of recursion known as the Droste effect . The woman in this image is holding an object which contains ... holding the same object, and so forth. Recursion is the process of repeating items in a Self similarity ... other the nested images that occur are a form of infinite recursion. The term has a variety of meanings ... of recursion is in mathematics and computer science , in which it refers to a method of defining ... of repeating objects in a self similar way. Formal definitions of recursion File Screenshot Recursion via vlc.png thumb Recursion in a screen recording program, where the smaller window contains a snapshot ... of one s ancestors are also one s ancestors recursion step . The Fibonacci sequence is a classic example of recursion Fib 0 is 0 base case Fib 1 is 1 base case For all integers n 1 Fib n is Fib n 1 Fib n 2 recursive definition A convenient mental model of recursion defines the recursive object whether ... natural numbers A more humorous illustration goes To understand recursion, you must first understand recursion. Or perhaps more accurate is the following, from Andrew Plotkin If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are then ask him or her what recursion is. Recursively defined mathematical objects include function mathematics function s, set mathematics sets , and especially fractal s. Recursion ... sentence. The idea that recursion is an essential property of human language as Chomsky suggests ... that cultural factors made recursion unnecessary in the development of the Pirah language . This concept, which challenges Chomsky s idea that recursion is the only trait which differentiates human ... Recursion in linguistics enables discrete infinity by embedding phrases within phrases of the same type in a hierarchical structure. Without recursion, language does not have discrete infinity and cannot ... that language must have discrete infinity, and that the Pirah language which he claims lacks recursion ... more details
Recursion theorem can refer to The Recursionrecursion theorem in set theory Kleene s recursion theorem , also called the fixed point theorem, in computability theory disambig ... more details
Mutual recursion is a form of recursion where two mathematical or computational functions are defined in terms of each other. ref Manuel Rubio S nchez, Jaime Urquiza Fuentes,Crist bal Pareja Flores 2002 , A Gentle Introduction to Mutual Recursion , Proceedings of the 13th annual conference on Innovation and technology in computer science education, June 30 July 2, 2008, Madrid, Spain. ref For instance, consider two functions code even? code and code odd? code defined as follows function even? number Integer if number 0 then return true else return odd? abs number 1 function odd? number Integer if number 0 then return false else return even? abs number 1 These functions are based on the realization that the question is three even is equivalent to the question, is two odd , which is the same as asking if 1 is even or 0 is odd. In the end, the answer is no, as realized by the function code odd? code . The code abs code function is used to ensure that code number code decrements towards zero even when it starts off as a negative value. Mutual recursion is very common in the functional programming style, and is often used for programs written in Lisp programming language LISP , Scheme programming language Scheme , ML programming language ML , and similar programming language languages . In languages such as Prolog programming language Prolog , mutual recursion is almost unavoidable. Some programming styles discourage mutual recursion, claiming that it can be confusing to distinguish the conditions which will return an answer from the conditions that would allow the code to run forever without producing an answer. It is usually possible to turn two mutually recursive functions into a single ... Any mutual recursion can be converted to direct recursion using procedural inlining. ref http delivery.acm.org ... 82873082&CFTOKEN 54657523 On the Conversion of Indirect to Direct Recursion by Owen Kaser, C. R ... are an example of a pair of integer sequences defined in a mutually recursive manner. See also Recursion ... more details
