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# Topic: Lambda calculus

 Lambda calculus - Wikipedia, the free encyclopedia The calculus can be used to cleanly define what a computable function is. The question of whether two lambda calculus expressions are equivalent cannot be solved by a general algorithm, and this was the first question, even before the halting problem, for which undecidability could be proved. In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument. A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see currying). en.wikipedia.org /wiki/Lambda_calculus   (2202 words)

 Typed lambda calculus - Wikipedia, the free encyclopedia Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell, they are closely related to intuitionistic logic via the Curry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. A more modern view is to consider typed lambda calculi as fundamental and to consider the untyped lambda calculus as a special case of a typed lambda calculus with only one type. Lambda calculi with dependent types are the base of Intuitionistic Type Theory, the Calculus of Constructions and the Logical Framework (LF), a pure lambda calculus with dependent types. en.wikipedia.org /wiki/Typed_lambda_calculus   (518 words)

 Knights of the Lambda Calculus - Wikipedia, the free encyclopedia The Knights of the Lambda Calculus is a semi-fictional organization of expert LISP and Scheme hackers. The name refers to the lambda calculus, a mathematical formalism invented by Alonzo Church, with which LISP is intimately connected, and references the Knights Templar. There is no actual organization that goes by the name Knights of the Lambda Calculus; it mostly only exists as a hacker culture in-joke with which to poke fun at conspiracy theories and secret societies. en.wikipedia.org /wiki/Knights_of_the_Lambda-Calculus   (266 words)

 Barendregt: Lambda Calculus   (Site not responding. Last check: 2007-10-21) In a lambda term MN, the occurrence of the subterm M is active and the occurrence of the subterm N is passive. Lambda with the eta conversion is the theory Lambda-eta. The lambda-I calculus corresponds to the combinatory logic with primitive combinators I, B, C, and S. For each lambda term M, we may construct the Bohm tree BT(M) of M. This tree may be infinite. www.andrew.cmu.edu /~cebrown/notes/barendregt.html   (21701 words)

 Lambda Calculus Summary - Lambda Calculus Information Even though Church's original motivation in developing the lambda calculus was to provide a foundation for mathematics by means of creating a special tool for mathematical logic, the results of his work have had considerable application in computer science. The lambda calculus has the power to represent all computable functions, but its uncomplicated syntax and semantics provides a fine basis for the specific context of the study of the semantics of programming language concepts. One of the many important points about the lambda notation is its lack of ambiguity; the number and order of the parameters of the function are strictly specified between the symbol and an expression. www.bookrags.com /sciences/computerscience/lambda-calculus-wcs.html   (705 words)

 Lambda Calculus Expressions in the pure lambda calculus are defined according to the following concrete syntax. A lambda expression is either a simple variable name (lower case letter), the lambda symbol followed by a variable name a dot and a lambda expression (this form is called a lambda abstraction) or one lambda expression following another (this is an application). In practice we enrich the lambda calculus with constants (mostly numeric) and operators (+,- etc.) as we need them. scom.hud.ac.uk /scomtlm/cas810/notes/lambda.html   (376 words)

 Introduction To Lambda Calculus (Tutorial) The Lambda Calculus was developed by Alonzo Church in the 1930s and published in 1941 as 'The Calculi Of Lambda Conversion'. In lambda calculus a function does not 'return' a result based on its parameters - instead the function and its parameters are 'reduced' to give an answer, which mathematically is equivalent to the question. Lambda calculus has the 'Church-Rosser property', so that if two methods of reduction lead to two normal forms, they can differ only by alpha conversion. www.safalra.com /science/lambdacalculus/introduction.html   (375 words)

