Making the Recursion article link to itself will not display correctly, and is considered to break WP ... A visual form of recursion known as the Droste effect . The woman in this image is holding an object ... image of herself holding the same object, and so forth. Recursion is the process of repeating items ... with each other the nested images that occur are a form of infinite recursion. The term has ... common application of recursion is in mathematics and computer science , in which it refers to a method ... a process 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 ... . The parents 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 ... 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 ... sets , and especially fractal s. Recursion in language Linguist Noam Chomsky theorizes that unlimited ..., to obtain a very verbose sentence. The idea that recursion is an essential property of human ... , in which he hypothesizes 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 ..., see Kaan et al. 2002 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 ... language which he claims lacks recursion is in fact finite. He likens it to the finite game of chess ... discovered throughout history. Recursion in plain English Recursion is the process a procedure goes ... 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
ref improve date September 2011 compu prog stub In computer programming, Walther recursion is a method of analysing recursive functions that can determine if the function is definitely terminating, given finite inputs. It allows a more natural style of expressing computation than simply using primitive recursive functions. Walther recursion does not solve the halting problem , as there are still classes of programs that will terminate, but which Walther recursion cannot prove to terminate. Walther recursion may be used in Total functional programming total functional languages in order to allow a more liberal style of showing primitive recursion. 1 http ttic.uchicago.edu dmcallester walther.ps Category Recursion ... 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 ... 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 computer science References references Category Theory of computation Category Recursion fr r cursion ... 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 Mathematical induction 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 ... that f is well defined. The idea is that one extends the sequence arbitrarily, using the recursion ... recursion are the intuitionistic equivalents of the axiom of dependent choice s. ref cite book authors ... ref References references DEFAULTSORT Bar Recursion Category Recursion ... more details
In computer science , polymorphic recursion refers to a recursion computer science recursive parametric polymorphism parametrically polymorph function computer science function where the type parameter changes with each recursive invocation made instead of staying constant. Type inference for polymorphic recursion is equivalent to semi unification and thefore undecidable problem undecidable and requires the use of a semi algorithm or programmer supplied type annotation s. sfn Henglein 1993 Example Consider the following nested datatype source lang haskell data Nested a a Nested a Epsilon infixr 5 nested 1 2,3,4 4,5 , 7 , 8,9 Epsilon source A length function defined over this datatype will be polymorphically recursive, as the type of the argument changes from code Nested a code to code Nested a code in the recursive call source lang haskell length Nested a Int length Epsilon 0 length xs 1 length xs source In Haskell programming language Haskell , unlike for most other functions definitions, the type signature cannot be omitted. Notes reflist 2 Further reading cite journal first Fritz last Henglein authorlink Fritz Henglein title Type inference with polymorphic recursion journal ACM Transactions on Programming Languages and Systems volume 15 issue 2 year 1993 doi 10.1145 169701.169692 ref harv cite journal first Alan last Mycroft authorlink Alan Mycroft title Polymorphic type schemes and recursive definitions journal International Symposium on Programming , Toulouse, France volume 167 series Lecture Notes in Computer Science pages 217&ndash 228 year 1984 doi 10.1007 3 540 12925 1 41 cite journal first1 Martin last1 Emms authorlink1 Martin Emms first2 Hans last2 Lei authorlink2 ... 2003 . Practical Type Inference for Polymorphic Recursion an Implementation in Haskell . Journal ... Standard ML with polymorphic recursion by Hans Lei , University of Munich DEFAULTSORT Polymorphic Recursion Category Polymorphism computer science Category Recursion 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
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 ... recursion is preferred for LALR parsers because it results in lower stack usage than right recursion ... Recursion in Polynomial Time. journal ACM SIGPLAN Notices volume 41 issue 5 pages 46 54 url http ... to a complete parsing algorithm to accommodate indirect as well as direct left recursion in polynomial ... 77442 6 12 series Lecture Notes in Computer Science isbn 978 3 540 77441 9 ref Removing left recursion Removing immediate left recursion The general algorithm to remove immediate left recursion follows ... 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 journal 6th Applied ... Expr , ,Expr , ,Int , ,String math This could be rewritten to avoid left recursion as math ... epsilon , , ,Expr math Removing indirect left recursion If the grammar has no math epsilon math ... algorithm may be applied to remove indirect left recursion Arrange the nonterminals in some ... recursion for math A i math Pitfalls The above transformations remove left recursion by creating a right recursive grammar but this changes the associativity of our rules. Left recursion makes left associativity right recursion makes right associativity. Example We start out with a grammar math ... rightarrow Expr , ,Int math After having applied standard transformations to remove left recursion ... 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