Bar recursion is a generalized form of recursion developed by Spector in his 1962 paper ref cite book author C. Spector chapter Provably recursive functionals of analysis a consistency proof of analysis by an extension of principles in current intuitionistic mathematics editor F. D. E. Dekker title Recursive Function Theory Proc. Symoposia in Pure Mathematics volume 5 pages 1 27 year 1962 publisher American Mathematical Society ref . It is related to bar induction in the same fashion that primitive recursion is related to ordinary induction , or transfinite recursion is related to transfinite induction . Technical Definition Let V , R , and O be type theory types , and i be any natural number, representing a sequence of parameters taken from V . Then the function sequence f of functions f sub n sub from V sup i n sup R to O is defined by bar recursion from the functions L sub n sub R O and B with B sub n sub V sup i n sup R x V sup n sup R O if f sub n sub V sup i n sup r L sub n sub r for any r long enough that L sub n k sub on any extension of r equals L sub n sub . Asusming L is a continuous sequence, there must be such r , because a continuous function can use only finitely much data. f sub n sub p B sub n sub p , x V f sub n 1 sub cat p , x for any p in V sup i n sup R . Here cat is the concatenation function, sending p , x to the sequence which starts with p , and has x as its last term. This definition is based on the one in ref cite journal authors Mart n Escard , Paulo Oliva title Selection functions, Bar recursion, and Backwards Induction journal Math. Struct. in Comp.Science url http www.cs.bham.ac.uk mhe papers selection escardo oliva.pdf ref . Provided that for every .... The idea is that one extends the sequence arbitrarily, using the recursion term B to determine the effect ... to invoke a bar induction. The principles of bar induction and bar recursion are the intuitionistic ... references DEFAULTSORT Bar Recursion Category Recursion ... more details
Context date December 2007 In computer science , left recursion is a special case of recursion . In terms ... ref Immediate left recursion Immediate left recursion occurs in rules of the form math A to A alpha ... would fall into infinite recursion when trying to parse a grammar which contains this rule. Indirect left recursion Indirect left recursion in its simplest form could be defined as math A to B alpha mid ..., A n math , indirect left recursion can be defined as being of the form math A 0 to A 1 alpha 1 mid .... Accommodating left recursion in top down parsing A formal grammar that contains left recursion cannot ... to a Weak equivalence weakly equivalent right recursive form. In contrast, left recursion is preferred for LALR parsers because it results in lower stack usage than right recursion . However, more .... Hafiz date 2006 title A New Top Down Parsing Algorithm to Accommodate Ambiguity and Left Recursion ... parsing algorithm to accommodate indirect as well as direct left recursion in polynomial time, and to generate ... left recursion Removing immediate left recursion The general algorithm to remove immediate left recursion follows. Several improvements to this method have been made, including the ones described in Removing Left Recursion from Context Free Grammars , written by Robert C. Moore. ref name Moore2000 cite journal last Moore first Robert C. title Removing Left Recursion from Context Free Grammars ... recursion as math Expr rightarrow Int ,ExprRest , ,String ,ExprRest math math ExprRest rightarrow ... math ExprRest rightarrow epsilon , , ,Expr math Removing indirect left recursion If the grammar has ... A , this general algorithm may be applied to remove indirect left recursion Arrange the nonterminals ... remove direct left recursion for math A i math Pitfalls The above transformations remove left recursion ... recursion makes left associativity right recursion makes right associativity. Example We start out with a grammar ... left recursion, we have the following grammar math Expr rightarrow Term Expr math math Expr ... more details
Multiple issues orphan October 2009 wikify October 2009 Expert subject Computer science date March 2011 Polymorphic code is code capable of changing its memory picture best looked at as its type . A common example of Type polymorphism is a polymorphic class which is capable of creating different extended versions of itself. Example syntaxhighlight lang java5 class Stack extends Object class ArrayStack extends Stack class LinkedListStack extends Stack syntaxhighlight Consider the above Stack class if Stack has a factory method or constructor capable of dynamically instantiating all of the above three classes the Stack would be called a polymorphic class. Recursion is code that calls upon itself. Consider the poor implementation of fibonacci numbers below syntaxhighlight lang java5 long fib long x if x 2 return x else return fib x 1 fib x 2 syntaxhighlight Finally polymorphic recursion would be a class that recursively defined itself in a polymorphic changing manner. This is most easily accomplished through generics, and defining your memory picture to be different