Tailrecursion Tupper s self referential formula Turtles all the way down Viable System Model div ...Making the Recursion article link to itself will not display correctly, and is considered to break WP ... page. Other uses No footnotes date February 2010 Image Droste.jpg thumb A visual form of recursion ... an identical 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 ... 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 ... can generate the set of all 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 ... fractal s. Recursion in language Linguist Noam Chomsky theorizes that unlimited extension ..., liquidated her with a pail of water. The idea that recursion is an essential property of human language ... , 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 ... 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
Wiktionary Recursion is the process of repeating items in a self similar way. Recursion may also refer to Recursion computer science , a method where the solution to a problem depends on solutions to smaller instances of the same problem Recursion novel Recursion novel 2004 , a science fiction novel by Tony Ballantyne Nyaya Sanskrit, literally recursion , the Hindu philosophical school of logic disambig ru ... more details
Tailrecursion References references External links http www.cs.umd.edu class fall2002 cmsc430 lec4.pdf ...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 ... created symbol is often called the tail , or the rest . As an example, consider the rule math Expr rightarrow 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 ... 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 ... Sudoku Puzzle ref Any mutual recursion can be converted to direct recursion by inlining the code ... to Direct Recursion by Owen Kaser, C. R. Ramakrishnan, and Shaunak Pawagi at State University ... in a mutually recursive manner. See also Recursion computer science References references Category Theory of computation Category Recursion fr r cursion mutuelle ja ... 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 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
In computer science , polymorphic recursion also referred to as Milner&ndash Mycroft typability or the Milner&ndash Mycroft calculus 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. See also Higher ranked polymorphism Notes reflist 2 Further reading cite journal first Lambert last Meertens authorlink Lambert Meertens title Incremental polymorphic type checking in B journal ACM Symposium on Principles of Programming Languages POPL , Austin, Texas year ... with polymorphic recursion journal ACM Transactions on Programming Languages and Systems volume 15 ... recursion journal ACM Transactions on Programming Languages and Systems date April 1993 ... Type Inference for Polymorphic Recursion an Implementation in Haskell . Journal of Universal ... Examples Needing Polymorphic Recursion journal Electronic Notes in Theoretical Computer Science ... recursion by Hans Lei , University of Munich DEFAULTSORT Polymorphic Recursion Category Polymorphism computer science Category Recursion Category Object oriented programming ... 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 James 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 ... recursion defines the n sup th sup forward vector , denoted math vec f n math , as the vector of length ..., the Levinson recursion can be derived in much the same way by regarding the block Toeplitz matrix ... of systems MIMO systems or cyclo stationary signals. See also Split Levinson recursion ..., T. 2004 . 2.2. Levinson Durbin Recursion. Linear Predictive Modelling of Speech Constraints and Line ... more details
all use this arrangement for this reason. T tail configuration also allows high performance aerodynamics and excellent glide ratio as the empennage is not affected by wing slipstream. Therefore the T tail configuration is especially popular on glider sailplane glider s. The tail surfaces are mounted .... This is why the T tail arrangement is also commonly found on airliner s with rear mounted engines ... Douglas MD 80 , McDonnell Douglas MD 90 , all used the T tail for this reason. Drawbacks ... composite materials are used, this inevitably makes it heavier as well. The T tail configuration can ... Flight 261 was directly attributed to lax maintenance due to the complexity of the T tail. Because of concerns about being able to clear the tail, the first high speed aircraft with a T tail, the Lockheed ... is an aircraft with this configuration see cruciform tail . See also Cruciform tail Pelikan tail Twin tail V tail References Reflist Aircraft components aviation lists Category Aircraft tail configurations es Cola en T nl T staart pt T tail ... more details
Other uses Refimprove article July 2008 date July 2008 This image may not be adequate a free image of the tail ... Cub Stalks Tail.jpg thumb A lion s tail The tail is the section at the rear end of an animal s body ... s, as well as snail s and slug s, have tail like appendages that are sometimes referred to as tails ... to the tail are given the adjective caudal . Function File Alopex lagopus IMG 9019.JPG thumb Alopex lagopus arctic fox sleeping with its tail wrapped as a blanket. Animal tails are used in a variety ..., such as New World monkey s and opossum s, have what are known as prehensile tail s, which are adapted ... them to escape from predator s, which are either distracted by the wriggling detached tail, or left with only the tail while the rest of the lizard flees. Tails cast in this manner generally grow back ... tail feathers play an important role in courtship Courtship in the animal kingdom courtship displays. The extra stiff tail feathers of other species, including woodpecker s and woodcreeper s, allow them to brace themselves firmly against tree trunks. The tails of grazing animals, such as tail ... is a short, erect tail. Hares, rabbits, and deer have scuts. p A prehensile tail is a tail that can ... s have a tail that measures about one sixth of the size of the embryo itself. ref cite news first ... accessdate 2009 04 28 language ref As the embryo develops into a fetus , the tail is absorbed by the growing body. The developmental tail is thus a human vestigial structure. ref cite news first last coauthors title Human tail caudal appendage tethered cord date February 1, 2008 publisher url http ... 2009 04 28 language ref ref cite news first last coauthors title The human tail causing tethered cervical ...? date February 2011 a child is born with a soft tail , which contains no vertebrae, but only blood ... j.1525 1470.1995.tb00174.x True Tail in a Newborn , Pediatric Dermatology 12 3 , pp 263 266, 2008 ref ... in West Bengal, a state in India, is famous because of his convert 33 cm adj on tail. It is not believed ... more details
