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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. First formulated by Alonzo Church to formalize the concept of effective computability, lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem. Lambda calculus is a conceptually simple universal model of computation (Turing showed in 1937 that Turing machines equaled the lambda calculus in expressiveness). The name derives from the Greek letter lambda (λ) used to denote binding a variable in a function. The letter itself is arbitrary and has no special meaning. Lambda calculus is taught and used in computer science because of its usefulness in showcasing functional thinking and iterative reduction.〔.〕 Because of the importance of the notion of variable binding and substitution, there is not just one system of lambda calculus, and in particular there are ''typed'' and ''untyped'' variants. Historically, the most important system was the untyped lambda calculus, in which 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 be applied only 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,〔Coquand, Thierry, ("Type Theory" ), ''The Stanford Encyclopedia of Philosophy'' (Summer 2013 Edition), Edward N. Zalta (ed.).〕 linguistics, and computer science. It is still used in the area of computability theory, although Turing machines are also an important model for computation. Lambda calculus has played an important role in the development of the theory of programming languages. Counterparts to lambda calculus in computer science are functional programming languages, which essentially implement the lambda 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. ==Lambda calculus in history of mathematics== The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.〔A. Church, "A set of postulates for the foundation of logic", ''Annals of Mathematics'', Series 2, 33:346–366 (1932).〕〔For a full history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006).〕 The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox. Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus.〔A. Church, "An unsolvable problem of elementary number theory", ''American Journal of Mathematics'', Volume 58, No. 2. (April 1936), pp. 345-363.〕 In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lambda calculus」の詳細全文を読む スポンサード リンク
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