The primitiverecursive functions are defined using primitive Recursion computer science recursion and function ... and Landweber, 1974 In fact, it is difficult to devise a function that is not primitiverecursive, although some are known see the section on Primitiverecursivefunction Limitations Limitations below ... theory complexity theory . Every primitiverecursivefunction is a general recursivefunction. Definition ... 0 is primitiverecursive. Successor function The 1 ary successor function S , which returns the successor ... Given f , a k ary primitiverecursivefunction, and k m ary primitiverecursive functions g sub 1 ... is primitiverecursive. Primitive recursion Given f , a k ary primitiverecursivefunction, and g , a k 2 ary primitiverecursivefunction, the k 1 ary function h is defined as the primitive recursion of f and g , i.e. the function h is primitiverecursive when math h 0, x 1, ldots, x k f x 1, ldots ... of one function to another function. For example, if g and h are 2 ary primitiverecursive ... recursivefunction is a partial recursivefunction that is defined for every input. Every primitiverecursivefunction is total recursive, but not all total recursive functions are primitiverecursive. The Ackermann function A m , n is a well known example of a total recursivefunction that is not primitive ... is primitiverecursive if and only if there is a natural number m such that the function ... of the arguments of the primitiverecursivefunction. Citation needed date February 2007 An important ... that there is a single computable function f e , n such that For every primitiverecursivefunction ... functionPrimitiverecursive functions tend to correspond very closely with our intuition of what ... provides a total computable function that is not primitiverecursive. A sketch of the proof ... it finds. This function is total and computable by the above , but clearly no primitiverecursivefunction exists that computes it as it differs from each possible primitiverecursivefunction ... more details
Recursivefunction may refer to Recursion computer science , a procedure or subroutine, implemented in a programming language, whose implementation references itself A total computable function , a function which is defined for all possible inputs See also recursivefunction , 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 cs Rekurzivn funkce ... more details
In mathematical logic , the primitiverecursive functionals are a generalization of primitiverecursive functions into higher type theory . They consist of a collection of functions in all pure finite types. The primitiverecursive functionals are important in proof theory and constructive mathematics They are a central part of the Dialectica interpretation of intuitionistic arithmetic developed by Kurt G del . In recursion theory , the primitiverecursive functionals are an example of higher type computability, as primitiverecursive functions are examples of Turing computability. Background Every primitiverecursive functional has a type, which tells what kind of inputs it takes and what kind of output it produces. An object of type 0 is simply a natural number it can also be viewed as a constant function that takes no input and returns an output in the set N of natural numbers. For any two types &sigma and &tau , the type &sigma &rarr &tau represents a function that takes an input of type &sigma and returns an output of type &tau . Thus the function f n n 1 is of type 0&rarr 0. The types 0&rarr 0 &rarr 0 and 0&rarr 0&rarr 0 are different by convention, the notation 0&rarr 0&rarr ... The primitiverecursive functionals are the smallest collection of objects of finite type such that The constant function f n 0 is a primitiverecursive functional The successor function g n n 1 is a primitiverecursive functional For any type &sigma &tau , the functional K x sup &sigma sup , y sup &tau sup x is a primitiverecursive functional For any types &rho , &sigma , &tau , the functional S r sup &rho &rarr &sigma &rarr &tau sup , s sup &rho &rarr &sigma sup , t sup &rho sup r t s t is a primitiverecursive functional For any type &tau , and f of type &tau , and any g of type 0&rarr ... f , g n 1 g n , R f , g n is a primitiverecursive functional References cite book title G del s functional ... as inputs a function f from N to N , and a natural number n , and returns f n . Then A has type 0 ... more details
