Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Recursion

Page semi-protected
From Wikipedia, the free encyclopedia
Process of repeating items in a self-similar way
For other uses, seeRecursion (disambiguation).

A visual form of recursion known as theDroste effect. The woman in this image holds an object that contains a smaller image of her holding an identical object, which in turn contains a smaller image of herself holding an identical object, and so forth. 1904 Drostecocoa tin, designed by Jan Misset.

Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself.[1] Recursion is used in a variety of disciplines ranging fromlinguistics tologic. The most common application of recursion is inmathematics andcomputer science, where afunction being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur.

A process that exhibits recursion isrecursive.Video feedback displays recursive images, as does aninfinity mirror.

Formal definitions

Ouroboros, an ancient symbol depicting a serpent or dragon eating its own tail

In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:

  • A simplebase case (or cases) — a terminating scenario that does not use recursion to produce an answer
  • Arecursive step — a set of rules that reduces all successive cases toward the base case.

For example, the following is a recursive definition of a person'sancestor. One's ancestor is either:

  • One's parent (base case),or
  • One's parent's ancestor (recursive step).

TheFibonacci sequence is another classic example of recursion:

Fib(0) = 0 as base case 1,
Fib(1) = 1 as base case 2,
For allintegersn > 1,Fib(n) = Fib(n − 1) + Fib(n − 2).

Many mathematical axioms are based upon recursive rules. For example, the formal definition of thenatural numbers by thePeano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."[2] By this base case and recursive rule, one can generate the set of all natural numbers.

Other recursively defined mathematical objects includefactorials,functions (e.g.,recurrence relations),sets (e.g.,Cantor ternary set), andfractals.

There are various more tongue-in-cheek definitions of recursion; seerecursive humor.

Informal definition

Sourdough starter being stirred into flour to produce sourdough: the recipe calls for some sourdough left over from the last time the same recipe was made.

Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be 'recursive'.[3]

To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, while the running of a procedure involves actually following the rules and performing the steps.

Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.

When a procedure is thus defined, this immediately creates the possibility of an endless loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the procedure can complete.

Even if it is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of the procedure; this requires some administration as to how far various simultaneous instances of the procedures have progressed. For this reason, recursive definitions are very rare in everyday situations.

In language

LinguistNoam Chomsky, among many others, has argued that the lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as the time available to utter one), can be explained as the consequence of recursion in natural language.[4][5]

This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence:Dorothy thinks witches are dangerous, in which the sentencewitches are dangerous occurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion.[clarification needed]

This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length:Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another.[6] Over the years, languages in general have proved amenable to this kind of analysis.

The generally accepted idea that recursion is an essential property of human language has been challenged byDaniel Everett on the basis of his claims about thePirahã language. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this.[7] Literaryself-reference can in any case be argued to be different in kind from mathematical or logical recursion.[8]

Recursion plays a crucial role not only in syntax, but also innatural language semantics. The wordand, for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that is suitably flexible,and is typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one.[9]

Arecursive grammar is aformal grammar that contains recursiveproduction rules.[10]

Recursive humor

Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving acircular definition orself-reference, in which the putative recursive step does not get closer to a base case, but instead leads to aninfinite regress. It is not unusual for such books to include a joke entry in their glossary along the lines of:

Recursion,see Recursion.[11]

A variation is found on page 269 in theindex of some editions ofBrian Kernighan andDennis Ritchie's bookThe C Programming Language; the index entry recursively refers to itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found inLet's talk Lisp by Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and inSoftware Tools by Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears inThe UNIX Programming Environment by Kernighan and Pike. It did not appear in the first edition ofThe C Programming Language. The joke is part of thefunctional programming folklore and was already widespread in the functional programming community before the publication of the aforementioned books.[12][13]

A plaque commemorates the Toronto Recursive History Project of Toronto's Recursive History.

Another joke is that "To understand recursion, you must understand recursion."[11] In the English-language version of the Google web search engine, when a search for "recursion" is made, the site suggests "Did you mean:recursion."[14] An alternative form is the following, fromAndrew Plotkin:"If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer toDouglas Hofstadter than you are; then ask him or her what recursion is."

