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{{Short description|Process of repeating items in a self-similar way}}
{{Other uses}}
{{pp-vandalism|small=yes}}
<!-- Making the Recursion article link to itself will not display correctly, and is considered to break [[WP:ASTONISH]]. The joke itself is already featured in the "Recursive humor" section. See discussion on the talk page. -->
[[File:Droste Cacao Alcalinise blikje, foto4.JPG|thumb|A visual form of recursion known as the [[Droste 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 Droste [[hot chocolate|cocoa]] tin, designed by Jan Misset.]]
'''Recursion''' occurs when the definition of a concept or process depends on a simpler or previous version of itself.<ref>{{Cite book |last=Causey |first=Robert L. |title=Logic, sets, and recursion |date=2006 |publisher=Jones and Bartlett Publishers |isbn=0-7637-3784-4 |edition=2nd|___location=Sudbury, Mass. |oclc=62093042}}</ref> Recursion is used in a variety of disciplines ranging from [[linguistics]] to [[logic]]. The most common application of recursion is in [[mathematics]] and [[computer science]], where a [[function (mathematics)|function]] 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 is ''recursive''. [[Video feedback]] displays recursive images, as does an [[infinity mirror]].
==Formal definitions==
[[File:Serpiente alquimica.jpg|thumb|[[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:
* {{anchor|base case}}A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer
* {{anchor|recursive step}}A ''recursive 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's ''ancestor''. One's ancestor is either:
*One's parent
*One's parent's ancestor (''recursive step'').
The [[Fibonacci sequence]] is another classic example of recursion:
:{{math|1=Fib(0) = 0}} as base case 1,
:{{math|1=Fib(1) = 1}} as base case 2,
:For all [[integer]]s {{math|''n'' > 1}}, {{math|1=Fib(''n'') = Fib(''n'' − 1) + Fib(''n'' − 2)}}.
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the [[natural number]]s by the [[Peano axioms]] can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."<ref>{{Cite web|url=https://www.britannica.com/science/Peano-axioms|title=Peano axioms {{!}} mathematics|website=Encyclopedia Britannica|language=en|access-date=2019-10-24}}</ref> By this base case and recursive rule, one can generate the set of all natural numbers.
Other recursively defined mathematical objects include [[factorial]]s, [[function (mathematics)|function]]s (e.g., [[recurrence relation]]s), [[set (mathematics)|sets]] (e.g., [[Cantor ternary set]]), and [[fractal]]s.
There are various more tongue-in-cheek definitions of recursion; see [[#Recursive humor|recursive humor]].
==Informal definition==
[[File:Mixing Sourdough starter into the flour.jpg|thumb|[[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'.<ref>{{Cite web|url=https://www.merriam-webster.com/dictionary/recursive|title=Definition of RECURSIVE|website=www.merriam-webster.com|language=en|access-date=2019-10-24}}</ref>
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==
Linguist [[Noam 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.<ref>{{cite book|last=Pinker|first=Steven|title=The Language Instinct|year=1994|publisher=William Morrow}}</ref><ref>{{cite journal | doi = 10.1016/j.cognition.2004.08.004 | title = The faculty of language: What's so special about it? | year = 2005 | last1 = Pinker | first1=Steven | last2 = Jackendoff | first2=Ray | journal = Cognition | volume = 95 | issue = 2 | pages = 201–236 | pmid=15694646| citeseerx = 10.1.1.116.7784 | s2cid = 1599505 }}</ref>
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 sentence ''witches 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|date=August 2025}}
