In mathematical logic , specifically recursion theory computability theory , a range function math f colon mathbb R to mathbb R math is sequentially computable if, for every computable sequence math x i i 1 infty math of real number s, the sequence math f x i i 1 infty math is also computable real number computable . A function math f colon mathbb R to mathbb R math is effectively uniformly continuous if there exists a primitive recursive function recursive function math d colon mathbb N to mathbb N math such that, if math x y 1 over d n math then math f x f y 1 over n math A real function is computable if it is both sequentially computable and effectively uniformly continuous. These definition s can be generalized to functions of more than one Variable mathematics variable or functions only defined on a subset of math mathbb R n. math The generalizations of the latter two need not be restated. A suitable generalization of the first definition is Let math D math be a subset of math mathbb R n. math A function math f colon D to mathbb R math is sequentially computable if, for every math n math tuplet math left x i , 1 i 1 infty, ldots x i , n i 1 infty right math of computable sequences of real numbers such that math forall i quad x i , 1 , ldots x i , n in D qquad , math the sequence math f x i i 1 infty math is also computable. PlanetMath attribution id 6248 title Computable real function Category Computable analysis mathlogic stub comp sci theory stub ... more details
In mathematics and computer science , computable analysis is the study of mathematical analysis from the perspective of computability theory . It is concerned with the parts of real analysis and functional analysis that can be carried out in a computability theory computable manner. The field is closely related to constructive analysis and numerical analysis . Basic results The computable real number s form a real closed field . The equality mathematics equality relation on computable real numbers is not computable, but for unequal computable real numbers the order relation is computable. Computable real function s map computable real numbers to computable real numbers. The function composition composition of computable real functions is again computable. Every computable real function is continuous function continuous . See also Specker sequence References Oliver Aberth 1980 , Computable analysis , McGraw Hill , 1980. Marian Pour El and Ian Richards, Computability in Analysis and Physics , Springer Verlag , 1989. Stephen G. Simpson 1999 , Subsystems of second order arithmetic . Klaus Weihrauch 2000 , Computable analysis , Springer, 2000. External Links http cca net.de Computability and Complexity in Analysis Network mathlogic stub Category Constructivism mathematics Category Computability theory Category Computable analysis ... more details
is not. Formal definition A real number a is said to be computable if it can be approximated by some computablefunction in the following manner given any integer math n ge 1 math , the function produces ... that are equivalent There exists a computablefunction which, given any positive rational error ... There is a computable sequence of rational numbers math q i math converging to math a math such that math q i q i 1 2 i , math for each i . There is another equivalent definition of computable numbers via computable Dedekind cut s. A computable Dedekind cut is a computablefunction math D math which .... In order to produce a computable real, a Turing machine must compute a total function , but the corresponding ...In mathematics , particularly theoretical computer science and mathematical logic , the computable numbers , also known as the recursive numbers or the computable reals , are the real numbers that can ... can be given using recursive function s, Turing machines or lambda calculus calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used ... fractions between 0 and 1 A computable number is one for which there is a Turing machine which ... p q mathrm true math math p 3 3 q 3 Rightarrow D p q mathrm false . math A real number is computable if and only if there is a computable Dedekind cut D converging to it. The function D is unique for each irrational computable number although of course two different programs may provide the same function . A complex number is called computable if its real and imaginary parts are computable. Properties While the set of real numbers is uncountable , the set of computable numbers is only countable and thus almost all real numbers are not computable. The computable numbers can be counted by assigning a G del number to each Turing machine definition. This gives a function from the naturals to the computable reals. Although the computable numbers are an ordered field, the set of G del numbers corresponding ... more details
Unreferenced stub auto yes date December 2009 In computability theory computer science computability theory two sets A and B are computably isomorphic or recursively isomorphic if there exists a bijective computable function f with f A B . Two numbering computability theory numbering s math nu math and are called computably isomorphic if there exists a bijective computable function math f math so that math nu mu circ f. , math Computably isomorphic numberings induce the same notion of computability on a set. DEFAULTSORT Computable Isomorphism Category Theory of computation Category Computability theory Comp sci stub mathlogic stub ... more details