because you encapsulate different things. Example syntaxhighlight lang java5 class BootstrappingStack E Recursive polymorphic datatype BootstrappingStack BootstrappingStack E stackOfStacks Anything concrete for holding the E s we are under contract to hold Stack E StackOfElements syntaxhighlight In some Strongly typed programming language strongly typed languages, such as C , polymorphic recursion is difficult to implement because all data type s are determined during compilation. Polymorphic recursion causes data types to be infinitely nested, forcing the compiler to throw an error. DEFAULTSORT Polymorphic Recursion Category Polymorphism computer science Category Object oriented programming ... more details
In recursive function theory , double recursion is an extension of primitive recursion which allows the definition of non primitive recursive functions like the Ackermann function . Raphael M. Robinson called functions of two natural number variables G n , x double recursive with respect to given functions , if G 0, x is a given function of x . G n 1, 0 is obtained by substitution from the function G n , and given functions. G n 1, x 1 is obtained by substitution from G n 1, x , the function G n , and given functions. ref cite journal author Raphael M. Robinson title Recursion and Double Recursion journal Bulletin of the American Mathematical Society year 1948 volume 54 pages 987 93 url http projecteuclid.org DPubS?verb Display&version 1.0&service UI&handle euclid.bams 1183512393&page record doi 10.1090 S0002 9904 1948 09121 2 ref Robinson goes on to provide a specific double recursive function originally defined by R zsa P ter G 0, x x 1 G n 1, 0 G n , 1 G n 1, x 1 G n , G n 1, x where the given functions are primitive recursive, but G is not primitive recursive. In fact, this is precisely the function now known as the Ackermann function . See also Primitive recursion Ackermann function References reflist mathlogic stub Category Computability theory ... more details
The Panjer recursion is an algorithm to compute the probability distribution of a compound random variable math S sum i 1 N X i. , math . where both math N , math and math X i , math are random variable s and of special types. In more general cases the distribution of S is a compound distribution . The recursion for the special cases considered was introduced in a paper of Harry Panjer ref cite journal last Panjer first Harry H. year 1981 title Recursive evaluation of a family of compound distributions. journal ASTIN Bulletin volume 12 issue 1 pages 22 26 publisher International Actuarial Association url http www.casact.org library astin vol12no1 22.pdf format PDF ref . It is heavily used in actuarial science . Preliminaries We are interested in the compound random variable math S sum i 1 N X i , math where math N , math and math X i , math fulfill the following preconditions. Claim size distribution We assume the math X i , math to be i.i.d. and independent of math N , math . Furthermore the math X i , math have to be distributed on a lattice math h mathbb N 0 , math with latticewidth math h 0 , math . math f k P X i hk . , math Claim number distribution The number of claims N is a random variable , which is said to have a claim number distribution , and which can take values 0, 1, 2, .... etc.. For the Panjer recursion , the probability distribution of N has to be a member of the Panjer class , otherwise known as the a,b,0 class of distributions . This class consists of all counting random variables which fulfill the following relation math P N k p k a frac b k cdot p k 1 , k ge 1. , math for some a and b which fulfill math a b ge 0 , math . The initial value math p 0 , math is determined such that math sum k 0 infty p k 1. , math The Panjer recursion makes use of this iterative ... the table in a,b,0 class of distributions . Recursion The algorithm now gives a recursion to compute ... distributions Aggregate modeling Panjer s recursive method.htm Panjer recursion and the distributions ... more details