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.<ref>{{Cite web|url=https://www.thoughtco.com/recursion-grammar-1691901|title=What Is Recursion in English Grammar?|last=Nordquist|first=Richard|website=ThoughtCo|language=en|access-date=2019-10-24}}</ref> 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 by [[Daniel Everett]] on the basis of his claims about the [[Pirahã language]]. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this.<ref>{{cite journal |doi=10.1353/lan.0.0140 |title=Evidence and argumentation: A reply to Everett (2009) |url=http://web.mit.edu/linguistics/people/faculty/pesetsky/Nevins_Pesetsky_Rodrigues_2_Evidence_and_Argumentation_Reply_to_Everett.pdf |year=2009 |last1=Nevins |first1=Andrew |last2=Pesetsky |first2=David |last3=Rodrigues |first3=Cilene |journal=Language |volume=85 |issue=3 |pages=671–681 |s2cid=16915455 |archive-url=https://web.archive.org/web/20120106154616/http://web.mit.edu/linguistics/people/faculty/pesetsky/Nevins_Pesetsky_Rodrigues_2_Evidence_and_Argumentation_Reply_to_Everett.pdf |archive-date=2012-01-06}}</ref> Literary [[self-reference]] can in any case be argued to be different in kind from mathematical or logical recursion.<ref name="Drucker2008">{{cite book |last=Drucker|first=Thomas |title=Perspectives on the History of Mathematical Logic |url=https://books.google.com/books?id=R70M4zsVgREC&pg=PA110 |date=4 January 2008 |publisher=Springer Science & Business Media |isbn=978-0-8176-4768-1 |page=110}}</ref>
Recursion plays a crucial role not only in syntax, but also in [[natural language semantics]]. The word ''and'', 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.<ref>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.</ref>
A [[recursive grammar]] is a [[formal grammar]] that contains recursive [[production (computer science)|production rules]].<ref name="ns02">{{citation
| last1 = Nederhof | first1 = Mark-Jan
| last2 = Satta | first2 = Giorgio
| contribution = Parsing Non-recursive Context-free Grammars
| doi = 10.3115/1073083.1073104
| ___location = Stroudsburg, PA, USA
| pages = 112–119
| publisher = Association for Computational Linguistics
| title = Proceedings of the 40th Annual Meeting on Association for Computational Linguistics (ACL '02)
| year = 2002| doi-access = free
}}.</ref>
===Recursive humor===
Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving a [[circular definition]] or [[self-reference]], in which the putative recursive step does not get closer to a base case, but instead leads to an [[infinite regress]]. It is not unusual for such books to include a joke entry in their glossary along the lines of:
:Recursion, ''see Recursion''.<ref name=Hunter>{{cite book|last=Hunter|first=David|title=Essentials of Discrete Mathematics|year=2011|publisher=Jones and Bartlett|pages=494|url=https://books.google.com/books?id=kuwhTxCVovQC&q=recursion+joke|isbn=9781449604424}}</ref>
A variation is found on page 269 in the [[Back-of-the-book index|index]] of some editions of [[Brian Kernighan]] and [[Dennis Ritchie]]'s book ''[[The C Programming Language]]''; the index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in ''Let's talk Lisp'' by Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and in ''Software Tools'' by Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears in ''The UNIX Programming Environment'' by Kernighan and Pike. It did not appear in the first edition of ''The C Programming Language''. The joke is part of the [[functional programming]] folklore and was already widespread in the functional programming community before the publication of the aforementioned books. <ref name="Grainger College">{{cite web |last1=Shaffer |first1=Eric |title=CS 173:Discrete Structures |url=https://courses.engr.illinois.edu/cs173/sp2009/Lectures/lect_19.pdf |publisher=University of Illinois at Urbana-Champaign |access-date=7 July 2023}}</ref><ref name="Columbia University">{{cite web |title=Introduction to Computer Science and Programming in C; Session 8: September 25, 2008 |url=http://www.cs.columbia.edu/~bert/courses/1003/lecture8.pdf |publisher=Columbia University |access-date=7 July 2023}}</ref>
[[File:Toronto recursive history plaque.jpg|thumb|A plaque commemorates the Toronto Recursive History Project of Toronto's Recursive History.]]
Another joke is that "To understand recursion, you must understand recursion."<ref name=Hunter/> 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''."<ref>{{Cite web|url=https://www.google.com/search?q=recursion|title=recursion - Google Search|website=www.google.com|access-date=2019-10-24}}</ref> An alternative form 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."''