Unreferenced date November 2008 In mathematics , computable measure theory is a version of measure theory which deals with computable number s, as opposed to real number s which are used in standard measure theory. Category Measure theory Category Computability theory Mathanalysis stub ... more details
Computable model theory is a branch of model theory which deals with questions of computability as they apply to model theoretical structures. It was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia , and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independently. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model theoretic conditions can be shown to exist. References citation last Harizanov first V. S. authorlink Valentina Harizanov contribution Pure Computable Model Theory pages 3 114 title Handbook of Recursive Mathematics, Volume 1 Recursive Model Theory series Studies in Logic and the Foundations of Mathematics volume 138 editor first Iurii Leonidovich editor last Ershov publisher North Holland year 1998 isbn 978 0444500038 mr 1673621 . Category Constructivism mathematics Category Model theory mathlogic stub ... more details
Logic of Computable Functions LCF is a deductive system for computable functions proposed by Dana Scott in 1969 in a memorandum unpublished until 1993. ref Dana S. Scott. http www.cs.cmu.edu kw scans scott93tcs.pdf A type theoretical alternative to ISWIM, CUCH, OWHY . Theoretical Computer Science , 121 411 440, 1993. Annotated version of the 1969 manuscript. ref It inspired Logic for Computable Functions LCF , theorem proving logic by Robin Milner . ref Robin Milner 1973 . ftp reports.stanford.edu pub cstr reports cs tr 73 332 CS TR 73 332.pdf Models of LCF ref Programming Computable Functions PCF , small theoretical programming language by Gordon Plotkin . ref cite journal first Gordon D. last Plotkin authorlink Gordon Plotkin title LCF considered as a programming language journal Theoretical Computer Science year 1977 pages 223 255 volume 5 doi 10.1016 0304 3975 77 90044 5 url http homepages.inf.ed.ac.uk gdp publications LCF.pdf ref harv ref References references Category Programming language theory ... more details
see also Logic of Computable Functions Logic for Computable Functions LCF is an interactive automated theorem prover developed at the universities of University of Edinburgh Edinburgh and Stanford University Stanford by Robin Milner and others in 1972. LCF introduced the general purpose programming language ML programming language ML to allow users to write theorem proving tactics. Theorems in the system are propositions of a special theorem abstract datatype . The ML type system ensures that theorems are derived using only the inference rule s given by the operations of the abstract type. Successors include Higher Order Logic HOL theorem prover HOL and Isabelle theorem prover Isabelle . References Reflist Refbegin cite web last Gordon first Michael J. C. authorlink Michael J. C. Gordon year 1996 title From LCF to HOL a short history url http www.cl.cam.ac.uk mjcg papers HolHistory.html accessdate 2007 10 11 cite manual author Milner, Robin title Logic for Computable Functions description of a machine implementation. publisher Stanford University date May 1972 url ftp reports.stanford.edu pub cstr reports cs tr 72 288 CS TR 72 288.pdf ref lcf Refend Mathlogic stub Category Logic in computer science Category Proof assistants es LCF ... more details
wiktionary functionFunction may refer to Diatonic function , a term in music theory Function E 40 song , a 2012 song by American rapper E 40 featuring YG rapper YG , iAmSu & Problem Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. An organised event such as a party or meeting See also Functionalism disambiguation Function hall Functional disambiguation Functionality in polymer chemistry see Structural unit Functor disambiguation bg bs Funkcija vor ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko id Fungsi it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon nn Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
In mathematics, S function may refer to sigmoid function Schur polynomials In physics, it may refer to Action physics action functional mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
In computer science , Programming Computable Functions , ref group note PCF is a programming language for computable functions, based on LCF, Scott s logic of computable functions harv Plotkin 1977 . Programming Computable Functions is used by harv Mitchell 1996 . It is also referred to as Programming with Computable Functions or Programming language for Computable Functions . ref or PCF , is a type system typed Functional programming functional language introduced by Gordon Plotkin in 1977. It is based on the Logic of Computable Functions LCF by Dana Scott . It can be considered as an extended version of the typed lambda calculus or a simplified version of modern typed functional languages such as ML programming language ML . A fully abstract model for PCF was first given by Robin Milner Milner 1977 . However, since Milner s model was essentially based on the syntax of PCF it was considered less than satisfactory Ong, 1995 . The first two fully abstract models not