unreferenced date July 2008 Recursion 2004 is Tony Ballantyne s first novel. It is in the science fiction genre and follows three separate characters and their stories in a futuristic dystopia . Of high import to the storyline is the concept of the Singularity , a point in the near future when the evolution of technology reaches such a speed that thinking machines outpace human minds, a point beyond which we cannot possibly predict what will happen and that of von Neumann machine s, self replicating robots that use available raw resources to make copies of themselves. The implication is that a system of such machines just one would suffice that is allowed to reproduce unchecked will in short order grey goo devour entire biosphere s, perhaps even entire solar systems or galaxies if these von Neumann machines are equipped with propulsion devices. An interesting corollary to this is that if two systems of von Neumann machines are battling for resources, the winner will not be decided by which group has the most members the victorious system will be the one that reproduces faster. Also brought up by Ballantyne is the intriguing possibility of being copied and inserted into a simulated reality simulation . Bostrom s tripartition tends to suggest that we ourselves are living in a simulated universe. Moments exist in Recursion in which the main character is rather unsure if he is in reality or a simulation &mdash certain computer bug bugs in the program make themselves apparent, such as blank spaces appearing between buildings and the ground, causing him to question reality. Eventually it becomes apparent that he has been copied multiple times, and inserted into various simulations, and that what he thinks of as his identity is truly not the original, but merely one of many copies. It can be argued, however, that perfect copies, at the moment of creation, are identical and indistinguishable from their originals. A copy is the original, so to speak. Category 2004 novels Category ... more details
In computing, Recursion termination is when certain conditions are met and the recursive algorithm ceases calling itself and begins to return values. ref http www.itl.nist.gov div897 sqg dads HTML recursiontrm.html ref This happens only if with every recursive call the recursive algorithm changes its state and moves towards the base case. Cases that satisfy the definition without being defined in terms of the definition itself are called base cases. They are small enough to solve directly. ref cite book title Recursion Lecture, Introduction to Computer Science pg. 3 url http www.cdf.toronto.edu csc148h winter stg lectures w3 1 m Recursion.pdf ref Examples Fibonacci function The Fibonacci function fibonacci n , which takes integer n n 0 as input, has three conditions 1. if n is 0, returns 0. br 2. if n is 1, returns 1. br 3. otherwise, return fibonacci n 1 fibonacci n 2 This recursive function terminates if either conditions 1 or 2 are satisfied. We see that the function s recursive call reduces the value of n by passing n 1 or n 2 in the function ensuring that n reaches either condition 1 or 2. C C Example ref cite book title An Introduction to the Imperative Part of C url http www.doc.ic.ac.uk wjk C Intro RobMillerL8.html ref pre int factorial int number else if number 0 return 1 else return number factorial number 1 pre Here we see that in the recursive call, the number passed in the recursive step is reduced by 1. This again ensures that the number will at some point reduce to 0 which in turn terminates the recursive algorithm. References reflist 2 External links http www.cs.princeton.edu courses archive spr05 cos126 lectures 07.pdf Princeton university An introduction to computer science in the context of scientific, engineering, and commercial applications http www.cdf.toronto.edu csc148h winter stg lectures w3 1 m Recursion.pdf University of Toronto Introduction to Computer Science Category Computer programming compu prog stub ... more details
Levinson recursion or Levinson Durbin recursion is a procedure in linear algebra to recursion recursively calculate the solution to an equation involving a Toeplitz matrix . The algorithm runs in Big O notation n sup 2 sup time, which is a strong improvement over Gauss Jordan elimination , which runs in n sup 3 sup . Newer algorithms, called asymptotically fast or sometimes superfast Toeplitz algorithms, can solve in n log sup p sup n for various p e.g. p 2, ref http wwwmaths.anu.edu.au brent pd rpb143tr.pdf ref ref http etd.gsu.edu theses available etd 04182008 174330 unrestricted kimitei symon k 200804.pdf ref p 3 ref http web.archive.org web 20070418074240 http saaz.cs.gsu.edu papers sfast.pdf ref . Levinson recursion remains popular for several reasons for one, it is relatively easy to understand in comparison for another, it can be faster than a superfast algorithm for small n usually n 256 . ref http www.math.niu.edu ammar papers amgr88.pdf ref The Levinson Durbin algorithm was proposed first by Norman Levinson in 1947, improved by J. Durbin in 1960, and subsequently improved to 4 n sup 2 sup and then 3 n sup 2 sup multiplications by W. F. Trench and S. Zohar, respectively. Other methods to process data include Schur decomposition and Cholesky decomposition . In comparison to these, Levinson recursion particularly Split Levinson recursion tends to be faster computationally, but more sensitive to computational inaccuracies like round off error s. Derivation Background Matrix equations follow the form math vec y mathbf M vec x. math The Levinson Durbin algorithm may ... for the second step, where they are used to build the solution desired. Levinson Durbin recursion ... block Toeplitz, the Levinson recursion can be derived in much the same way by regarding the block ... recursion Linear prediction Notes reflist References Defining sources Levinson, N. 1947 . The Wiener ... . 2.2. Levinson Durbin Recursion. Linear Predictive Modelling of Speech Constraints and Line Spectrum ... more details