[[Recursive acronym]]s are other examples of recursive humor. [[PHP]], for example, stands for "PHP Hypertext Preprocessor", [[Wine (software)|WINE]] stands for "WINE Is Not an Emulator", [[GNU]] stands for "GNU's not Unix", and [[SPARQL]] denotes the "SPARQL Protocol and RDF Query Language".
==In mathematics==
[[File:Sierpinski triangle.svg|thumb|250px|The [[Sierpiński triangle]]—a confined recursion of triangles that form a fractal]]
===Recursively defined sets===
{{Main|Recursive definition}}
====Example: the natural numbers====
{{See also|Closure (mathematics)}}
The canonical example of a recursively defined set is given by the [[natural numbers]]:
:
:if ''n'' is in <math>\mathbb{N}</math>, then ''n'' + 1 is in <math>\mathbb{N}</math>
:The set of natural numbers is the smallest set satisfying the previous two properties.
In mathematical logic, the [[Peano axioms]] (or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician [[Richard Dedekind]] and by the Italian mathematician [[Giuseppe 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 an [[axiomatic system]] that are defined in terms of a [[proof 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|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 the [[Cantor set]] is a subdivision rule, as is [[barycentric subdivision]].
===Functional recursion===
A [[function (mathematics)|function]] may be
===Proofs involving recursive definitions===
Applying the standard technique of [[proof by cases]] to recursively defined sets or functions, as in the preceding sections, yields [[structural induction]] — a powerful generalization of [[mathematical induction]] widely used to derive proofs in [[mathematical logic]] and computer science.
===Recursive optimization===
[[Dynamic programming]] is an approach to [[optimization (mathematics)|optimization]]
===The recursion theorem===
In [[set theory]], this is a theorem guaranteeing that recursively defined functions exist. Given a set {{mvar|X}}, an element {{mvar|a}} of {{mvar|X}} and a function {{math|''f'': ''X'' → ''X''}}, the theorem states that there is a unique function <math>F: \N \to X</math> (where <math>\N</math> denotes the set of natural numbers including zero) such that
:<math>F(0) = a</math>
:<math>F(n + 1) = f(F(n))</math>
for any natural number {{mvar|n}}.
Dedekind was the first to pose the problem of unique definition of set-theoretical functions on <math>\mathbb N</math> by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?" <ref>A. Kanamori, "[https://math.bu.edu/people/aki/20.pdf In Praise of Replacement]", pp.50--52. Bulletin of Symbolic Logic, vol. 18, no. 1 (2012). Accessed 21 August 2023.</ref>
====Proof of uniqueness====
Take two functions <math>F: \N \to X</math> and <math>G: \N \to X</math> such that:
:<math>F(0) = a</math>
:<math>G(0) = a</math>
:<math>F(n + 1) = f(F(n))</math>
:<math>G(n + 1) = f(G(n))</math>
where {{mvar|a}} is an element of {{mvar|X}}.
It can be proved by [[mathematical induction]] that {{math|1=''F''(''n'') = ''G''(''n'')}} for all natural numbers
{{mvar|n}}:
:'''Base Case''': {{math|1=''F''(0) = ''a'' = ''G''(0)}} so the equality holds for {{math|1=''n'' = 0}}.
:'''Inductive Step''': Suppose {{math|1=''F''(''k'') = ''G''(''k'')}} for some {{nowrap|<math>k \in \N</math>.}} Then {{math|1=''F''(''k'' + 1) = ''f''(''F''(''k'')) = ''f''(''G''(''k'')) = ''G''(''k'' + 1)}}.
::Hence {{math|1=''F''(''k'') = ''G''(''k'')}} implies {{math|1=''F''(''k'' + 1) = ''G''(''k'' + 1)}}.
By induction, {{math|1=''F''(''n'') = ''G''(''n'')}} for all <math>n \in \N</math>.