 [No title] ;; ;; The lambda calculus, invented by Alonzo Church in the 1930s, is ;; a mathematical model of computation based around the operations of ;; "abstraction" (defining functions) and "application" (evaluating ;; them at a point). For example: ;; ;; 0 is (lambda (f) (lambda (x) x)) ;; 1 is (lambda (f) (lambda (x) (f x))), or just (lambda (f) f) ;; 2 is (lambda (f) (lambda (x) (f (f x)))) ;; 3 is (lambda (f) (lambda (x) (f (f (f x))))) ;; ;;... It is possible to ;; write recursive functions within the lambda calculus, but it is ;; impossible to stop at the base case without relying on knowledge of ;; the underlying language's evaluation order. www.cs.cmu.edu /~dst/DeCSS/Gallery/css-descramble-lambda.txt   (1392 words)

 Lambda Calculus   (Site not responding. Last check: 2007-10-21) Lambda calculus is a theory of functions that is central to (theoretical) computer science. It is well known that all recursive functions are representable as lambda terms: the representation is so compelling that definability in the calculus may as well be regarded as a definition of computability. Lambda calculus is the commonly accepted basis of functional programming languages; and it is folklore that the calculus is the prototypical functional language in purified form. users.comlab.ox.ac.uk /luke.ong/teaching/lambda   (295 words)

 The Lambda Calculus Mail Series Lambda calculus was invented long before there were any computers (in the modern sense, as programmable machines), or computer languages. Lambda calculus was created to model the notion of functions that mathematics has. But in lambda calculus, this definition of natural numbers, is just a nencoding that we have made up, because we think it satisfies the notion of natural numbers, that we want it to satisfy. www24.brinkster.com /srineet/lambda\lambda.html   (15635 words)

 Lambda calculus The lambda calculus was invented by Alonzo Church in the 1930s to study the interaction of functional abstraction and function application from an abstract, purely mathematical point of view. To code this in the lambda calculus, we can take two terms A and B and produce a function that will apply another function to them. The idea is to encode the number n as a lambda term n representing a function that takes another function f as input and produces the n-fold composition of f with itself. www.cs.unc.edu /~stotts/COMP204/Lambda/overview.html   (1423 words)

 [No title] Functions are created from lambda terms which are written in the form \a(expr)' which is the function taking argument a' and having value expr'. It is fairly well known that the lambda calculus (and hence SK combinators) can compute anything, but mere computation is no use if you cannot communicate with the world. For example, if I had the appropriate arithmetic operators defined, I could write a factorial function in the lambda calculus like this: Y \f\n((= n 0) 1 (* n (f (- n 1)))) OFL provides a facility for naming expressions so that they can be used more than once without writing them in full. www.ioccc.org /1998/fanf.hint   (1937 words)

 Lambda Calculus and A++: Basic Concepts The Lambda Calculus has been created by the American logician Alonzo Church in the 1930's and is documented in his works published in 1941 under the title The Calculi of Lambda Conversion'. In this article the importance of the Lambda Calculus is extended to non-functional languages like Java, C, and C++ as well. The article uses A++, a programming language directly derived from the Lambda Calculus, as a vehicle to demonstrate the application of the basic ideas of the Lambda Calculus in a multi-paradigm environment. www.lambda-bound.com /book/lambdacalc/lcalconl.html   (290 words)

 Lambda Calculus   (Site not responding. Last check: 2007-10-21) For instance, (lambda(x)(+x1)) is a function that takes in an evaluated argument, binds it with x, and then computes the body of the lambda form with the understanding that any occurrence of parameter x in the body will refer to the value of x bound by the lambda form. Consider it a question of ownership: if a lambda form references a parameter it does not own, it is said to be referencing a free variable; if a lambda form references a parameter it does own, it is said to be referencing a bound variable. It’s called lexical scoping: The reference to the free variable fact-wrong inside the lambda form that was being bound to fact-wrong must refer to a variable named fact-wrong in a parent environment. www.mactech.com /articles/mactech/Vol.07/07.05/LambdaCalculus   (6336 words)

 lambda-calculus The lambda calculus is basically function theory, there are only three things available in the pure lambda calculus, these are: This page is not intended to be an introduction to the lambda calculus; there are lots of good books for that. Recursion in the lambda calculus can be done with a fixpoint operator. frmb.org /lambda-plain.html   (449 words)

 Lecture 25: Introduction to the Lambda Calculus   (Site not responding. Last check: 2007-10-21) This is the basis of the inductive definition of lambda terms. In the lambda calculus, a reduction rule can be performed at any time to any subterm of a lambda term at which the rule is applicable. The lambda calculus is powerful enough to encode other data types and operations. www.cs.cornell.edu /Courses/cs312/2001FA/lecture/lec25.htm   (1043 words)

 Lambda tutorial   (Site not responding. Last check: 2007-10-21) The following is too fun not to mention: David Keenan, To dissect a mockingbird: a graphical notation for the lambda calculus with animated reduction. Here, "var" is the variable bound by the lambda operator, and "body" is the constituent that the lambda operator has scope over. Line 6 (beta reduction): if f has the form ((lambda variable body) argument), it is a lambda form being applied to an argument, so perform lambda conversion. ling.ucsd.edu /~barker/Lambda   (2439 words)

 Lambda calculus plus letrec   (Site not responding. Last check: 2007-10-21) The scoped lambda-graphs are represented by terms defined over lambda calculus extended with the letrec construct. We conclude by presenting a variant of our cyclic lambda calculus, which follows the tradition of the explicit substitution calculi. Since most implementations of non-strict functional languages rely on sharing to avoid repeating computations, we develop a variant of our cyclic lambda calculus that enforces the sharing of computations and show that the two calculi are observationally equivalent. www.cs.uoregon.edu /~ariola/cycles.html   (357 words)

 Notes: Lambda Calculus Lambda Calculus as a basis for functional programming languages Lambda calculus is a formal model of computation. (lambda x M A) can be reduced by substituting A into M for all free occurrances of x. www.cs.unc.edu /~stotts/COMP204/Lambda   (821 words)

 No Title   (Site not responding. Last check: 2007-10-21) Lambda Calculus is intended to capture, in a very simple way, the idea of creating a function and then calling it on an argument. Miraculously, the Lambda Calculus model is both simpler than the Turing machine model and more practical for real computation. Our goals are to develop Lambda Calculus versions of if-then tests, boolean and integer constants, arithmetic operators, and some simple data structures. perl.plover.com /lambda/tpj.html   (3637 words)

 Infinitary Lambda Calculus - Kennaway, Klop, Sleep, de Vries (ResearchIndex)   (Site not responding. Last check: 2007-10-21) In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Bohm model of the lambda calculus can be seen as an infinitary term model. 8: The Lambda Calculus: Its Syntax and Semantics (context) - Barendregt - 1984 citeseer.ist.psu.edu /351096.html   (519 words)

 Lambda Calculus   (Site not responding. Last check: 2007-10-21) The Lambda Calculus is a method for investigating and resolving identifier bindings. The form of the expressions in the untyped Lambda Calculus can be found in any elementary programming languages text. Extending the untyped Lambda Calculus by associating a type with each constant and binding results in the formalism described in the text. ugrad-www.cs.colorado.edu /~csci5535/HW/lamb.html   (375 words)

 Church's lambda calculus Doubtful, since C doesn't lend itself well to working with lambda calculus (the world-view of the two models is too different). If you want to find a treatment of the lambda calculus that uses a programming language rather than mathematical notation, you're better off looking at a functional language like Scheme. http://groups.google.com/groups?sel...ian.ccrwest.org That post has the term "lambda calculus" in it, in reference to a program called loader.c, but I don't understand a lot of that post. www.codecomments.com /message276723.html   (876 words)

 lambda calculus - a Whatis.com definition The language deals with the application of a function to its arguments (a function is a set of rules) and expresses any entity as either a variable, the application of one function to another, or as a "lambda abstraction" (a function in which the Greek letter�lambda is defined as the abstraction operator). Lambda calculus, and the closely related theories of combinators and type systems, are important foundations in the study of mathematics, logic, and computer programming language. Roger Bishop Jones provides an overview of Lambda calculus, combinatory logic, and type systems. whatis.techtarget.com /definition/0,,sid9_gci341298,00.html   (195 words)

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