wiktionary tailTail is the section at the rear end of an animal s body, a distinct, flexible appendage to the torso. Tail s may also refer to Tail Unix , a Unix program used to display the last few lines of a file Tails album Tails album , an album by Lisa Loeb Miles Tails Prower , a character in the Sonic the Hedgehog video games and comics The Obverse and reverse reverse side of a coin Tails or tailcoat or tailsuit, a type of coat suit used for evening dress white tie formalwear Tail, the empennage of an aircraft Tail, the luminous matter anti sunwards from a comet Tail, the final batsmen in the Batting order cricket batting order for cricket Tail, one of the extreme ends of a probability density function Tail or fee tail , an obsolescent term in common law T.A.I.L.S. The Amnesic Incognito Live System , a Linux distribution designed for anonymity and privacy, which includes Tor. Tail, a character in the Kaiketsu Zorori movie Terminal amine isotopic labeling of substrates See also Tailing disambiguation Tailrecursion , a type of recursion in computer programming Tail sequence , an unbounded sequence of contiguous ordinals Tail rotor , a small vertical propeller mounted at the rear of a helicopter disambig zh yue Tail de Tail es Cola desambiguaci n fr Tail it Coda pl Ogon ru ... more details
case of tailrecursion . There are some types of problems whose solutions are inherently recursive ... recursionTail recursive functions are functions in which all recursive calls are tail call s and hence ... language control structures like the for and while loops. class wikitable Tailrecursion Augmenting ... 1 return 1 else return n fact n 1 source The significance of tailrecursion is that when making a tail ... that support tailrecursion optimization, tailrecursion saves both space and time. Order of execution ...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 ... s is a string, and tail s is a stream of strings. This is very similar to an inductive definition ... of the data structure namely, via the accessor functions code head code and code tail code and what ... of lazy evaluation lazy programming languages, and can be preferable to recursion when ... criterion that establishes the base case. Structural versus generative recursion Some authors classify recursion as either generative or structural . The distinction is related to where a recursive procedure ... Recursion ref blockquote Thus, the defining characteristic of a structurally recursive function is that the argument ... recursion includes nearly all tree traversals, including XML processing, binary tree creation and search ... as structural recursion. Generative recursion is the alternative blockquote Many well known ... 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
procedure although a tail call in an implementation of quicksort , say, might not . The call site is then said to be in tail position , i.e. at the end of the calling procedure. If a subroutine performs a tail call to itself, it is called tail recursive . This is a special case of recursion computer science recursion . Tail calls are significant because they can be implemented without adding ... is called tail call elimination , or tail call optimization . Traditionally, tail call elimination is optional. However, in functional programming language s, tail call elimination is often guaranteed by the language standard, and this guarantee allows using recursion computer science recursion , in particular tailrecursion, in place of loop computing loop s. In such cases, it is not correct though ... 1 source This program is not written in a tailrecursion style. Now take this Scheme programming language ... cases such as filtering lists and in some languages, full tailrecursion may require a function that was previously ... problems but this is the basic idea. Tailrecursion modulo cons Tailrecursion Modulo jargon modulo cons is a generalization of tailrecursion optimization introduced by David H. D. Warren ref D. H. D ... 7 105 doi 10.1023 A 1010051815785 ref Implementation methods Tailrecursion is important to some high ..., tailrecursion is the most commonly used way and sometimes the only way available of implementing ... tailrecursion in C without growing the stack. Many implementations achieve this by using a device ... The garbage collection ensures that mutual tailrecursion can continue indefinitely. However, this approach ... Inline expansion Leaf subroutine References Wiktionary tailrecursion references FOLDOC DEFAULTSORT ...In computer science , a tail call is a subroutine call that happens inside another procedure as its final ... more, and it can be replaced by the frame of the tail call, modified as appropriate similar to Overlay ... were reached. For tail calls, there is no need to remember the place we are calling from &mdash ... 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
Butterfly tail may refer to V tail , an arrangement of the tail on aircraft . Butterfly tail goldfish , a breed of goldfish . disambig ... 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
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