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Lambda Calculus Definition
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- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Application
In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction. Representation Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses. For example, the following expression represents the application of the function ''ƒ'' to its argument ''x''. :f(x) In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by juxtaposition. For example, the following expression can be considered the same as the previous one: :f\; x The latter notation is especially useful in combination with the currying isomorphism. Given a function f : (X \times Y) \to Z, its application is represented as f(x, y) by the former notation and f\;(x,y) (or f \; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Reduction (capture Avoiding)
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] |
Definition Of Semantics
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Changes To The Substitution Operator
Changes may refer to: Books * ''Changes'', the 12th novel in Jim Butcher's ''The Dresden Files'' Series * ''Changes'', a novel by Danielle Steel * ''Changes'', a trilogy of novels on which the BBC TV series was based, written by Peter Dickinson Film and television * ''Changes'' (1991 film), a 1991 television film * ''Changes'' (1969 film), a 1969 American drama film * ''Changes'' (advertisement), a 1987 advertisement * ''The Changes'' (TV series), produced by the BBC in 1975 * "Changes" (''House''), a 2011 episode of the American medical drama ''House'' * "Changes", a 1984 episode of the American TV sitcom ''Silver Spoons'' * "Changes", the name of five episodes of the TV sitcom ''Punky Brewster'' * "Changes", the name of the ''You Can't Do That on Television'' 2004 reunion episode Music * A jazz term for chord progression * An algorithmic Change ringing, pattern for ringing tuned bells * ''Changes'' (Godsmack video album), a 2004 documentary and live DVD by the heavy met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free And Bound Variable Sets
Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decisions * Free of charge, also known as gratis. See Gratis vs libre. Computing * Free (programming), a function that releases dynamically allocated memory for reuse * Free format, a file format which can be used without restrictions * Free software, software usable and distributable with few restrictions and no payment * Freeware, a broader class of software available at no cost Mathematics * Free object ** Free abelian group ** Free algebra ** Free group ** Free module ** Free semigroup * Free variable People * Free (surname) * Free (rapper) (born 1968), or Free Marie, American rapper and media personality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution Operator
Substitution may refer to: Arts and media *Chord substitution, in music, swapping one chord for a related one within a chord progression * Substitution (poetry), a variation in poetic scansion * "Substitution" (song), a 2009 song by Silversun Pickups * Substitution (theatre), an acting methodology *Tritone substitution, in music, reinterpreting a chord via a new root note located an augmented fourth or diminished fifth distant from the root of the original interpretation Science and mathematics Biology and chemistry *Base-pair substitution or point mutation, a type of mutation *Substitution reaction, where a functional group in a chemical compound is replaced by another group *Substitution, a process in which an allele arises and undergoes fixation Mathematics and computing *Substitution (algebra), replacing occurrences of some symbol by a given value *Substitution (logic), a syntactic transformation on strings of symbols of a formal language *String substitution, a mapping of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same. Example Consider the two functions ''f'' and ''g'' mapping from and to natural numbers, defined as follows: * To find ''f''(''n''), first add 5 to ''n'', then multiply by 2. * To find ''g''(''n''), first multiply ''n'' by 2, then add 10. These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same. Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionally identical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town. Then, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language (see e.g. Church 1970). Alongside his student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a Justice of the Peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829-1909), United States Senate Librarian from 1881-1901, and great grandson of Alonzo Church, a Professor of Mathematics and Astronomy and 6th Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scope (programming)
In computer programming, the scope of a name binding (an association of a name to an entity, such as a variable) is the part of a program where the name binding is valid; that is, where the name can be used to refer to the entity. In other parts of the program, the name may refer to a different entity (it may have a different binding), or to nothing at all (it may be unbound). Scope helps prevent name collisions by allowing the same name to refer to different objects – as long as the names have separate scopes. The scope of a name binding is also known as the visibility of an entity, particularly in older or more technical literature—this is from the perspective of the referenced entity, not the referencing name. The term "scope" is also used to refer to the set of ''all'' name bindings that are valid within a part of a program or at a given point in a program, which is more correctly referred to as ''context'' or ''environment''. Strictly speaking and in practice for most pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
