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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]{{cbignore, bot=medic. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arity
In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. Examples In general, functions or operators with a given arity follow the naming conventions of ''n''-based numeral systems, such as binary and hexadecimal. A Latin prefix is combined with the -ary suffix. For example: * A nullary function takes no arguments. ** Example: f()=2 * A unary function takes one argument. ** Example: f(x)=2x * A binary function takes two arguments. ** Example: f(x,y)=2xy * A ternary function takes three arguments. ** Example: f(x,y,z)=2xyz * An ''n''-ary function takes ''n'' arguments. ** Example: f(x_1, x_2, \ldots, x_n)=2\prod_^n x_i Nullary A constant can be treated as the output of an operation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Memoization
In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls to pure functions and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts (and for purposes other than speed gains), such as in simple mutually recursive descent parsing. It is a type of caching, distinct from other forms of caching such as buffering and page replacement. In the context of some logic programming languages, memoization is also known as tabling. Etymology The term ''memoization'' was coined by Donald Michie in 1968 and is derived from the Latin word ('to be remembered'), usually truncated as ''memo'' in American English, and thus carries the meaning of 'turning he results ofa function into something to be remembered'. While ''memoization'' might be confused with ''memorization'' (because they are etymological cognates), ''memoization'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Explicit Substitution
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness" conditions in such implicit calculi are a notorious source of errors. The concept has appeared in a large number of published papers in quite different fields, such as in abstract machines, predicate logic, and symbolic computation. Overview A simple example of a lambda calculus with explicit substitution is "λx", which adds one new form of term to the lambda calculus, namely the form M⟨x:=N⟩, which reads "M where x will be substituted by N". (The meaning of the new term is the same as the common idiom let x:=N in M from many programming languages.) λx can be written with the following rewriting rules: # (λx.M) N → M&la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Rewrite System
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Explicit Substitution
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness" conditions in such implicit calculi are a notorious source of errors. The concept has appeared in a large number of published papers in quite different fields, such as in abstract machines, predicate logic, and symbolic computation. Overview A simple example of a lambda calculus with explicit substitution is "λx", which adds one new form of term to the lambda calculus, namely the form M⟨x:=N⟩, which reads "M where x will be substituted by N". (The meaning of the new term is the same as the common idiom let x:=N in M from many programming languages.) λx can be written with the following rewriting rules: # (λx.M) N → M&la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatory Algebra
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments. In mathematics Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory logic typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Reduction
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computation based on function abstraction and application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A #validLambdaVar, variable is a character or string representing a pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