==
{{Main|Recursion (computer science)}}
A common method of simplification is to divide a problem into subproblems of the same type. As a [[computer programming]] technique, this is called [[divide and conquer algorithm|divide 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 is [[dynamic 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 the [[factorial]] function, given here in
<
if n > 0:
return n * factorial(n - 1)
return 1
</syntaxhighlight>
The function calls itself recursively on a smaller version of the input {{code|(n - 1)}} and multiplies the result of the recursive call by {{code|n}}, until reaching the [[base case (recursion)|base 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 in [[parser]]s 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 relation]]s are equations
Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity
==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 is [[Romanesco broccoli]].<ref>{{cite web |title=Picture of the Day: Fractal Cauliflower |date=28 December 2012 |url=https://twistedsifter.com/2012/12/fractal-cauliflower-romanesco-broccoli/ |access-date=19 April 2020}}</ref>
== In the social sciences ==
Authors use the concept of ''recursivity'' to foreground the situation in which specifically ''social'' scientists find themselves when producing knowledge about the world they are always already part of.<ref>{{Cite journal |last=Bourdieu |first=Pierre |year=1992 |title=Double Bind et Conversion |journal=Pour Une Anthropologie Réflexive |publisher=Le Seuil |publication-place=Paris}}</ref><ref>{{Cite book |last=Giddens |first=Anthony |title=Social Theory and Modern Sociology |publisher=Polity Press |year=1987}}</ref> 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).”<ref name="Alejandro2021">{{Cite journal |last=Alejandro |first=Audrey |date=2021 |title=Reflexive discourse analysis: A methodology for the practice of reflexivity |journal=[[European Journal of International Relations]] |language=en |volume=27 |issue=1 |page=171 |doi=10.1177/1354066120969789 |s2cid=229461433 |issn=1354-0661|doi-access=free }}</ref> From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise of [[Reflexivity (social theory)|reflexive]] efforts:{{quote|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| {{Harvp|Alejandro|2021}}}}
==In business==
{{Further information|Management cybernetics}}
Recursion is sometimes referred to in [[management science]] as the process of iterating through levels of abstraction in large business entities.<ref>{{cite journal |title=The Canadian Small Business–Bank Interface: A Recursive Model |date=1994 |url=https://journals.sagepub.com/doi/pdf/10.1177/104225879401800401 |publisher=SAGE Journals|doi=10.1177/104225879401800401 |last1=Riding |first1=Allan |last2=Haines |first2=George H. |last3=Thomas |first3=Roland |journal=Entrepreneurship Theory and Practice |volume=18 |issue=4 |pages=5–24 |url-access=subscription }}</ref> A common example is the recursive nature of management [[hierarchy|hierarchies]], ranging from [[line management]] to [[senior management]] via [[middle management]]. It also encompasses the larger issue of [[capital structure]] in [[corporate governance]].<ref>{{cite book |last1=Beer |first1=Stafford |title=Brain Of The Firm |date=1972 |publisher=John Wiley & Sons |isbn=978-0471948391}}</ref>
==In art==
[[File:First matryoshka museum doll open.jpg|thumb|Recursive dolls: the original set of [[Matryoshka doll]]s by [[Vasily Zvyozdochkin|Zvyozdochkin]] and [[Sergey Malyutin|Malyutin]], 1892]]
[[File:Giotto. The Stefaneschi Triptych (verso) c.1330 220x245cm. Pinacoteca, Vatican..jpg|thumb|left|Front face of [[Giotto]]'s ''[[Stefaneschi Triptych]]'', 1320, recursively contains an image of itself (held up by the kneeling figure in the central panel).]]
