Lambda calculus

In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and substitution. Untyped lambda calculus, the topic of this article, is a universal machine, i.e. a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was logically consistent, and documented it in 1940.

Definition

The lambda calculus consists of a language of lambda terms, which are defined by a formal syntax, and a set of transformation rules for manipulating those terms. In BNF, the syntax is

e

::=

x

∣

λ

x

.

e

∣

e

e

,

{\displaystyle e::=x\mid \lambda x{\text{.}}e\mid e\,e,}

where variables

x

,

y

,

z

{\displaystyle x,y,z}

range over an infinite set of names. Terms

M

,

N

,

t

,

s

,

e

,

f

{\displaystyle M,N,t,s,e,f}

range over all lambda terms. This corresponds to the following inductive definition:

A variable

x

{\displaystyle x}

is a valid lambda term.