Recursive acronyms are other examples of recursive humor.PHP, for example, stands for "PHP Hypertext Preprocessor",WINE stands for "WINE Is Not an Emulator",GNU stands for "GNU's not Unix", andSPARQL denotes the "SPARQL Protocol and RDF Query Language".

In mathematics

TheSierpiński triangle—a confined recursion of triangles that form a fractal

Recursively defined sets

Main article:Recursive definition

Example: the natural numbers

See also:Closure (mathematics)

The canonical example of a recursively defined set is given by thenatural numbers:

0 is inN{\displaystyle \mathbb {N} }
ifn is inN{\displaystyle \mathbb {N} }, thenn + 1 is inN{\displaystyle \mathbb {N} }
The set of natural numbers is the smallest set satisfying the previous two properties.

In mathematical logic, thePeano axioms (or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematicianRichard Dedekind and by the Italian mathematicianGiuseppe Peano. The Peano Axioms define the natural numbers referring to a recursive successor function and addition and multiplication as recursive functions.

Example: Proof procedure

Another interesting example is the set of all "provable" propositions in anaxiomatic system that are defined in terms of aproof procedure which is inductively (or recursively) defined as follows:

  • If a proposition is an axiom, it is a provable proposition.
  • If a proposition can be derived from true reachable propositions by means of inference rules, it is a provable proposition.
  • The set of provable propositions is the smallest set of propositions satisfying these conditions.

Finite subdivision rules

Main article:Finite subdivision rule

Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with a collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating theCantor set is a subdivision rule, as isbarycentric subdivision.

Functional recursion

Afunction may be recursively defined in terms of itself. A familiar example is theFibonacci number sequence:F(n) =F(n − 1) +F(n − 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this caseF(0) = 0 andF(1) = 1.

Proofs involving recursive definitions

Applying the standard technique ofproof by cases to recursively defined sets or functions, as in the preceding sections, yieldsstructural induction — a powerful generalization ofmathematical induction widely used to derive proofs inmathematical logic and computer science.

Recursive optimization

Dynamic programming is an approach tooptimization that restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is theBellman equation, which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step).

The recursion theorem

Inset theory, this is a theorem guaranteeing that recursively defined functions exist. Given a setX, an elementa ofX and a functionf:X →X, the theorem states that there is a unique functionF:N→X{\displaystyle F:\mathbb {N} \to X} (whereN{\displaystyle \mathbb {N} } denotes the set of natural numbers including zero) such that

F(0)=a{\displaystyle F(0)=a}
F(n+1)=f(F(n)){\displaystyle F(n+1)=f(F(n))}

for any natural numbern.

Dedekind was the first to pose the problem of unique definition of set-theoretical functions onN{\displaystyle \mathbb {N} } by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?"[15]

Proof of uniqueness

Take two functionsF:N→X{\displaystyle F:\mathbb {N} \to X} andG:N→X{\displaystyle G:\mathbb {N} \to X} such that:

F(0)=a{\displaystyle F(0)=a}
G(0)=a{\displaystyle G(0)=a}
F(n+1)=f(F(n)){\displaystyle F(n+1)=f(F(n))}
G(n+1)=f(G(n)){\displaystyle G(n+1)=f(G(n))}

wherea is an element ofX.

It can be proved bymathematical induction thatF(n) =G(n) for all natural numbersn:

Base Case:F(0) =a =G(0) so the equality holds forn = 0.
Inductive Step: SupposeF(k) =G(k) for somek∈N{\displaystyle k\in \mathbb {N} }. ThenF(k + 1) =f(F(k)) =f(G(k)) =G(k + 1).
HenceF(k) =G(k) impliesF(k + 1) =G(k + 1).

By induction,F(n) =G(n) for alln∈N{\displaystyle n\in \mathbb {N} }.

In computer science

Main article:Recursion (computer science)

A common method of simplification is to divide a problem into subproblems of the same type. As acomputer programming technique, this is calleddivide and conquer and is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach isdynamic programming. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.

