LYCOS RETRIEVER 
Lambda Calculus: Alonzo Church
built 606 days ago
In mathematical logic and computer science, lambda calculus... λ-calculus, is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding obstacles like Russell's Paradox). The Kleene-Rosser paradox shows that the lambda calculus is unable to avoid set-theoretic paradoxes, but the lambda calculus emerged as a useful tool in the investigation of problems in computability or recursion theory, and forms the basis of a paradigm of computer programming called functional programming. [1]
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The Church-Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even concurrently. (Indeed, the lambda calculus is referentially transparent.) While this means the lambda calculus can model the various nondeterministic evaluation strategies, it does not offer any richer notion of parallelism, nor can it express any concurrency issues. The Actor model and Process calculi such as CSP, the CCS, the π calculus and the ambient calculus have been designed for such purposes.
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The Church-Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even concurrently. This means that various nondeterministic evaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs for parallelism. Various process calculi have been proposed as minimal languages for concurrency and distributed computation.
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Church invented lambda-calculus in order to set up a foundational project restricting mathematics to quantities with "effective procedures". Unfortunately, the resulting system admits Russell's paradox in a particularly nasty way; Church couldn't see any way to get rid of it, and gave the project up.
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The lambda calculus was created by Alonzo Church as part of an effort to provide a foundation for mathematics. Church's attempt failed, as his system was vulnerable to an analogue of Russel's Paradox.
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This module allows simple experimentation with the lambda calculus, first developed by Church. It understands the different types of lambda expressions, can extract lists of variables (both free and bound) and subterms, and can simplify complicated by expression by means of application.
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Alonzo Church