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 Category Recursion compu prog stub ... 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
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
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 ... 1 . The first recursion theorem in particular, part 1 states that there is a set F such that &Phi ... 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 0 1, n 1 n 1 n . This recursion is a primitive recursion because it computes the next value n 1 of the function based on the value of n and the previous value n of the function. On the other hand, the function Fib n , which returns the n th Fibonacci number , is defined with the recursion equations Fib 0 0, Fib 1 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 ... recursion. In general, a function f is defined by course of values recursion if there is a fixed ... the initial value of the recursion. The function h might test its first argument to provide ... an appropriate G del numbering is used . Equivalence to primitive recursion In order to convert a definition by course of values recursion into a primitive recursion, an auxiliary helper function ... math f using primitive recursion, first define the auxiliary course of values function that should ... recursion because math bar f n 1 math is obtained by appending to math bar f n math the new element ..., 1989, Classical Recursion Theory , North Holland second edition, 1999. Category Computability theory Category Recursion ... 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 math alpha math 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 langle H,J,K rangle in R 0 wedge H subseteq B wedge J subseteq alpha B , math math K subseteq alpha A leftrightarrow exists H exists J langle H,J,K rangle 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 ... such that math scriptstyle A alpha B alpha C math . References Gerald Sacks, Higher recursion 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
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 ... to an unfortunate notational confusion where some recursion theorists reverse the direction of the forcing ... forcing construction. References Melvin Fitting 1981 , Fundamentals of generalized recursion theory . Piergiorgio Odifreddi 1999 , Classical Recursion Theory , v. 2. Category Computability theory ... more details
about recursive approaches to solving problems recursion in computer science acronyms Recursive acronym Computer related examples Programming paradigms Recursion in computer science is a method where the solution ... edition 2nd page 427 ref blockquote The power of recursion evidently lies in the possibility of defining ... Prentice Hall year 1976 page 126 ref blockquote Most computer programming languages support recursion ... programming languages do not define any looping constructs but rely solely on recursion to repeatedly ... language Logo programming language and relying heavily on recursion. Recursive data types Many computer program s must process or generate an arbitrarily large quantity of data. Recursion is one ... of lazy evaluation lazy programming languages, and can be preferable to recursion when the desired ... with a corecursive program. Structural versus generative recursion Some authors classify recursion ... be described without using recursion by making use of the typical looping constructs found in imperative ... is a particularly poor example of recursion, because each time the function is executed on a number ... is much greater, being a linear recursion. The recursion in condition 4, shows that OneNumber back ... recursion. Implemented in the Java or the C programming language source lang java public static ... this clause merely stops one additional else recursion from occurring return fibonacciOf times 1, oneNumberBack ... for. Recursion is used in this algorithm because with each pass a new array is created by cutting ... data, toFind, mid 1, end source Recursive data structures structural recursion main Recursive data type An important application of recursion in computer science is in defining dynamic data structures ... blockquote The examples in this section illustrate what is known as structural recursion . This term ... recursion. That is, the recursions in a function s body consume some immediate piece of a given ... can be implemented using recursion. Note that because there are two self referencing pointers ... more details
one is reducible to the other. In recursion theory, these equivalence classes are called the degrees ... time decidable. This is acceptable in the study of recursion theory, which is interested in theoretical ... . Unpublished 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 ... 1 paperback , ISBN 0 07 053522 1 G Sacks, 1990. Higher Recursion Theory , Springer Verlag. ISBN ... more details
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 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 Category Recursion cs Rekurzivn funkce ru ... 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
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 Kashima Rinkai Railway , Japanese railway company disambig ... 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 bs Odlu ivost vor hr Odlu ivost ... 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 computer hardware and software 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 Endless loop . See Loop, endless Loop, endless . See Endless loop Recursion . See Recursion See also The Devil s Dictionary Wickedictionary Category Satirical books ... more details
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