employing syntax were formulated during the 1990s. These models are based on game semantics Hyland and Ong, 2000 Abramsky, Jagadeesan, and Malacaria, 2000 and Kripke logical relations O Hearn and Riecke, 1995 . For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effectively presentable fully abstract model could exist, since the question of program equivalence in the finitary fragment of PCF is not decidable. Syntax The types of PCF are inductively defined as nat is a type For types and , there is a type A context is a list of pairs x , where x is a variable name and is a type, such that no variable name is duplicated. One then defines typing judgments of terms in context in the usual way for the following syntactical constructs Variables if x is part of a context , then x Application of a term of type to a term of type abstraction The Y combinator Y fixed point combinator ... more details
Infobox file format name Computable Document Format logo File WolframCDFLogoSmall.png icon iconcaption screenshot caption extension .cdf mime application cdf typecode uniform type com.wolfram.cdf magic owner Wolfram Research released Start date 2011 07 21 Start date YYYY mm dd df yes latest release version latest release date Start date and age YYYY mm dd df yes genre container for contained by extended from extended to standard free url http www.wolfram.com cdf Computable Document Format Computable Document Format CDF is an electronic document format ref http www.telegraph.co.uk technology news 8561619 Wolfram Alpha creator plans to delete the PDF.html Wolfram Alpha Createor plans to delete the PDF The Telegraph UK ref designed to allow easy authoring ref http www.pcworld.com businesscenter article 236202 wolfram makes data interactive.html Wolfram makes data interactive PC World ref of dynamically generated interactive content. It was created by Wolfram Research . Features Computable document format supports GUI elements such as sliders, menus and buttons. Content is updated using embedded computation in response to GUI interaction. Contents can include formatted text, tables, images, sounds and animations. CDF supports Mathematica typesetting and technical notation. ref http www.zdnet.com blog btl wolfram launches new document format meet cdf 52917 Wolfram Launches new document format. Meet CDF ZDNet ref Paginated layout, structured drill down layout and slide show mode are supported. Styles can be controlled using a cascading style sheet. Reading CDF files can be read using a proprietary CDF Player with a restrictive license, which can be downloaded free of charge from Wolfram Research. In contrast to static formats such as PDF and pre generated interactive content provided ... Computable Document Format has been used in electronic books by Pearson Education , ref http www.schoollibraryjournal.com ... Digital press de Computable Document Format ja pt Computable Document Format ru ... more details
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally vulnerable to guess and determine attacks, well chosen bitwise transposition mathematics transposition ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
Computable general equilibrium CGE models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors. CGE models are also referred to as AGE applied general equilibrium models. Overview A CGE model consists of a equations describing model variables and b a database usually very detailed consistent with the model equations. The equations tend to be Neo classical economics neo classical in spirit, often assuming cost minimizing behaviour by producers, average cost pricing, and household demands based on optimizing behaviour. However, most CGE models conform only loosely to the theoretical general equilibrium paradigm. For example, they may allow for non market clearing, especially for labour unemployment or for commodities inventories imperfect competition e.g., monopoly pricing demands not influenced by price e.g., government demands a range of taxes externalities, such as pollution A CGE model database consists of tables of transaction values, showing, for example, the value of coal used by the iron industry. Usually the database is presented as an input output analysis input output table or as a social accounting matrix . In either case, it covers the whole economy of a country or even the whole world , and distinguishes a number of sectors, commodities, primary factors and perhaps types of household. elasticities dimensionless parameters that capture behavioural response. For example, export demand elasticities specify by how much export volumes might fall if export prices went up. Other elasticities may belong to the Constant Elasticity of Substitution class. Amongst these are Armington ... Computable General Equilibrium CGE in GAMS, Microcomputers in Policy Research, vol.5, International ... Decision Making in Australia and the Development of Computable General Equilibrium Modelling, CoPS IMPACT ... workingpapers SCEPA 20Working 20Paper 202008 1 20Kahn.pdf Debunking the Myths of Computable ... more details
Recursive function may refer to Recursion computer science , a procedure or subroutine, implemented in a programming language, whose implementation references itself A total computablefunction , a function which is defined for all possible inputs See also recursive function , defined from a particular formal model of computable functions using primitive recursion and the operator Recurrence relation , in mathematics, an equation that defines a sequence recursively disambig Category Recursion cs Rekurzivn funkce ru ... more details
In recursion theory computability theory , a semicomputable function is a partial function math f mathbb Q rightarrow mathbb R math that can be approximated either from above or from below by a computablefunction . More precisely a partial function math f mathbb Q rightarrow mathbb R math is upper semicomputable , meaning it can be approximated from above, if there exists a computablefunction math phi x,k mathbb Q times mathbb N rightarrow mathbb Q math , where math x math is the desired parameter for math f x math and math k math is the level of approximation, such that math lim k rightarrow infty phi x,k f x math math forall k in mathbb N phi x,k 1 leq phi x,k math Completely analogous a partial function math f mathbb Q rightarrow mathbb R math is lower semicomputable iff math f x math is upper semicomputable or equivalently if there exists a computablefunction math phi x,k math such that math lim k rightarrow infty phi x,k f x math math forall k in mathbb N phi x,k 1 geq phi x,k math If a partial function is both upper and lower semicomputable it is called computable. See also recursion theory computability theory References Ming Li and Paul Vit nyi, An Introduction to Kolmogorov Complexity and Its Applications , pp 37&ndash 38, Springer, 1997. DEFAULTSORT Semicomputable Function Category Mathematical logic mathlogic stub ... more details
Non recursive function might refer to Recursion computer science a procedure or subroutine, implemented in a programming language, whose implementation references itself recursive function , defined from a particular formal model of computable functions using primitive recursion and the operator Computablefunction , or total recursive function, a functioncomputable by a turing machine Turing machine See also Recursive function disambig cs Rekurzivn funkce ... more details
domain theory domains to model programming language concepts such as function types. It turns out that restricting ourselves to the set of computablefunction s is not sufficient either if the programming ...In computer science , a function type also arrow type or exponential is the type of a variable computer science variable or parameter computer science parameter to which a function computer science function has or can be assigned or the result type of a higher order function returning a function. A function type depends on the type of the parameters and the result type of the function it, or more accurately the unapplied type constructor · &rarr · , is a higher kinded type . In theoretical settings and languages where functions are defined in curried form , such as the simply typed lambda calculus , a function type depends on exactly two types, the domain of a function domain A and the range of a function range B . Here a function type is often denoted A &rarr B , following ... exponentially many set theoretic function s mapping A to B . Programming languages The following table summarized the syntax used for function types in several programming languages, including an example type signature for the higher order function composition computer science function composition function class wikitable colspan 2 Language Notation Example type signature rowspan 3 With first class function s, br parametric polymorphism C Sharp programming language C code Func &alpha sub 1 ... code def compose A, B, C f B C, g A B A C code Without first class function s, br parametric polymorphism ... C , one should note that the type of the function code compose code is actually code Func Func A,B ,Func< B,C ,Func A,C code . Denotational semantics The function type in programming languages ... function contains the parallel or function, which cannot be correctly defined in all programming ... First class functionFunction space , set theoretic equivalent References cite book first ... more details
The ramp function is an elementary function elementary unary function unary real function , easily computable as the arithmetic mean mean of its independent variable and its absolute value . This function is applied in engineering e.g., in the theory of Digital signal processing DSP . The name ramp function can be derived by the look of its graph. Definitions Image Ramp function.svg Graph of a function Graph of the ramp function thumb 260px right The ramp function math R x mathbb R rightarrow mathbb R math may be defined analytically in several ways. Possible definitions are math R x begin cases x, & x ge 0 0, & x 0 end cases math The mean of a straight line with unity gradient and its modulus math R x frac x x 2 math this can be derived by noting the following definition of math operatorname max a,b math , math operatorname max a,b frac a b a b 2 math for which math a x math and math b 0 math The Heaviside step function multiplied by a straight line with unity gradient math R left x right xH left x right math The convolution of the Heaviside step function with itself math R left x right H left x right H left x right math The integral of the Heaviside step function math R x int infty x H xi , mathrm d xi math Analytic properties Non negativity In the whole domain of a function domain the function is non negative, so its absolute value is itself, i.e. math forall x in mathbb R R x geqslant 0 math and math left R left x right right R left x right math Proof by the mean of definition 2 it is non negative in the I. quarter, and zero in the II. so everywhere it is non negative. Derivative Its derivative is the Heaviside function math R x H x mathrm if x ne 0 math Ugyanis ha x 0, akkor R x 0 konstans, teh t ezen a tartom nyon sup sup R x 0 konstans deriv ltja 0 ami megegyezik a Heaviside f ggv nnyel. ha x 0, akkor R x x, teh t ezen a tartom nyon sup sup R x 1 a val s sz mokon ... invariance Every iterated function of the ramp mapping is itself, as br center math R left R left ... more details
