Editing Ackermann function (section)

==Bibliography==
{{Refbegin}}
*{{cite journal
 |last= Ackermann
 |first= Wilhelm
 |author-link= Wilhelm Ackermann
 |journal= [[Mathematische Annalen]]
 |title= Zum Hilbertschen Aufbau der reellen Zahlen
 |language=de
 |trans-title=On the Hilbertian construction of the real numbers
 |year= 1928
 |volume= 99
 |pages= 118–133
 |url= http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN235181684_0099&DMDID=DMDLOG_0009
 |doi= 10.1007/BF01459088
 |s2cid= 123431274
}}
*{{cite journal 
 |last= Buck 
 |first= R. C. 
 |author-link= Robert Creighton Buck
 |title= Mathematical Induction and Recursive Definitions 
 |journal= [[American Mathematical Monthly]] 
 |year= 1963 
 |volume= 70 
 |issue= 2 
 |pages= 128–135 
 |doi= 10.2307/2312881 
 |jstor= 2312881 
 |doi-access=  
}}
*{{cite journal 
 |last1= Calude
 |first1= Cristian
 |author-link1= Cristian S. Calude
 |last2= Marcus
 |first2= Solomon
 |author-link2= Solomon Marcus
 |last3= Tevy 
 |first3= Ionel
 |journal= Historia Math. 
 |title= The first example of a recursive function which is not primitive recursive
 |date= November 1979 
 |pages= 380–84 
 |volume= 6 
 |issue= 4 
 |doi= 10.1016/0315-0860(79)90024-7
 |doi-access= free
}}
*{{cite book
 |last= Cohen
 |first= Daniel E.
 |title= Computability and logic
 |publisher= Halsted Press
 |date= January 1987
 |isbn=9780745800349
}}
*{{cite journal 
 |last1= Cornelius
 |first1= B. J. 
 |last2= Kirby
 |first2= G. H.
 |journal = [[BIT Numerical Mathematics]] 
 |title= Depth of recursion and the Ackermann function 
 |year= 1975 
 |pages= 144–150 
 |volume= 15 
 |issue= 2 
 |doi= 10.1007/BF01932687
|s2cid= 120532578 
 }}
*{{cite conference 
 |last1= Czerwiński
 |first1= Wojciech 
 |last2= Orlikowski
 |first2= Łukasz
 |title= Reachability in Vector Addition Systems is Ackermann-complete
 |conference= Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science
 |date= 7 February 2022
 |url= https://ieeexplore.ieee.org/document/9719806
 |doi= 10.1109/FOCS52979.2021.00120
 |arxiv= 2104.13866
 }}
*{{cite book|first1=M.|last1=Fredman|author-link=Michael Fredman|first2=M.|last2=Saks|title=Proceedings of the twenty-first annual ACM symposium on Theory of computing – STOC '89 |chapter=The cell probe complexity of dynamic data structures |pages=345&ndash;354|date=May 1989|doi=10.1145/73007.73040|isbn=0897913078|s2cid=13470414|doi-access=free}}
*{{cite journal 
 |last1= Grossman
 |first1= Jerrold W.