No footnotes date April 2009 In computability theory , course of values recursion is a technique for defining number theoretic function s by Recursion computer science recursion . In a definition of a function f by course of values recursion, the value of f n 1 is computed from the sequence math langle f 1 ,f 2 , ldots,f n rangle math . The fact that such definitions can be converted into definitions using a simpler form of recursion is often used to prove that functions defined by course of values recursion are primitive recursive . This article uses the convention that the natural number s are the set 1,2,3,4,... . Definition and examples The factorial function n is recursively defined by the rules 1 1, n 1 n 1 n . This recursion is a primitive recursion because it only requires the previous value n in order to compute the next value n 1 . On the other hand, the function Fib n , which returns the n th Fibonacci number , is defined with the recursion equations Fib 1 1, Fib 2 1, Fib n 2 Fib n 1 Fib n . In order to compute Fib n 2 , the last two values of the Fib function are required. Finally, consider the function g defined with the recursion equations g 1 2, math g n 1 sum i 1 n g ... Fib and g are examples of functions defined by course of values recursion. In general, a function f is defined by course of values recursion if there is a fixed function h such that for all n , math ... , math and thus the initial value s of the recursion must be hard coded into h . Equivalence to primitive recursion In order to convert a definition by course of values recursion into a primitive recursion ... rangle math . To redefine f using primitive recursion, first define the auxiliary course of values ... recursion because math bar f n 1 math is the concatenation of math bar f n math and the one ... . Thus f can be defined without a course of values recursion in any setting where it is possible to handle ... Logic , A K Peters. Odifreddi, P.G., 1989, Classical Recursion Theory , North Holland second ... more details
In computability theory , Kleene s recursion theorems are a pair of fundamental results about the application ... Cole Kleene Stephen Kleene in 1938. The two recursion theorems can be applied to construct fixed ... quines , and to construct functions defined via recursive definition s. The second recursion theorem The second recursion theorem is closely related to definitions of computable functions using recursion . Because it is better known than the first recursion theorem, it is sometimes called just the recursion theorem . Its statement refers to the standard G del numbering &phi of the partial recursive ... recursion theorem . If F is a total computable function then there is an index e such that math varphi ... to the original recursion equations. The recursion theorem establishes the existence of such a fixed .... The theorem does not guarantee that e is an index for the smallest fixed point of the recursion equations this is the role of the first recursion theorem described below. Proof of the second recursion theorem The proof uses a particular total computable function h , defined as follows ... recursion theorem is that any partial computable function can guess an index for itself. This follows ... varphi F e math for all e is called fixed point free . The recursion theorem shows that no computable ... a fixed point free function is 0 , the degree of the halting problem. The first recursion theorem The first recursion theorem is related to fixed points determined by enumeration operators, which ... obtained if the enumeration operator itself is computable. First recursion theorem . The following .... Example Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator. Consider the recursion equations ... it defines a value of f n 1 . The first recursion theorem in particular, part 1 states that there is a set ... more details
In recursion theory , recursion theory is a generalisation of recursion theory to subsets of admissible ordinal s math alpha math . An admissible ordinal is closed under math Sigma 1 L alpha math functions. Admissible ordinals are models of Kripke Platek set theory . In what follows math alpha math is considered to be fixed. The objects of study in math alpha math recursion are subsets of math alpha math . A is said to be math alpha math recursively enumerable if it is math Sigma 1 math definable over math L alpha math . A is recursive if both A and math alpha A math its complement in math alpha math are recursively enumerable. Members of math L alpha math are called math alpha math finite and play a similar role to the finite numbers in classical recursion theory. We say R is a reduction procedure if it is recursively enumerable and every member of R is of the form math langle H,J,K rangle math where H , J , K are all finite. A is said to be recusive in B if there exist math R 0,R 1 math reduction procedures such that math K subseteq A leftrightarrow exists H exists J H,J,K in R 0 wedge H subseteq B wedge J subseteq alpha B , math math K subseteq alpha A leftrightarrow exists H exists J H,J,K in R 1 wedge H subseteq B wedge J subseteq alpha B . math If A is recursive in B this is written math scriptstyle A le alpha B math . By this definition A is recursive in math scriptstyle varnothing math the empty set if and only if A is recursive. However A being recursive in B is not equivalent to A being math Sigma 1 L alpha B math . We say A is regular if math forall beta in alpha A cap beta in L alpha math or in other words if every initial portion of A is finite. Results in math alpha math recursion Shore s splitting theorem Let A be math alpha math recursively enumerable and regular. There exist math alpha math recursively enumerable math B 0,B 1 math such that math ... C math . References Gerald Sacks, Higher recursion theory , Springer Verlag, 1990 Robert Soare, Recursively ... more details
Forcing in recursion theory is a modification of Paul Cohen mathematician Paul Cohen s original set theory set theoretic technique of forcing set theory forcing to deal with the effective concerns in recursion theory . Conceptually the two techniques are quite similar, in both one attempts to build generic set generic objects intuitively objects that are somehow typical by meeting dense sets. Also both techniques are elegantly described as a relation customarily denoted math Vdash math between conditions and sentences. However, where set theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore some of the more difficult machinery used in set theoretic forcing can be eliminated or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different recursion theoretic and set theoretic forcing are properly regarded as an application of the same technique to different classes of formulas. Terminology In this article we use the following terminology. real an element of math 2 omega math . In other words a function that maps each integer to either 0 or 1. string an element of math 2 omega math . In other words a finite approximation to a real. notion of forcing A notion of forcing is a set math P math and a partial order on math P math , math succ P math with a greatest element math 0 P math . condition An element in a notion of forcing. We say a condition math p math is stronger than a condition math q math just when math q succ P p math . math ... where some recursion theorists reverse the direction of the forcing partial order exchanging ... Melvin Fitting 1981 , Fundamentals of generalized recursion theory . Piergiorgio Odifreddi 1999 , Classical Recursion Theory , v. 2. Category Computability theory ... more details
In the field of recursion theory , index sets describe classes of partial recursive function s, specifically they give all indices of functions in that class according to a fixed enumeration of partial recursive functions a G del number ing . Definition Fix an enumeration of all partial recursive functions, or equivalently of recursively enumerable sets whereby the e th such set is math W e math and the e th such function whose domain is math W e math is math phi e math . Let math mathcal A math be a class of partial recursive functions. If math A x phi x in mathcal A math then math A math is the index set of math mathcal A math . In general math A math is an index set if for every math x,y in mathbb N math with math phi x simeq phi y math i.e. they index the same function , we have math x in A leftrightarrow y in A math . Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index. Index sets and Rice s theorem Most index sets are incomputable non recursive aside from two trivial exceptions. This is stated in Rice s theorem blockquote Let math mathcal C math be a class of partial recursive functions with index set math C math . Then math C math is recursive if and only if math C math is empty, or math C math is all of math omega math . blockquote where math omega math is the set of natural numbers , including zero . Rice s theorem says any nontrivial property of partial recursive functions is undecidable ref name odifreddi cite book title Classical Recursion Theory, Volume 1 author Odifreddi, P. G. page 151 ref Notes reflist References cite book title Classical Recursion Theory, Volume 1 author Odifreddi, P. G. publisher Elsevier year 1992 isbn 0 444 89483 7 pages 668 cite book title Theory of Recursive Functions and Effective Computability author Rogers Jr., Hartley publisher MIT Press isbn 0 262 68052 1 pages 482 year 1987 Category Computability theory ... more details
one is reducible to the other. In recursion theory, these equivalence classes are called the degrees .... This is acceptable in the study of recursion theory, which is interested in theoretical computability ... preprint. P. Odifreddi, 1989. Classical Recursion Theory , North Holland. ISBN 0 444 87295 7 P. Odifreddi, 1999. Classical Recursion Theory, Volume II , Elsevier. ISBN 0 444 50205 X E. Post, 1944, Recursively ... , ISBN 0 07 053522 1 G Sacks, 1990. Higher Recursion Theory , Springer Verlag. ISBN 3 540 19305 7 ... more details
otheruses Wiktionarypar recursive Recursive may refer to Recursion , the technique of functions calling themselves Recursive function, a total computable function Recursive 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 Primitive recursive function Recursion computer science Recursive definition Recursive disambig Category Recursion fr R cursif zh ... 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 computable function , 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 cs Rekurzivn funkce ... 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 Computable function , or total recursive function, a function computable by a turing machine Turing machine See also Recursive disambiguation disambig cs Rekurzivn funkce ... more details
Wiktionary The word decidable may refer to Decidable language Decidability logic for the equivalent in mathematical logic G del s incompleteness theorem , a theorem on the indecidability of languages consisting of true statements in mathematical logic. Recursive set , a decidable set in recursion theory See also List of undecidable problems disambig hr Odlu ivost ... more details
Italic text KRT may stand for The Kanawha Valley Regional Transportation Authority in West Virginia, United States. Khartoum International Airport , with IATA code KRT. Kleene s recursion theorem in mathematical logic. Kanoo Rapid Transport KRT is a courier service based in the Middle east. disambig ... more details
The Computer Contradictionary by Stan Kelly Bootle is a satirical list of definitions of computer industry terms. It is an example of cynical lexicography in the tradition of Ambrose Bierce s The Devil s Dictionary . It was originally published as The Devil s DP Dictionary , in New York, by McGraw Hill in 1981, ISBN 0070340226. DP stood for data processing , a term formerly used to describe the software and hardware industries. It was re edited under the new title in Boston by MIT Press , in 1995, ISBN 0262611120. Kelly Bootle adds in the Guide The meaning of an entry should always be ascertained before consulting this dictionary . Examples Infinite loop . See Loop, infinite Loop, infinite . See Infinite loop Recursion . See Recursion Category Satirical books ... more details
Unreferenced date December 2009 Cyclic history is a theory which dictates that the major forces that motivate human actions return in a cycle. Among these forces are religion spirituality , politics , science , philosophy , curiosity , and creativity . Religion recurs whenever a new sect reaches a large population. Christianity peaked three times around the 2nd century AD, when the core of believers gained political power in the Middle Ages , when the Church controlled almost all knowledge in Europe during the Protestant Reformation reformation , where the religion split and the many branches modernized themselves. The theory of cyclic history was considered in A. E. van Vogt s 1950 science fiction novel, The Voyage of the Space Beagle . Recursion of historical cycles For more articles about the concept of recursion of historical cycles see The ricorso of Giambattista Vico Major works and their reception Giambattista Vico . The Decline of the West by Oswald Spengler . DEFAULTSORT Cyclic History Category Historiography Category Theories of history ... more details
Encyclopedia
top