In computability theory , the Ackermann function , named after Wilhelm Ackermann , is one of the simplest and earliest discovered examples of a total function total computablefunction that is not Primitive recursive function primitive recursive . All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive ... 0315 0860 79 90024 7 ref with discovering total function total computablefunction s termed simply recursive in some references that are not primitive recursive function primitive recursive . Sudan published the lesser known Sudan function , then shortly afterwards and independently, in 1928, Ackermann published his function math varphi , math . Ackermann s three argument function, math varphi ... . , math Aside from its historic role as a total computable but not primitive recursive function, Ackermann s original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann s function that are specifically designed for that purpose ... and so is a computablefunction , grows faster than any primitive recursive function and is therefore ... 133 doi 10.1007 BF01459088 ref of his function which had three nonnegative integer arguments , many authors modified it to suit various purposes, so that today the Ackermann function may refer to any of numerous variants of the original function. One common version, the two argument Ackermann P ter function , is defined as follows for nonnegative integers m and n math A m, n begin cases n 1 & mbox ..., Solomon Marcus and Ionel Tevy journal Historia Math. title The first example of a recursive function ... Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann ... Robinson later developed a two variable version of the Ackermann function that became preferred by many ... function math varphi m, n, p , math is defined recursion recursively as follows for nonnegative integers ... more details
In mathematics, omega function or function may refer to Pearson Cunningham function Lambert W function Wright Omega function Mathematical disambiguation ... more details
Barnes G function , related to the Gamma function Meijer G function , a generalization of the hypergeometric function Siegel G function , a class of functions in transcendence theory mathdab ... more details
group G given by a finite presentation has computable Dehn function Dehn n , then the word problem ... function whose binary representation is computable in time math O sqrt 4 f n math by a Turing ...In the mathematical subject of geometric group theory , a Dehn function , named after Max Dehn , is an optimal function associated to a finitely presented group finite group presentation which bounds the area ... name Gersten . The growth type of the Dehn function is a quasi isometry quasi isometry invariant of a finitely presented group . The Dehn function of a finitely presented group is also closely connected ... for groups word problem if and only if the Dehn function for a finitely presented group finite presentation of this group is recursive function recursive see Theorem 2.1 in ref name Gersten . The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function ... curve of that surface. History The idea of an isoperimetric function for a finitely presented ... consequence of this fact is that for this presentation the Dehn function satisfies Dehn n ... notion of an isoperimetric function and a Dehn function as it is used today appeared in late 1980s ... if the Dehn function of this group is equivalent to the function f n n . Gromov s proof was in large ... cells in a van Kampen diagram over with boundary cycle labelled by w . Isoperimetric function An isoperimetric function for a finite presentation is a monotone non decreasing function math f mathbb ... 1 in G , then Area w &le f w , where w is the length of the word w . Dehn function Then the Dehn function of a finite presentation is defined as math rm Dehn n max rm Area w w 1 text in G, w le n, w text freely reduced . math Equivalently, Dehn n is the smallest isoperimetric function for , that is, Dehn n is an isoperimetric function for and for any other isoperimetric function ... more details
In mathematics, by sigma function one can mean one of the following The Divisor function sum of divisors function sub a sub n , an arithmetic function Weierstrass sigma function , related to elliptic functions Kronecker s sigma function Rado s sigma function, see busy beaver See also sigmoid function . mathdab de Teilersumme fr Fonction sigma ... more details
Function test or functional test can refer to Thyroid function tests Liver function tests Lung function test Functional testing disambig ... more details
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