propositions involving natural number s and any primitiverecursivefunction , including the operations ... connectives The equality symbol , the constant symbol 0 , and the primitiverecursivefunction successor symbol S meaning add one A symbol for each primitiverecursivefunction . The logical axioms ... 0 math math S x S y to x y, math and recursive defining equations for every primitiverecursivefunction ... order arithmetic , the only primitiverecursivefunction s that need to be explicitly axiomatized are addition and multiplication . All other primitiverecursive predicates can be defined using these two primitiverecursive functions and quantification over all natural numbers . Defining primitiverecursivefunction s in this manner is not possible in PRA, because it lacks quantifier s. Logic free ... v . Negation can be expressed as math 1 dot x y 0 math . See also Elementary recursive arithmetic Heyting arithmetic Peano arithmetic Second order arithmetic primitiverecursivefunction References ...Primitiverecursive arithmetic , or PRA , is a quantifier free formalization of the natural numbers . It was first ... a sentence of PRA is just an equation between two terms. In this setting a term is a primitiverecursivefunction of zero or more variables. In 1941 Haskell Curry gave the first such system ref Haskell Curry , http www.jstor.org stable 2371522 A Formalization of Recursive Arithmetic . American Journal ... operation, and F , G , and H are any primitiverecursive functions which may have parameters other than ... , and C are any terms primitiverecursive functions of zero or more variables . Finally, there are symbols for any primitiverecursive functions with corresponding defining equations, as in Skolem ... can be extended to forms of recursion beyond primitive recursion, up to epsilon 0 &epsilon sub ... resolveppn GDZPPN002343355 Logic free formalisations of recursive arithmetic , Mathematica ... numbers can be defined by primitive recursion math begin align P 0 0 quad & quad P S x x x dot 0 x quad ... more details
Non recursivefunction might refer to Recursion computer science a procedure or subroutine, implemented in a programming language, whose implementation references itself recursivefunction , defined from a particular formal model of computable functions using primitive recursion and the operator Computable function , or total recursivefunction, a function computable by a turing machine Turing machine See also Recursive disambiguation disambig cs Rekurzivn funkce ... more details
recursion i.e. without minimisation is the class of primitiverecursive functions . While all primitiverecursive functions are total, this is not true of partial recursive functions for example, the minimisation of the successor function is undefined. The set of total recursive functions is a subset of the partial recursive functions and is a superset of the primitiverecursive functions functions like the Ackermann function can be proven to be total recursive, and not primitive. The first ... f . A consequence of this result is that any recursivefunction can be defined using a single instance of the operator applied to a total primitiverecursivefunction. Minsky 1967 observes as does ...lowercase title recursivefunction In mathematical logic and computer science , the recursive functions are a class of partial function s from natural number s to natural number s which are computable ... partial recursivefunction. The unbounded search operator is not definable by the rules ... form theorem A normal form theorem due to Kleene says that for each k there are primitiverecursive functions math U y math and math T y,e,x 1, ldots,x k math such that for any recursivefunction ... recursivefunction U n, x that correctly interprets the number n and computes the appropriate ... University Press, Cambridge, UK. Cf pp. 70 71. DEFAULTSORT Mu RecursiveFunction Category Computability ... that the recursive functions are precisely the functions that can be computed by Turing machine s. The recursive functions are closely related to primitiverecursivefunction s, and their inductive definition below builds upon that of the primitiverecursive functions. However, not every recursivefunction is a primitiverecursivefunction &mdash the most famous example is the Ackermann function . Other equivalent classes of functions are the lambda recursivefunction &lambda recursive functions and the functions that can be computed by Markov algorithm s. The set of all recursive functions ... more details
otheruses Wiktionarypar recursiveRecursive may refer to Recursion , the technique of functions calling themselves Recursivefunction, a total computable functionRecursive language , a language which is decidable in mathematics, logic and computer science Recursive set , a set which is decidable Recursive acronym , an acronym which refers to itself Recursive filter See also Recursively enumerable language Recursively enumerable set Primitiverecursivefunction Recursion computer science Recursive definition Recursive disambig Category Recursion fr R cursif zh ... more details