A classic example of recursion is the definition of thefactorial function, given here inPython code:

deffactorial(n):ifn>0:returnn*factorial(n-1)else:return1

The function calls itself recursively on a smaller version of the input(n - 1) and multiplies the result of the recursive call byn, until reaching thebase case, analogously to the mathematical definition of factorial.

Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to the problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is inparsers for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program.

Recurrence relations are equations which define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain a non-recursive definition (e.g., aclosed-form expression).

Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity of instructions. The main disadvantage is that the memory usage of recursive algorithms may grow very quickly, rendering them impractical for larger instances.

In biology

Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example isRomanesco broccoli.[16]

In the social sciences

Authors use the concept ofrecursivity to foreground the situation in which specificallysocial scientists find themselves when producing knowledge about the world they are always already part of.[17][18] According to Audrey Alejandro, “as social scientists, the recursivity of our condition deals with the fact that we are both subjects (as discourses are the medium through which we analyse) and objects of the academic discourses we produce (as we are social agents belonging to the world we analyse).”[19] From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise ofreflexive efforts:

we are socialised into discourses and dispositions produced by the socio-political order we aim to challenge, a socio-political order that we may, therefore, reproduce unconsciously while aiming to do the contrary. The recursivity of our situation as scholars – and, more precisely, the fact that the dispositional tools we use to produce knowledge about the world are themselves produced by this world – both evinces the vital necessity of implementing reflexivity in practice and poses the main challenge in doing so.

— Audrey Alejandro,Alejandro (2021)

In business

Further information:Management cybernetics

Recursion is sometimes referred to inmanagement science as the process of iterating through levels of abstraction in large business entities.[20] A common example is the recursive nature of managementhierarchies, ranging fromline management tosenior management viamiddle management. It also encompasses the larger issue ofcapital structure incorporate governance.[21]

In art

Recursive dolls: the original set ofMatryoshka dolls byZvyozdochkin andMalyutin, 1892
Front face ofGiotto'sStefaneschi Triptych, 1320, recursively contains an image of itself (held up by the kneeling figure in the central panel).
See also:Mathematics and art andInfinity mirror

TheMatryoshka doll is a physical artistic example of the recursive concept.[22]

Recursion has been used in paintings sinceGiotto'sStefaneschi Triptych, made in 1320. Its central panel contains the kneeling figure of Cardinal Stefaneschi, holding up the triptych itself as an offering.[23][24] This practice is more generally known as theDroste effect, an example of theMise en abyme technique.

M. C. Escher'sPrint Gallery (1956) is a print which depicts a distorted city containing a gallery whichrecursively contains the picture, and soad infinitum.[25]

In culture

The filmInception has colloquialized the appending of the suffix-ception to a noun to jokingly indicate the recursion of something.[26]

