Lecture Note 2

IST 597A: Software Security and Analysis
(Spring 2015)
Lecture Note 2: Lambda Calculus
Dinghao Wu
1
Untyped λ-calculus
Terms.
The set of lambda expressions or terms can be defined inductively.
1. If x is a variable, then x is a lambda expression;
2. If x is a variable and e is a lambda expression, then λx.e is a lambda
expression. This is called λ-abstraction.
3. If e1 and e2 are lambda expressions, then (e1 e2 ) is a lambda expression.
This is called applicaiton.
e ::= x | λx.e | e1 e2
Free and bound variables.
FV(x) = {x}
FV(λx.e) = FV(e)\{x}
FV(e1 e2 ) = FV(e1 ) ∪ FV(e2 )
Capture-avoiding substitutions.
e
x[e/y] =
x
(λx.e)[t/y] =
if x = y
if x =
6 y
λx.e
if x = y
λx.(e[t/y]) if x 6= y and x 6∈ FV(t)
(e1 e2 )[t/y] = (e1 [t/y]e2 [t/y])
1
α-conversion. Also called alpha renaming, or α-equivalence, allows bound
variable names to be changed.
α
λx.e = λy.(e[y/x])
where y is a fresh variable, or y 6= x and y 6∈ FV(e).
In a nameless notation called De Bruijn index [1], any two α-equivalent terms
are literally identical.
beta-reduction
beta-reduction is a kind of function application.
β
(λx.e)e0 = e[e0 /x]
η-conversion η-conversion defines function extensionality: Two functions are
extensionally equivalent if and only if they compute the same result for all the
inputs.
η
λx.(f x) = f (x 6∈ FV(f ))
2
Simply typed λ-calculus
Terms.
e ::= x|λx : τ.e|e1 e2
Typing
x:τ ∈Γ
Γ`x:τ
TyVar
Γ, x : τ1 ` e : τ2
Γ ` λx.τ1 e : τ1 → τ2
TyAbs
Γ ` e1 : τ1 → τ2
Γ ` e2 : τ1
Γ ` (e1 e2 ) : τ2
3
TyApp
Homework 2
The project in Rojas [3].
4
Bibliography Notes and Further reading
Chapter 5–7, 9–10 of Pierce [2].
A Short Tutorial [3] by Raul Rojas of FU Berlin summarizes how λ-calculus
encodes typical programming objects and controls. Chapter 5 of Slonneger and
Kurtz [4] is also easy to read.
2
References
[1] N.G. de Bruijn. Lambda calculus notation with nameless dummies, a tool
for automatic formula manipulation, with application to the Church-Rosser
theorem. Indagationes Mathematicae (Proceedings), 75(5):381–392, 1972.
[2] Benjamin C. Pierce. Types and Programming Languages. MIT Press, Cambridge, MA, USA, 2002.
[3] Ra´
ul Rojas. A tutorial introduction to the lambda calculus, 1997. Online.
[4] Kenneth Slonneger and Barry Kurtz. Formal Syntax and Semantics of Programming Languages: A Laboratory Based Approach. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition, 1995.
3