{{See also|Mathematics and art|Infinity mirror}}
The [[Matryoshka doll]] is a physical artistic example of the recursive concept.<ref>{{cite web |last1=Tang |first1=Daisy |title=CS240 -- Lecture Notes: Recursion|date=March 2013|publisher=California State Polytechnic University, Pomona |url=http://www.cpp.edu/~ftang/courses/CS240/lectures/recursion.htm|archive-url=https://web.archive.org/web/20180317100946/https://www.cpp.edu/~ftang/courses/CS240/lectures/recursion.htm|archive-date=2018-03-17 |access-date=24 September 2015 |quote=More examples of recursion: Russian Matryoshka dolls. Each doll is made of solid wood or is hollow and contains another Matryoshka doll inside it. }}</ref>
Recursion has been used in paintings since [[Giotto]]'s ''[[Stefaneschi Triptych]]'', made in 1320. Its central panel contains the kneeling figure of Cardinal Stefaneschi, holding up the triptych itself as an offering.<ref>{{cite web |title=Giotto di Bondone and assistants: Stefaneschi triptych |url=http://mv.vatican.va/3_EN/pages/PIN/PIN_Sala02_03.html |publisher=The Vatican |access-date=16 September 2015}}</ref><ref>{{Cite book |title=Physical (A)Causality: Determinism, Randomness and Uncaused Events |url=https://books.google.com/books?id=gxBMDwAAQBAJ&pg=PA12 |first=Karl |last=Svozil |year=2018 |publisher=Springer |pages=12| isbn=9783319708157 }}</ref> This practice is more generally known as the [[Droste effect]], an example of the [[Mise en abyme]] technique.
[[M. C. Escher]]'s ''[[Print Gallery (M. C. Escher)|Print Gallery]]'' (1956) is a print which depicts a distorted city containing a gallery which [[recursive]]ly contains the picture, and so ''[[ad infinitum]]''.<ref>{{cite web |last1=Cooper |first1=Jonathan |title=Art and Mathematics |url=https://unwrappingart.com/art/art-and-mathematics/ |access-date=5 July 2020 |date=5 September 2007}}</ref>
{{Clear}}
==
The film ''[[Inception]]'' has colloquialized the appending of the suffix ''[[wiktionary:-ception|-ception]]'' to a noun to jokingly indicate the recursion of something.<ref>{{cite web |title=-ception – The Rice University Neologisms Database |url=http://neologisms.rice.edu/index.php?a=term&d=1&t=17573 |url-status=live |archive-url=https://web.archive.org/web/20170705153941/http://neologisms.rice.edu/index.php?a=term&d=1&t=17573 |archive-date=July 5, 2017 |access-date=December 23, 2016 |publisher=Rice University}}</ref>
== See also ==<!-- Making the Recursion article link to itself will not display correctly, and is considered to break [[WP:ASTONISH]]. The joke itself is already featured in the "Recursive humor" section. See discussion on the talk page. -->
* {{Annotated link|Corecursion}}
* {{Annotated link|Course-of-values recursion}}
* {{Annotated link|Digital infinity}}
* {{Annotated link|A Dream Within a Dream (poem)}}
* {{Annotated link|Droste effect}}
* {{Annotated link|False awakening}}
* {{Annotated link|Fixed point combinator}}
* {{Annotated link|Infinite compositions of analytic functions}}
* {{Annotated link|Infinite loop}}
* {{Annotated link|Infinite regress}}
* {{Annotated link|Infinitism}}
* {{Annotated link|Infinity mirror}}
* {{Annotated link|Iterated function}}
* {{Annotated link|Mathematical induction}}
* {{Annotated link|Mise en abyme}}
<!-- {{Annotated link|Recursion}} and [[Recursion]] will not display correctly in this list, and including it
is considered to break [[WP:ASTONISH]]. See discussion on the talk page.-->
* {{Annotated link|Reentrant (subroutine)}}
* {{Annotated link|Self-reference}}