wiktionary primitivePrimitive may refer to Anarcho primitivism , an anarchist critique of the origins and progress of civilization Primitive culture , one that lacks major signs of economic development or modernity Noble savage , uncorrupted by the influences of civilization Primitive communism , a pre agrarian form of communism according to Karl Marx and Friedrich Engels Primitive Church, another name for early Christianity Primitive Baptist , a religious movement seeking to retain or restore early Christian practices Primitive phylogenetics , also premative, characteristic of an early stage of development or evolution, cf. Basal phylogenetics basal Mathematics Geometric primitive , the simplest kinds of figures Simple extension Primitive element field theory Primitive element finite field Primitive cell crystallography Primitive polynomial , one of two concepts Primitivefunction or Antiderivative , F &prime f Primitive group Primitive permutation group Computer science Language primitive , the simplest element provided by a programming language Machine code , instructions and data directly understandable by a CPU Primitive data type , a datatype provided by a programming language Geometric primitive , the simplest kinds of figures in computer graphics Cryptographic primitive s, low level cryptographic algorithms frequently used to build computer security systems Art Na ve art , created by untrained artists Neo primitivism , looks to early human history, folk art and non Western ... Primitive , a novel by J. F. Gonzalez Primitive , a novel by Mark Nykanen Music Primitive Cyndi Lauper song , by Cyndi Lauper Primitive The Groupies song , by The Groupies and covered by The Cramps The Primitives , a British indie rock band Primitive album , by Soulfly Primitive Radio Gods , an American alternative rock band Primitive , a song by Annie Lennox, from her album Diva Annie Lennox album ... Primitive ... more details
recursive if there exists a total function total computable function math f such that math ... S var . In other words, the set math var S var is recursive if and only if the indicator function math 1 sub var S var sub is computable function computable . Examples Every finite or cofinite subset ...In computability theory , a Set mathematics set of natural number s is called recursive , computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. A more general class of sets consists of the recursively enumerable set s, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number is in the set the algorithm may give .... A recursive language is a recursive subset of a formal language . The set of G del numbers ... Mathematica and related systems I see G del s incompleteness theorems . Properties If A is a recursive set then the complement set theory complement of A is a recursive set. If A and B are recursive sets then A B , A B and the image of A × B under the Cantor pairing function are recursive sets. A set A is a recursive set if and only if A and the complement set theory complement of A are both recursively enumerable set s. The preimage of a recursive set under a total function total computable function is a recursive set. The image of a computable set under a total computable bijection is computable. A set is recursive if and only if it is at level math &Delta su p 0 b 1 of the arithmetical hierarchy . A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set. The image of a computable set under a nondecreasing total computable function is computable. References Cutland, N. Computability. Cambridge University Press, Cambridge New York, 1980. ISBN 0 521 22384 9 ISBN 0 521 29465 7 Rogers, H. The Theory of Recursive ... more details
In graph theory , a discipline within mathematics, a recursive tree i.e., unordered tree is a non planar labeled rooted tree graph theory tree . A size n recursive tree is labeled by distinct integers 1, 2, ..., n , where the labels are strictly increasing starting at the root labeled 1. Recursive trees are non planar, which means that the children of a particular node are not ordered. E.g. the following two size three recursive trees are the same. pre 1 1 2 3 3 2 pre Recursive trees also appear in the literature under the name Increasing Cayley trees. Properties The number of size n recursive trees is given by math T n n 1 . , math Hence the exponential generating function T z of the sequence T sub n sub is given by math T z sum n ge 1 T n frac z n n log left frac 1 1 z right . math Combinatorically a recursive tree can be interpreted as a root followed by an unordered sequence of recursive trees. Let F denote the family of recursive trees. math F circ frac 1 1 cdot circ times F frac 1 2 cdot circ times F F frac 1 3 cdot circ times F F F cdots circ times exp F , math where math circ math denotes the node labeled by 1, × the Cartesian product and math math the partition product for labeled objects. By translation of the formal description one obtains the differential equation for T z math T z exp T z , math with T 0 0. Bijections There are bijection bijective correspondences between recursive trees of size n and permutation s of size n &minus 1. Applications Recursive trees can be generated using a simple stochastic process. Such random recursive trees are used as simple models for epidemics. References Analytic Combinatorics , Philippe Flajolet and Robert Sedgewick, Cambridge University Press, 2008 Varieties of Increasing Trees , Francois ... of random recursive trees and binary search trees Michael Drmota and Hsien Kuei Hwang, Adv. Appl. Prob., 37, 1 21, 2005. Profiles of random trees Limit theorems for random recursive trees and binary ... more details