 |last2= Zeitman
 |first2= R. Suzanne
 |journal = [[Theoretical Computer Science]] 
 |title= An inherently iterative computation of ackermann's function 
 |date= May 1988 
 |pages= 327–330 
 |volume= 57
 |issue= 2–3 
 |doi= 10.1016/0304-3975(88)90046-1
 |doi-access= 
}}
*{{cite book
 |last= van Heijenoort
 |first= Jean
 |title= From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931
 |publisher= Harvard University Press
 |orig-year= reprinted with corrections, first published in 1967
 |year= 1977
}}
*{{cite journal 
 |last= Hilbert
 |first= David
 |author-link= David Hilbert
 |journal= [[Mathematische Annalen]] 
 |title= Über das Unendliche
 |language=de
 |trans-title=On the infinite
 |year= 1926 
 |volume= 95
 |pages= 161–190 
 |doi= 10.1007/BF01206605
 |s2cid= 121888793
}}
*{{cite conference 
 |last= Leroux
 |first= Jérôme
 |title= The Reachability Problem for Petri Nets is Not Primitive Recursive
 |conference= Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science
 |date= 7 February 2022
 |url= https://ieeexplore.ieee.org/document/9719763
 |doi= 10.1109/FOCS52979.2021.00121
 |arxiv= 2104.12695
 }}
*{{cite web
 |last= Matos 
 |first= Armando B
 |title= The inverse of the Ackermann function is primitive recursive 
 |date= May 7, 2014 
 |url= http://www.dcc.fc.up.pt/~acm/PRinv.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.dcc.fc.up.pt/~acm/PRinv.pdf |archive-date=2022-10-09 |url-status=live
}}
*{{cite report 
 |last1= Meeussen
 |first1= V. C. S.
 |last2= Zantema
 |first2= H.
 |title= Derivation lengths in term rewriting from interpretations in the naturals 
 |publisher= University of Utrecht Department of Computer Science 
 |url= https://research.tue.nl/files/4245011/398270.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://research.tue.nl/files/4245011/398270.pdf |archive-date=2022-10-09 |url-status=live 
 |year= 1992 
 |issn= 0924-3275
}}
*{{cite conference 
 |last1= Meyer 
 |first1= Albert R. 
 |author-link1= Albert R. Meyer 
 |last2= Ritchie 
 |first2= Dennis MacAlistair 
 |title= Proceedings of the 1967 22nd national conference 
 |author-link2= Dennis Ritchie 
 |chapter= The complexity of loop programs 
 |conference= ACM '67: Proceedings of the 1967 22nd national conference 
 |year= 1967 
 |pages= 465–469 
 |doi= 10.1145/800196.806014
|doi-access= free 
 }}
*{{cite book
 |last1= Monin
 |first1= Jean-Francois
 |last2= Hinchey
 |first2= M. G.
 |title= Understanding Formal Methods
 |publisher= Springer
 |year= 2003
 |isbn= 9781852332471
 |page= 61
 |url= https://books.google.com/books?id=rUudIPZD-B0C&pg=PA61
}}
*{{cite web 
 |last= Munafo 
 |first= Robert 
 |title= Versions of Ackermann's Function 
 |url= http://www.mrob.com/pub/math/ln-2deep.html#ack
 |work= Large Numbers at MROB 
 |date= 1999a 
 |access-date= 2021-11-06 
}}
*{{cite web 
 |last= Munafo 
 |first= Robert 
 |title= Inventing New Operators and Functions 
 |url= http://www.mrob.com/pub/math/largenum-3.html 
 |work= Large Numbers at MROB 
 |date= 1999b 
 |access-date= 2021-11-06 
}}
*{{cite web 
 |last= Paulson
 |first= Lawrence C.
 |title= Ackermann's Function in Iterative Form: A Proof Assistant Experiment 
 |url= https://www.researchgate.net/publication/351063906 
 |date= 2021
 |access-date= 2021-10-19 
}}
*{{cite journal 
 |last= Péter 
 |first= Rózsa
 |author-link= Rózsa Péter
 |journal= [[Mathematische Annalen]] 
 |title= Konstruktion nichtrekursiver Funktionen
 |language=de
 |trans-title=Construction of non-recursive functions
 |year= 1935 
 |volume= 111
 |pages= 42–60 
 |doi= 10.1007/BF01472200
 |s2cid= 121107217 
}}
*{{cite book
 |last= Pettie
 |first= S.
 |title= The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
 |chapter= An inverse-Ackermann style lower bound for the online minimum spanning tree verification problem
 |year= 2002
 |pages= 155–163
 |doi= 10.1109/SFCS.2002.1181892
 |isbn= 0-7695-1822-2
 |s2cid= 8636108
}}
*{{cite journal 
 |last1= Porto 
 |first1= António 
 |last2= Matos
 |first2= Armando B.