* {{Annotated link|Spiegel im Spiegel}}
*
* {{Annotated link|Tail recursion}}
* {{Annotated link|Tupper's self-referential formula}}
* {{Annotated link|Turtles all the way down}}
==References==
{{Reflist}}
==Bibliography==
{{refbegin}}
* {{cite journal|first=Edsger W.|last=Dijkstra|author-link=Edsger W. Dijkstra|title=Recursive Programming|journal=Numerische Mathematik|volume=2|issue=1|year=1960|pages=312–318|doi=10.1007/BF01386232|s2cid=127891023}}
*{{cite book | author=Johnsonbaugh, Richard|author-link=Richard Johnsonbaugh | title=Discrete Mathematics | publisher=Prentice Hall | year=2004 | isbn=978-0-13-117686-7 }}
*{{cite book | author=Hofstadter, Douglas | author-link=Douglas Hofstadter | title=Gödel, Escher, Bach: an Eternal Golden Braid | publisher=Basic Books | year=1999 | isbn=978-0-465-02656-2 | url=https://archive.org/details/gdelescherbachet00hofs }}
*{{cite book | author=Shoenfield, Joseph R. |author-link=Joseph R. Shoenfield| title=Recursion Theory | url=https://archive.org/details/recursiontheory0000shoe | url-access=registration | publisher=A K Peters Ltd | year=2000 | isbn=978-1-56881-149-9 }}
*{{cite book | author=[[Causey, Robert L.]] | title=Logic, Sets, and Recursion | url=https://archive.org/details/logicsetsrecursi0000caus | url-access=registration | publisher=Jones & Bartlett | year=2001 | isbn=978-0-7637-1695-0 }}
*{{cite book |author1=Cori, Rene |author2=Lascar, Daniel |author3=Pelletier, Donald H. | title=Recursion Theory, Gödel's Theorems, Set Theory, Model Theory | publisher=Oxford University Press | year=2001 | isbn=978-0-19-850050-6 }}
*{{cite book | author1=Barwise, Jon| author2=Moss, Lawrence S. |author1-link=Jon Barwise| title=Vicious Circles | publisher=Stanford Univ Center for the Study of Language and Information | year=1996 | isbn=978-0-19-850050-6 }} - offers a treatment of [[corecursion]].
*{{cite book | author=Rosen, Kenneth H. | title=Discrete Mathematics and Its Applications | publisher=McGraw-Hill College | year=2002 | isbn=978-0-07-293033-7 }}
*{{cite book | last1=Cormen |first1=Thomas H. |first2=Charles E. |last2=Leiserson |first3=Ronald L. |last3=Rivest |first4=Clifford |last4=Stein | title=Introduction to Algorithms | publisher=Mit Pr | year=2001 | isbn=978-0-262-03293-3 }}
*{{cite book |author1=Kernighan, B. |author2=Ritchie, D. |title=The C programming Language |publisher=Prentice Hall |year=1988 |isbn=978-0-13-110362-7 |url=https://archive.org/details/cprogramminglang00bria }}
*{{cite book |author1=Stokey, Nancy |author2=Robert Lucas |author3=Edward Prescott | title=Recursive Methods in Economic Dynamics | publisher=Harvard University Press | year=1989 | isbn=978-0-674-75096-8}}
*{{cite book | author=Hungerford |title=Algebra | publisher=Springer|year=1980|isbn=978-0-387-90518-1}}, first chapter on set theory.
{{refend}}
==External links==
{{Commons category}}
{{Wiktionary|recursion|recursivity}}
* [https://web.archive.org/web/20050206051223/http://www.freenetpages.co.uk/hp/alan.gauld/tutrecur.htm Recursion] - tutorial by Alan Gauld
* [http://research.swtch.com/2010/03/zip-files-all-way-down.html Zip Files All The Way Down]
*[http://www.ucl.ac.uk/psychlangsci/staff/linguistics-staff/nevins-publications/npr09b Nevins, Andrew and David Pesetsky and Cilene Rodrigues. Evidence and Argumentation: A Reply to Everett (2009). Language 85.3: 671--681 (2009)]
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{{Mathematical logic}}
{{Authority control}}
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