Orphan date February 2009 Cleanup date February 2008 Recursive economics is a branch of modern economics , which is based on highly complex mathematical function . The mathematical model model s are based on dynamical system dynamic differential equation s. This approach has been popularised by Robert Lucas, Jr. and Edward Prescott . Books Recursive Methods in Economic Dynamics by Nancy L. Stokey, Robert E. Lucas, Jr., Edward C. Prescott, Harvard 1989 economics stub Category Mathematical economics ... more details
case, and define the value of a function in terms of that value itself, rather than on other values of the function. Such a situation would lead to an infinite regress . Examples of recursive definitions ... that follows the recursive definition. For example, the definition of the natural numbers presented ... holds of all natural numbers Aczel 1978 742 . Form of recursive definitions Most recursive definition ... between a circular definition and a recursive definition is that a recursive definition must ... formulas It is chiefly in logic or computer programming that recursive definitions are found. For example ... K means both are true , so Kpq may mean Connor is a lawyer and Mary likes music. The value of such a recursive ... and Np is a wff, etc. See also Recursive data type s Recursion Mathematical induction References P ... . ISBN 0 763 77206 2 DEFAULTSORT Recursive Definition Category Definition Category Mathematical ... more details
citations missing article date March 2007 Recursive Recycling is a technique where a function, in order to accomplish a task, calls itself with some part of the task or output from a previous step. In municipal solid waste and waste reclamation processing it is the process of extracting and converting materials from recycled materials derived from the previous step until all subsequent levels of output are extracted or used. Example Level 1 Recursive Solid waste or municipal solid waste can be treated, sanitized and separated under steam in a pressure vessel waste autoclave . Following the processing under steam and removal of toxic materials via condensate filtering, usable recyclables are immediately extracted for reuse plastics, ferrous metals, aluminum , glass, wood, etc. . Level 2 Recursive Organic material s from the original waste stream are converted to a fiber using steam at 60 psi and 160 C. The converted organics sanitary fiber is size reduced by 85 and can be used to produce bio fuels using acidic hydrolysis or enzymatic hydrolysis as Ethanol or may be used as Refuse Derived Fuel RDF . Level 3 Recursive After the monosacrides are extracted for distillation , the remaining residue used fiber can be used as a feed stock for electricity production. Level 4 Recursive Finally, the non toxic ash from the combusted fiber can be collected and used as a filler for preparation in super concrete and then reused in combination with similar materials gravel, stones, pottery, glass to form aggregate for construction materials. In true recursive recycling and conservation processing ... delivery of the derivatives. Analyst Commentary The concept of Recursive Recycling has been ... release. Since that pilot commercial facility stopped operating, the concept of Recursive Recycling ... of technologies to achieve full recursive levels has not been accepted. A number of companies ... wtd 679004 679032 679093 ?lang e RecyclingByMaterial DEFAULTSORT Recursive Recycling Category ... more details
In mathematics , specifically set theory , an ordinal number ordinal math alpha math is said to be recursive if there is a recursive set recursive binary relation math R math that well order s a subset of the natural numbers and the order type of that ordering is math alpha math . It is trivial to check that math omega math is recursive, the successor ordinal successor of a recursive ordinal is recursive, and the Set mathematics set of all recursive ordinals is closure mathematics closed downwards. We call the supremum of all recursive ordinals the Church Kleene ordinal and denote it by math omega CK 1 math . Since the recursive relations are parameter parameterized by the natural numbers, the recursive ordinals are also parameterized by the natural numbers. Therefore, there are only countable countably many recursive ordinals. Thus, math omega CK 1 math is countable. The recursive ordinals are exactly the ordinals that have an ordinal notation in Kleene s O Kleene s math mathcal O math . See also Arithmetical hierarchy Large countable ordinals Ordinal notation References Rogers, H. The Theory of Recursive Functions and Effective Computability , 1967. Reprinted 1987, MIT Press, ISBN 0 262 68052 1 paperback , ISBN 0 07 053522 1 Sacks, G. Higher Recursion Theory . Perspectives in mathematical logic, Springer Verlag, 1990. ISBN 0 387 19305 7 Category Set theory Category Computability theory Category Ordinal numbers settheory stub ... more details
