, ∂ : GF → TG and σ : G → T be two natural
transformations (e.g. ∂(x, a, b) = η(x, b/a), σ(x, b) = η(x, o(b))).
In particular, G lifts to CT and ∂, σ lift to ∂ : GF → G and σ : G → Id.
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• Let f : X → TFX be a coalgebra (e.g. FX = X × A);
• Let G a functor distributing over T (e.g. GX = X × B);
• Let, additionally, ∂ : GF → TG and σ : G → T be two natural
transformations (e.g. ∂(x, a, b) = η(x, b/a), σ(x, b) = η(x, o(b))).
id ×δ
hσ,Gf i
id ×∂∗
g : GX −−−→ TX × GTFX −−→ TX × TGFX −−−→ TX × TGX
For any h : X → Y let f∂h,σ : GX → TX be the least solution of
f∂h,σ = Th ⊕ (f∂h,σ)∗ · g
and R ⊆ X × X is a ∂-σ-⊕-bisimulation equivalence if R ⊆ ker f∂π,σ
with canonical π : X → R/X.
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Let f : X → TFX, ∂ : GF → TG and σ : G → T.
id ×δ
hσ,Gf i
id ×∂∗
g : GX −−−→ TX × GTFX −−→ TX × TGFX −−−→ TX × TGX
For any h : X → Y let f∂h,σ : GX → TX be the least solution of
f∂h,σ = Th ⊕ (f∂h,σ)∗ · g
and R ⊆ X × X is a ∂-σ-⊕-bisimulation equivalence if R ⊆ ker f∂π,σ
with canonical π : X → R/X.
• This covers the previous treatment with FX = X × A, GX = X × B;
• It also covers trace equivalence with FX = X × A + 1, B = A∗,
∂(inl(x, a), s) = η(x, s/a), σ(inr ?, ) = ⊥, σ(inr ?, s) = η(x) (s 6= ).
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Weak Bisimulation from Strong Bisumulation
Unless ⊕ is algebraic w/b does not reduce to s/b of the same type.
But it can (always?) reduce to a s/b of a different type!
Theorem: Assume
b an injective monad morphism,
• κ : T → T,
b , a lifting of ⊕ : T2 → T along κ.
• an algebraic operation ⊕
Then E is a B-⊕-equivalence on (X, f ) iff E is a strong equivalence
on (X, (κf )idB ).
c
Example: κ : T → (Id → T1) →
− T1 is a monad morphism to a
c
submonad of continuation monad (→
− denotes the function space of
continuous functions); κ is an injection e.g. for TX = X → [0, ∞].
Hence, weak bisimulation for probabilistic system is expressible as
strong bisimulation of systems of type T → (Id ×A → T1) →c T1.
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Further Work
• Elaborate the generalization from T(X × A) to TFX
• Coalgebraic modal logic with weak modalities [[a]], hhaii
• Generic algorithms for checking weak bisimulation
• Further variants of weak bisimulation, e.g. branching bisimulation
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Coalgebraic Weak Bisimulation from Recursive Equations over

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