Lambda calculus

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any single-taped Turing machine and was first introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.

Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any single-taped Turing machine. Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.

Lambda calculus may be typed and untyped. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data.

Lambda calculus has applications in many different areas in mathematics, philosophy,linguistics, and computer science. Lambda calculus has played an important role in the development of the theory of programming languages. Functional programming languages implement the lambda calculus. Lambda calculus also is a current research topic in Category theory.

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