Unreferenced date March 2009 A recursive call is a system call that must be completed before the completion of user s SQL statement. Basically, recursive calls are generated by Oracle Database Oracle internal sql statements to maintain changes to tables for internal processing. Reasons for Recursive Calls Recursive calls can be generated due to following reasons Dictionary cache is too small resulting in misses on cache Database Trigger Firing Performing DDL PL SQL blocks containing sql statements Category Computing terminology ... more details
In signal processing , a recursive filter is a type of filter signal processing filter which re uses one or more of its outputs as an input. This feedback typically results in an unending impulse response commonly referred to as infinite impulse response IIR , characterised by either exponential growth exponentially growing , exponential decay decaying , or sinusoid al signal output components. However, a recursive filter does not always have an infinite impulse response. Some implementations of moving average filter are recursive filters but with a Finite impulse response. Examples of recursive filters Kalman filter Category Signal processing electronics stub statistics stub economics stub ... more details
A recursive language in mathematics , logic and computer science is a type of formal language which is also called decidable or Turing decidable . The class of all recursive languages is often called R complexity R , although this name is also used for the class RP complexity RP . This type of language was not defined in the Chomsky hierarchy of Harv Chomsky 1959 . Definitions There are two equivalent major definitions for the concept of a recursive language A recursive formal language is a recursive set recursive subset in the set mathematics set of all possible words over the alphabet of the formal language language . A recursive language is a formal language for which there exists a Turing machine which will, when presented with any finite input literal string string , halt and accept if the string is in the language, and halt and reject otherwise. The Turing machine always halts it is known as a Machine that always halts decider and is said to decide the recursive language. By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs. An undecidable problem is a problem that is not decidable. All recursive languages are also recursively enumerable language recursively enumerable . All regular language regular , context free language context free and context sensitive language context sensitive languages are recursive. Closure properties Recursive languages are closure mathematics closed under the following operations. That is, if L and P are two recursive languages, then the following languages are recursive as well The Kleene star math L math The image L under a Homomorphism Homomorphisms and e free homomorphisms in formal language theory e free homomorphism The concatenation math L circ P math The union math L cup P math The intersection math L cap P math The complement of L The set ... Recursive acronym Formal languages and grammars Category Computability theory Category Formal languages ... more details
Unreferenced date December 2006 Orphan date February 2009 The recursive join is an operation used in relational databases , also sometimes called a fixed point join . It is a compound operation that involves repeating the join SQL join operation, typically accumulating more records each time, until a repetition makes no change to the results as compared to the results of the previous iteration . For example, if a database of family relationships is to be searched, and the record for each person has mother and father fields, a recursive join would be one way to retrieve all of a person s known ancestors first the person s direct parents records would be retrieved, then the parents information would be used to retrieve the grandparents records, and so on until no new records are being found. In this example, as in many real cases, the repetition involves only a single database table, and so is more specifically a recursive self join . Recursive joins can be very time consuming unless optimized through indexing, the addition of extra key fields, or other techniques. Recursive joins are highly characteristic of hierarchical data, and therefore become a serious issue with XML data. In XML, operations such as determining whether one element contains another are extremely common, and the recursive join is perhaps the most obvious way to implement them when the XML data is stored in a relational database. See also Join SQL Join Category Database theory Comp sci stub ... more details
