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Lambda Calculus
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Lambda Calculus is a formal language which helped define the notion of a computable funcion. The languages lisp, scheme and haskell are very closely related to lamda calculus. The language defines means for abstraction and invocation, which are its axioms. All other programming constructs are derived from these basic axioms.
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As a mathematical logical system the Lambda Calculus is covered in detail in [Bar81] and less comprehensively but in a more readable form in [Sto81]. A clear account of the historical origins and basic properties of the lambda calculus is presented by Curry and Fey in their book [CF58]. This view is taken from [Jon87] page 23.
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The Lambda Calculus was developed by the logician Alonzo Church in the 1930s as a foundation for mathematics. It is the inspiration for the programming language LISP, and the basis for much of the field of functional programming. It ... turns out to be quite useful for denotational semantics, so it is clearly important for this class.
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The lambda calculus, as is very well known, is a deceptively simple system that is Turing-complete, theoretically capable of computing whatever can be computed. It is not the simplest complete system (see here and here), but the others are directly related to it, and some are deliberately designed to hurt the contents of your head. The calculus itself is very completely described in the articles linked to above.
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The goal of the Penn Lambda Calculator project is to provide teachers and students with a software that assists them in mastering the centerpiece of introductory courses to formal semantics: the lambda calculus. The most important feature of this software is its sophisticated routines that detect common errors and return "smart" feedback to the student user. It ... has a module for performing derivations of natural language phrases up a syntactic tree in the style of Heim & Kratzer (1998).
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This system implements and visualizes various reduction strategies for the pure untyped lambda calculus. It is intended as a pedagogical tool, and as an experiment in the programming of visual user interfaces using Standard ML and HTML.
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Lambda Calculus