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Lambda Calculus: Functions
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The Lambda Calculus was developed by Alonzo Church in the 1930s and published in 1941 as ‘The Calculi Of Lambda Conversion’. It became important, along with Turing machines, in the development of computation theory, and is the theoretical basis of all functional programming languages, such as Lisp, Haskell and ML.
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Lambda calculus can be called the smallest universal programming language. It consists of a single transformation rule (variable substitution) and a single function definition scheme. Lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. It is ... the same as the Turing machine formalism. However, lambda calculus emphasizes the use of transformation rules. It does not care about the actual machine that implements them.
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Lambda Animator is a tool for demonstrating and experimenting with alternative reduction strategies in the lambda calculus. Eager languages reduce arguments before function application. Lazy languages reduce arguments, if needed, after function application.
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Lambda calculus is not only an abstract calculus, but an Abstract VM: a kind of calculus of very basic abstraction. It was invented even before Turing Machines... as a universal computational framework. Check pages and books about functional programming, functional programming languages, computer science laboratories, theoretical computer science research, etc, for more about lambda calculus.
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The lambda calculus can be thought of as an idealized, minimalistic programming language. It is a close cousin of the Turing machine, another minimalist abstraction capable of expressing any algorithm. The difference between the two is that the lambda calculus takes a functional view of algorithms, while the original Turing machine takes an imperative view. That is, a Turing machine maintains 'state' - a 'notebook' of symbols that can change from one instruction to the next. The imperative paradigm can be seen in programming languages like C or BASIC. By contrast, the lambda calculus is stateless, it deals exclusively with functions which accept and return data (including other functions), but produce no side effects in 'state' and do not make alterations to incoming data (immutability.) The functional paradigm can be seen in modern languages like Lisp, Scheme and Haskell.
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In mathematical logic and computer science, the lambda calculus (... λ-calculus) is a formal system designed to investigate functions and recursion. Although it was originally intended as a mathematical formalism and predates the electronic computer, it can be seen as the world's first formalized programming language. The programming language Lisp was created in large part from the λ-calculus. In mathematics, the λ-calculus was central to some of the first theorems of computability - theorems about the limits of systematic computation which had broad philosophical impact ouside of mathematics.
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