Recursive partitioning is a statistics statistical method for multivariable analysis . ref name isbn0 412 04841 8 cite book author Breiman, Leo title Classification and Regression Trees publisher Chapman & Hall CRC location Boca Raton year 1984 pages isbn 0 412 04841 8 oclc doi ref Recursive partitioning creates a Decision tree learning decision tree that strives to correctly classify members of the population based on several dichotomous dependent variable s. This article focuses on recursive partitioning for medical diagnostic tests, but the technique has far wider applications. See Decision tree learning decision tree . As compared to regression analyses that creates a formula that health care providers can use to calculate the probability that a patient has a disease, recursive partition creates a rule such as If a patient has finding x, y, or z they probably have disease q. A variation is Cox linear recursive partitioning . ref name pmid6501544 Advantages and disadvantages Compared to other multivariable methods, recursive partitioning has advantages and disadvantages. Advantages are Generates clinically more intuitive models that do not require the user to perform calculations. ref name pmid16149128 cite journal author James KE, White RF, Kraemer HC title Repeated split sample validation to assess logistic regression and recursive partitioning an application to the prediction ... analytic techniques advantages and disadvantages of recursive partitioning analysis journal Journal ... KR, Beck JR title Experiments to determine whether recursive partitioning CART or an artificial neural ... modeling and recursive partitioning journal Methods of information in medicine volume 45 issue 1 pages ... recursive partitioning in research of diagnostic tests. ref name pmid15687312 cite journal author Fonarow ... set by recursive partitioning methodology new insights into the relative merit of individual criteria ... pages 588 96 year 1982 pmid 7110205 doi 10.1056 NEJM198209023071004 ref Goldman used recursive partitioning ... more details
refimprove date July 2010 A recursive acronym synonymous with metacronym ref cite web url http www.urbandictionary.com define.php?term metacronym title UrbanDictionary.com Metacronym accessdate 2010 10 24 ref , recursive initialism , and recursive backronym is an acronym that recursion refers to itself in the expression for which it stands. The term was first used in print in April 1986. ref cite web url http www.wordspy.com words recursiveacronym.asp title WordSpy Recursive Acronym accessdate 2008 ... Isn t a Recursive Acronym Wine software Wine Wine Is Not an Emulator ref name wine cite web url http ... Language Zinf Zinf Is Not Freeamp Mutually recursive or otherwise special The GNU Hurd project is named with a mutually recursive acronym Hurd stands for Hird of Unix Replacing Daemon computer software ... mutually recursive acronym Brain stands for Brian Relates Any Independent Node and Brian stands ... of being the first recursive anti acronym. Jini Is Not Initials . It might, however, be more properly ... recent Microsoft XNA XNA , on the other hand, was deliberately designed that way. Most recursive acronyms are recursive on the first letter, which is therefore an arbitrary choice, often selected ... home page . However Yopy YOPY , Your own personal YOPY is recursive on the last letter hence the last letter had to be the same as the first . Non technical examples Recursive acronyms are not limited to computing terminology. For example TIARA TIARA is a recursive acronym ref .EXE magazine, November .... RAS stands for Redundant Acronym Syndrome and is not recursive. The redundancy is in the name RAS ... Multiple Operation Systems, from the video game series Xenosaga . A recursive initialism appeared ..., recursive or otherwise, the may appear to be recursive acronyms, because of mnemonic devices ... Spelling of the Word GENE u A u LOGY accessdate 2010 10 07 ref See also Wiktionary recursive acronym ... Reflist 2 JargonFile DEFAULTSORT Recursive Acronym Category Acronyms Recursive Category Self reference ... more details
Orphan date November 2006 When number generally large number is represented in a finite alphabet set, and it cannot be represented by just one member of the set, Recursive indexing is used. Recursive indexing itself is a method to write the successive differences of the number after extracting the maximum value of the alphabet set from the number, and continuing recursively till the difference falls in the range of the set. Recursive indexing with a 2 letter alphabet is called Unary code . Encoding To encode a number N , keep reducing the maximum element of this set S sub max sub from N and output S max for each such difference, stopping when the number lies in the half closed half open range 0 S sub max sub . Example Let set S 0 1 2 3 4 10 , be a 11 element