 |title= Ackermann and the superpowers 
 |journal= [[ACM SIGACT|ACM SIGACT News]] 
 |date= 1 September 1980
 |volume= 12 
 |issue= 3 
 |pages= 90–95 
 |url= https://www.dcc.fc.up.pt/~acm/ack.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.dcc.fc.up.pt/~acm/ack.pdf |archive-date=2022-10-09 |url-status=live 
 |doi= 10.1145/1008861.1008872
|s2cid= 29780652 
 }} Original version 1980, published in ''ACM SIGACT News'', modified on 20 October 2012 and 23 January 2016 (working paper)
*{{cite journal 
 |last= Ritchie 
 |first= Robert Wells 
 |title= Classes of recursive functions based on Ackermann's function 
 |journal= [[Pacific Journal of Mathematics]] 
 |date= November 1965
 |volume= 15 
 |issue= 3 
 |pages= 1027–1044 
 |url= https://msp.org/pjm/1965/15-3/p25.xhtml 
 |doi= 10.2140/pjm.1965.15.1027
 |doi-access= free 
}}
*{{cite journal 
 |last= Robinson 
 |first= Raphael Mitchel 
 |author-link= 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 
 |issue= 10
 |doi-access= free 
}}
*{{cite journal
 |last= Sundblad
 |first= Yngve
 |date= March 1971
 |title= The Ackermann function. A theoretical, computational, and formula manipulative study
 |journal= BIT Numerical Mathematics
 |volume= 11
 |issue= 1
 |pages= 107–119
 |doi= 10.1007/BF01935330
 |s2cid= 123416408
}}
*{{cite journal|last1=Tarjan|first1=Robert Endre|author1-link=Robert E. Tarjan|year=1975|title=Efficiency of a Good But Not Linear Set Union Algorithm|journal=Journal of the ACM|volume=22|issue=2|pages=215&ndash;225|doi=10.1145/321879.321884|hdl=1813/5942|s2cid=11105749|hdl-access=free }}
*{{cite journal
 |last= Vaida
 |first= Dragoș
 |date= 1970
 |title= Compiler Validation for an Algol-like Language
 |journal= Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie 
 |series= Nouvelle série
 |volume= 14 (62)
 |issue= 4
 |pages= 487–502
 |jstor= 43679758
}}
*{{citation|mode=cs1
 |last= Ward
 |first= Martin P.
 |date= 16 July 1993
 |title= Iterative Procedures for Computing Ackerman's Function
 |citeseerx= 10.1.1.35.9907
 }}
*{{cite journal
 |last= Wichmann
 |first= Brian A.
 |date= March 1976
 |title= Ackermann's function: A study in the efficiency of calling procedures
 |journal= BIT Numerical Mathematics
 |volume= 16
 |pages= 103–110
 |doi= 10.1007/BF01940783
 |s2cid= 16993343
|citeseerx= 10.1.1.108.4125
 }}
*{{cite journal
 |last= Wichmann
 |first= Brian A.
 |date= July 1977
 |title= How to call procedures, or second thoughts on Ackermann's function
 |journal= BIT Numerical Mathematics
 |volume= 16
 |issue= 3
 |pages= 103–110
 |doi= 10.1002/spe.4380070303
 |s2cid= 206507320
}}
*{{cite web
 |last= Wichmann
 |first= Brian A.
 |date= July 1982
 |title= Latest results from the procedure calling test, Ackermann's function
 |url= http://history.dcs.ed.ac.uk/archive/docs/Imp_Benchmarks/acklt.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://history.dcs.ed.ac.uk/archive/docs/Imp_Benchmarks/acklt.pdf |archive-date=2022-10-09 |url-status=live}}
{{Refend}}