Lambda Calculus

Lambda calculus (also written as λ-calculus or called "the lambda calculus") is a formal system in mathematical logic for expressing computation by way of variable binding and substitution. It was first formulated by Alonzo Church as a way to formalize mathematics through the notion of functions, in contrast to the field of set theory. Although not very successful in that respect, the lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem.

Because of the importance of the notion of variable binding and substitution, there is not just one system of lambda calculus. Historically, the most important system was the untyped lambda calculus. In the untyped lambda calculus, function application has no restrictions (so the notion of the domain of a function is not built into the system). In the Church–Turing Thesis, the untyped lambda calculus is claimed to be capable of computing all effectively calculable functions. The typed lambda calculus is a variety that restricts function application, so that functions can only be applied if they are capable of accepting the given input's "type" of data.

Today, the lambda calculus has applications in many different areas in mathematics, philosophy, and computer science. It is still used in the area of computability theory, although Turing machines are arguably the preferred model for computation. Lambda calculus has played an important role in the development of the theory of programming languages. The most prominent counterparts to lambda calculus in computer science are functional programming languages, which essentially implement the calculus (augmented with some constants and datatypes). Beyond programming languages, the lambda calculus also has many applications in proof theory. A major example of this is the Curry–Howard correspondence, which gives a correspondence between different systems of typed lambda calculus and systems of formal logic.