See also

References

  1. ^Causey, Robert L. (2006).Logic, sets, and recursion (2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers.ISBN 0-7637-3784-4.OCLC 62093042.
  2. ^"Peano axioms | mathematics".Encyclopedia Britannica. Retrieved2019-10-24.
  3. ^"Definition of RECURSIVE".www.merriam-webster.com. Retrieved2019-10-24.
  4. ^Pinker, Steven (1994).The Language Instinct. William Morrow.
  5. ^Pinker, Steven; Jackendoff, Ray (2005). "The faculty of language: What's so special about it?".Cognition.95 (2):201–236.CiteSeerX 10.1.1.116.7784.doi:10.1016/j.cognition.2004.08.004.PMID 15694646.S2CID 1599505.
  6. ^Nordquist, Richard."What Is Recursion in English Grammar?".ThoughtCo. Retrieved2019-10-24.
  7. ^Nevins, Andrew; Pesetsky, David; Rodrigues, Cilene (2009)."Evidence and argumentation: A reply to Everett (2009)"(PDF).Language.85 (3):671–681.doi:10.1353/lan.0.0140.S2CID 16915455. Archived fromthe original(PDF) on 2012-01-06.
  8. ^Drucker, Thomas (4 January 2008).Perspectives on the History of Mathematical Logic. Springer Science & Business Media. p. 110.ISBN 978-0-8176-4768-1.
  9. ^Barbara Partee and Mats Rooth. 1983. In Rainer Bäuerle et al.,Meaning, Use, and Interpretation of Language. Reprinted in Paul Portner and Barbara Partee, eds. 2002.Formal Semantics: The Essential Readings. Blackwell.
  10. ^Nederhof, Mark-Jan; Satta, Giorgio (2002), "Parsing Non-recursive Context-free Grammars",Proceedings of the 40th Annual Meeting on Association for Computational Linguistics (ACL '02), Stroudsburg, PA, USA: Association for Computational Linguistics, pp. 112–119,doi:10.3115/1073083.1073104.
  11. ^abHunter, David (2011).Essentials of Discrete Mathematics. Jones and Bartlett. p. 494.ISBN 9781449604424.
  12. ^Shaffer, Eric."CS 173:Discrete Structures"(PDF). University of Illinois at Urbana-Champaign. Retrieved7 July 2023.
  13. ^"Introduction to Computer Science and Programming in C; Session 8: September 25, 2008"(PDF). Columbia University. Retrieved7 July 2023.
  14. ^"recursion - Google Search".www.google.com. Retrieved2019-10-24.
  15. ^A. Kanamori, "In Praise of Replacement", pp.50--52. Bulletin of Symbolic Logic, vol. 18, no. 1 (2012). Accessed 21 August 2023.
  16. ^"Picture of the Day: Fractal Cauliflower". 28 December 2012. Retrieved19 April 2020.
  17. ^Bourdieu, Pierre (1992). "Double Bind et Conversion".Pour Une Anthropologie Réflexive. Paris: Le Seuil.
  18. ^Giddens, Anthony (1987).Social Theory and Modern Sociology. Polity Press.
  19. ^Alejandro, Audrey (2021)."Reflexive discourse analysis: A methodology for the practice of reflexivity".European Journal of International Relations.27 (1): 171.doi:10.1177/1354066120969789.ISSN 1354-0661.S2CID 229461433.
  20. ^Riding, Allan; Haines, George H.; Thomas, Roland (1994)."The Canadian Small Business–Bank Interface: A Recursive Model".Entrepreneurship Theory and Practice.18 (4). SAGE Journals:5–24.doi:10.1177/104225879401800401.
  21. ^Beer, Stafford (1972).Brain Of The Firm. John Wiley & Sons.ISBN 978-0471948391.
  22. ^Tang, Daisy (March 2013)."CS240 -- Lecture Notes: Recursion". California State Polytechnic University, Pomona. Archived fromthe original on 2018-03-17. Retrieved24 September 2015.More examples of recursion: Russian Matryoshka dolls. Each doll is made of solid wood or is hollow and contains another Matryoshka doll inside it.
  23. ^"Giotto di Bondone and assistants: Stefaneschi triptych". The Vatican. Retrieved16 September 2015.
  24. ^Svozil, Karl (2018).Physical (A)Causality: Determinism, Randomness and Uncaused Events. Springer. p. 12.ISBN 9783319708157.
  25. ^Cooper, Jonathan (5 September 2007)."Art and Mathematics". Retrieved5 July 2020.
  26. ^"-ception – The Rice University Neologisms Database". Rice University.Archived from the original on July 5, 2017. RetrievedDecember 23, 2016.

Bibliography

External links

Wikimedia Commons has media related toRecursion.
Look uprecursion orrecursivity in Wiktionary, the free dictionary.
Characteristics
Iterated function
system
Strange attractor
L-system
Escape-time
fractals
Rendering techniques
Random fractals
People
Other
General
Theorems (list)
 and paradoxes
Logics
Traditional
Propositional
Predicate
Set theory
Types ofsets
Maps and cardinality
Set theories
Formal systems (list),
language and syntax
Example axiomatic
systems (list)
Proof theory
Model theory
Computability theory
Related
Authority control databasesEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1319339463"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp