Fall 2001, CSE 520: Lecture 1

The Lambda Calculus

Some history

The Type-Free Lambda Calculus


The terms of the lambda calculus are constructed over variables and two basic operations:

Formal syntax

Variables: V ::= x | y | z | ...
Lambda-Terms: M ::= V | (\V M) | (M M)

Notational convention

M1M2 ... Mk stands for ( ... (M1M2) ... Mk)
\x1x2 ... xk.M stands for (\x1(\x2 ... (\xk M) ... ))

The Semantics of the Lambda Calculus

The theory of the lambda calculus is an equational theory called "lambda-conversion", namely, a set of rules which allows to derive that certain terms are "equal" (i.e. lambda-convertible to each other).

This theory defines a semantics for lambda-terms in the sense that it defines the meaning of a term M as the (equivalence) class of all terms lambda-convertible to M.

It will be useful to introduce some preliminary notions.

Free and Bound variables

Given a lambda abstraction \x.M, we say that the operator "\x" is a binder for x, and that the occurrences of x in M are bound. The scope of this binder is the whole term M, except for the subterms of M which contain another binder for x, namely, the subterms which are again of the form \x.N.

These concepts of binder and scope are similar to the concepts of declaration and scope in programming langauges like for instance C, Pascal, or ML.

For example, consider the term

+++ (\x. ... (\x. --- ) *** ) ^^^
The parts ... and *** are in the scope of the outermost \x. The part --- is in the scope of the innermost \x. The parts +++ and ^^^ are out of the scope of both \x.

An occurrence of x is free in M if it is not in the scope of any binder for x.

For example, consider the term

(\x. x y (\x.\y. y z x ) x ) y x
We have 4 occurrences of x, of which only the last is free. The first and the third are bound by the outermost \x and the second is bound by the innermost \x. As for the other variables, we have 3 occurrences of y, of which the first and the last are free, and we have only one occurrence of z, free.

The free variables of M are all the variables which occur free in M. Formally:

FV(x) = {x}
FV(\x.M) = FV(M) - {x}
FV(M N) = FV(M) union FV(N)


Given a term M, a variable x, and a term N which does not contain any free variable which is bound in M (in a subterm containing x), the term M[N/x] (also denoted by M[x:=N]) is the term obtained from M by replacing every free occurrence of x by N. Formally:
x[N/x] = N
y[N/x] = y
(\y.M)[N/x] = \y.(M[N/x])
(\x.M)[N/x] = \x.M
(M P)[N/x] = (M[N/x])(P[N/x])

The condition "N does not contain free variables which are bound in M" is meant to avoid "variable capture". For example, given M = \y.zx, and N = y, if we simply replaced x by N in M, we would obtain the term \y.zy, which is intuitively incorrect since y, which was free in N, would become bound in M[N/x]. (Or in other words, M was constant on y and would become not constant.)

Alpha conversion

The alpha conversion (or alpha renaming) axiom formalizes the idea that "the names of bound variables don't matter". Formally:
\x.M =alpha \y.M[y/x]
provided that y is neither free nor bound in M.

Beta conversion

The beta axiom formalizes the notion of transformation which takes place when a function is applied to an argument. Formally:
(\x.M) N =beta M[N/x]
provided that N does not contain free variables which is bound in M (in a subterm containing x), so to avoid variable capture.

Example: (\x. x+1) 5 =beta 5+1 = 6

Variable assumption: To avoid the "variable capture" problem, and more in general, confusion with names, we will assume that all variables that occur free have names different from those of the bound variables, and that all bound variables have different names as well. Note that this assumption is not restrictive since we can always alpha-rename the bound variables.

For example, when we have a term like

x (\x.x(\x.x)x)
we will write, instead
x (\y.y(\z.z)y)

Another solution would be to define the notion of substitution so that it renames the variables automatically when needed. For this approach see for instance Hindley and Seldin, Definition 1.1.

The theory of the lambda calculus

We will say that M = N is provable in the lambda calculus, or that M and N are lambda-convertible (notation: |- M = N or simply M = N) if M = N can be derived using the alpha axiom, the beta axiom, reflexivity, symmetry, transitivity, and the following rules: