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Beta Normal Form
In the lambda calculus, a term is in beta normal form if no ''beta reduction'' is possible. A term is in beta-eta normal form if neither a beta reduction nor an ''eta reduction'' is possible. A term is in head normal form if there is no ''beta-redex in head position''. Beta reduction In the lambda calculus, a beta redex is a term of the form: : (\mathbf x . A) M. A redex r is in head position in a term t, if t has the following shape (note that application has higher priority than abstraction, and that the formula below is meant to be a lambda-abstraction, not an application): : \lambda x_1 \ldots \lambda x_n . \underbrace_ M_2 \ldots M_m , where n \geq 0 and m \geq 1. A beta reduction is an application of the following rewrite rule to a beta redex contained in a term: : (\mathbf x . A) M \longrightarrow A := M where A := M/math> is the result of substituting the term M for the variable x in the term A. A ''head'' beta reduction is a beta reduction applied in head position ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
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. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TheFreeDictionary
''The Free Dictionary'' is an American online dictionary and encyclopedia that aggregates information from various sources. Content The site cross-references the contents of '' The American Heritage Dictionary of the English Language'', the ''Collins English Dictionary'', the ''Columbia Encyclopedia'', the ''Computer Desktop Encyclopedia'', the '' Hutchinson Encyclopedia'' (subscription), and Wikipedia, as well as the Acronym Finder database, several financial dictionaries, legal dictionaries, and other content. It has a feature that allows a user to preview an article while positioning the mouse cursor over a link. One can also double-click on any word to look it up in the dictionary. Site operator The site is run by Farlex, Inc., located in Huntingdon Valley, Pennsylvania. Farlex also maintains a companion title, ''The Free Library'', an online library of out-of-copyright classic books as well as a collection of periodicals of over four million articles dating back to 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Head Normal Form
Lambda calculus is a formal mathematical system based on lambda abstraction and function application. Two definitions of the language are given here: a standard definition, and a definition using mathematical formulas. Standard definition This formal definition was given by Alonzo Church. Definition Lambda expressions are composed of * variables v_, v_, ..., v_, ... * the abstraction symbols lambda '\lambda ' and dot '.' * parentheses ( ) The set of lambda expressions, \Lambda , can be defined inductively: #If x is a variable, then x \in \Lambda #If x is a variable and M \in \Lambda , then (\lambda x . M) \in \Lambda #If M, N \in \Lambda , then (M \ N) \in \Lambda Instances of rule 2 are known as abstractions and instances of rule 3 are known as applications. Notation To keep the notation of lambda expressions uncluttered, the following conventions are usually applied. * Outermost parentheses are dropped: M \ N instead of (M \ N) * Applications are assumed to be left-a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Strategy (lambda Calculus)
In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. Some authors use the term to refer to an evaluation strategy. Definitions Formally, for an abstract rewriting system (A, \to), a reduction strategy \to_S is a binary relation on A with \to_S \subseteq \overset , where \overset is the transitive closure of \to (but not the reflexive closure). In addition the normal forms of the strategy must be the same as the normal forms of the original rewriting system, i.e. for all a, there exists a b with a\to b iff \exists b'. a\to_S b'. A ''one step'' reduction strategy is one where \to_S \subseteq \to. Otherwise it is a ''many step'' strategy. A ''deterministic'' strategy is one where \to_S is a partial function, i.e. for each a\in A there is at most one b such that a \to_S b. Otherwise it is a ''nondeterministic'' strategy. Term rewriting In a term rewriting system a rewr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standardization Theorem
Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardization can help maximize compatibility, interoperability, safety, repeatability, or quality. It can also facilitate a normalization of formerly custom processes. In social sciences, including economics, the idea of ''standardization'' is close to the solution for a coordination problem, a situation in which all parties can realize mutual gains, but only by making mutually consistent decisions. History Early examples Standard weights and measures were developed by the Indus Valley civilization.Iwata, Shigeo (2008), "Weights and Measures in the Indus Valley", ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition)'' edited by Helaine Selin, pp. 2254–2255, Springer, . The centralized wei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Director String
In mathematics, in the area of lambda calculus and computation, directors or director strings are a mechanism for keeping track of the free variables in a term. Loosely speaking, they can be understood as a kind of memoization for free variables; that is, as an optimization technique for rapidly locating the free variables in a term algebra or in a lambda expression. Director strings were introduced by Kennaway and Sleep in 1982 and further developed by Sinot, Fernández and MackieF.-R. Sinot, M. Fernández and I. Mackie. [ftp://nozdr.ru/biblio/kolxoz/Cs/CsLn/R/Rewriting%20Techniques%20and%20Applications,%2014%20conf.,%20RTA%202003(LNCS2706,%20Springer,%202003)(ISBN%203540402543)(526s)_CsLn_.pdf#page=57 Efficient Reductions with Director Strings]. In ''Proc. Rewriting Techniques and Applications''. Springer LNCS vol 2706, 2003 as a mechanism for understanding and controlling the computational complexity cost of beta reduction. Motivation In beta reduction, one defines the value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
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. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Form (other)
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Normal form may refer to: * Normal form (databases) * Normal form (game theory) *Canonical form * Normal form (dynamical systems) *Hesse normal form * Normal form in music *Jordan normal form in formal language theory: *Chomsky normal form *Greibach normal form *Kuroda normal form *Normal form (abstract rewriting), an element of a rewrite system which cannot be further rewritten in logic: * Normal form (natural deduction) * Algebraic normal form *Canonical normal form * Clausal normal form *Conjunctive normal form *Disjunctive normal form *Negation normal form *Prenex normal form *Skolem normal form in lambda calculus: * Beta normal form See also * Normalization (other) *Normalization property In abstract rewriting, an object is in normal form if it cannot be rewritten any further, i.e. it is irreducible. Depending on the rewriting system, an object may rewrite to several normal forms or none at all. Many properties of rewriting systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