set, and we have to recursively index the value N 49. According to this method, we need to keep removing 10 from 49, and keep proceeding till we reach a number in the 0 10 range. So the values are 10 N 49 10 39 , 10 N 39 10 29 , 10 N 29 10 19 , 10 N 19 10 9 , 9. Hence the recursively indexed sequence for N 49 with set S , is 10,10,10,10,9. Decoding Keep adding all the elements of the index, stopping when the index value is between inclusive of ends the least and penultimate elements of the set S . Example Continuing from above example we have 10 10 10 10 9 49. Uses This technique is most commonly used in RLE Run Length Encoding systems to encode longer runs than the alphabet sizes permit. References Khalid Sayood, Data Compression 3rd ed, Morgan Kaufmann . Category Coding theory Category Data compression Category Lossless compression algorithms ... more details
wiktionarypar functionFunction may refer to Diatonic function , a term in music theory 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. A formal event such as a party or meeting See also Function hall Functional disambiguation Functionalism disambiguation Functor disambig bs Funkcija vor bg ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko it Funzione lt Funkcija lmo Funziun nl Functie ja no 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
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
In mathematics , logic , and formal system s, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to Intuition knowledge intuition and everyday experience. In an axiomatic theory or other formal system , the role of a primitive notion is analogous to that of axiom . In axiomatic theories, the primitive notions are sometimes said to be defined by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress . Alfred Tarski explained the role of primitive notions as follows When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle not to employ any of the other expression of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions ... the fundamental concept is an example of a primitive notion. As Mary Tiles wrote The definition of set ... of a primitive, undefined, term. As evidence, she quotes Felix Hausdorff A set is formed by the grouping ... system begins with its axiom s, the primitive notions may be forgotten. Susan Haak 1978 wrote, A set of axioms is sometimes said to give an implicit definition of its primitive terms. Examples . In Naive set theory , the empty set is a primitive notion. To assert that it exists would be an implicit axiom . Peano arithmetic , the successor function and the number zero are primitive notions. Euclidean geometry , the primitive notions were discussed by Alessandro Padoa at the International ... Primitive Notion Category Mathematical logic Category Set theory de Grundbegriff es Noci n primitiva ... more details
Unreferenced date May 2010 In field theory mathematics field theory , a branch of mathematics , a primitive polynomial is the minimal polynomial field theory minimal polynomial of a primitive element finite field primitive element of the finite field finite extension field GF p sup m sup . In other words, a polynomial math F X math with coefficients in GF p Z p Z is a primitive polynomial if it has ... having math alpha math as root. In ring theory , the term primitive polynomial is used for a different ... minimal polynomials are irreducible polynomial irreducible , all primitive polynomials are also irreducible. A primitive polynomial must have a non zero constant term, for otherwise it will be divisible by x . Over the field of two elements, x 1 is a primitive polynomial and all other primitive polynomials ... by x 1 . An irreducible polynomial of degree m , F x over GF p for prime p , is a primitive polynomial ... 1. Over GF p sup m sup there are exactly p sup m sup &minus 1 m primitive polynomials of degree m , where is Euler s totient function . The roots of a primitive polynomial all have order p sup m sup &minus 1. Usage Field element representation Primitive polynomials are used in the representation of elements of a finite field . If in GF p sup m sup is a root of a primitive polynomial F x ... , a primitive polynomials f is a polynomial such that x is a generator of the multiplicative group in GF p x f x Pseudo Random bit generation Primitive polynomials define a recurrence relation that can ... shift register with maximum cycle that is 2 sup lfsr length sup 1 is related with primitive polynomial. For example, given the primitive polynomial x sup 10 sup x sup 3 sup 1, we start with a user ... can be repeated to generate 2 sup 10 sup &minus 1 1023 pseudo random bits. In general, for a primitive ... 2 and dividing it by a fixed generator polynomial also over GF 2 see Mathematics of CRC . Primitive ... n sup 1 for a degree n primitive polynomial. Primitive trinomials The most useful kind of primitive ... more details
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