Ackermann function (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Ackermann function" in English language version.

Last modified: 19:52:00, 28 August 2026 (UTC)

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2521doi.org

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ams.org (Global: 434^th place; English: 249^th place)

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Odifreddi 1999, p. 298. Odifreddi, Piergiorgio (1999). Classical recursion theory. Vol. II. Studies in Logic and the Foundations of Mathematics. Vol. 143. Amsterdam: North-Holland. ISBN 978-0-444-50205-6. MR 1718169.
Wiernik & Sharir 1988. Wiernik, Ady; Sharir, Micha (1988). "Planar realizations of nonlinear Davenport–Schinzel sequences by segments". Discrete & Computational Geometry. 3 (1): 15–47. doi:10.1007/BF02187894. MR 0918177.

arxiv.org (Global: 49^th place; English: 31^st place)

Czerwiński & Orlikowski 2022. Czerwiński, Wojciech; Orlikowski, Łukasz (7 February 2022). Reachability in Vector Addition Systems is Ackermann-complete. Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science. arXiv:2104.13866. doi:10.1109/FOCS52979.2021.00120.
Leroux 2022. Leroux, Jérôme (7 February 2022). The Reachability Problem for Petri Nets is Not Primitive Recursive. Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science. arXiv:2104.12695. doi:10.1109/FOCS52979.2021.00121.

books.google.com (Global: 3^rd place; English: 3^rd place)

Monin & Hinchey 2003, p. 61. Monin, Jean-Francois; Hinchey, M. G. (2003). Understanding Formal Methods. Springer. p. 61. ISBN 978-1-85233-247-1.

doi.org (Global: 2^nd place; English: 2^nd place)

The maximum depth of recursion refers to the number of levels of activation of a procedure that exist during the deepest call of the procedure. Cornelius & Kirby (1975) Cornelius, B. J.; Kirby, G. H. (1975). "Depth of recursion and the Ackermann function". BIT Numerical Mathematics. 15 (2): 144–150. doi:10.1007/BF01932687. S2CID 120532578.
Ackermann 1928. Ackermann, Wilhelm (1928). "Zum Hilbertschen Aufbau der reellen Zahlen" [On the Hilbertian construction of the real numbers]. Mathematische Annalen (in German). 99: 118–133. doi:10.1007/BF01459088. S2CID 123431274.
Calude, Marcus & Tevy 1979. Calude, Cristian; Marcus, Solomon; Tevy, Ionel (November 1979). "The first example of a recursive function which is not primitive recursive". Historia Math. 6 (4): 380–84. doi:10.1016/0315-0860(79)90024-7.
Hilbert 1926, p. 185. Hilbert, David (1926). "Über das Unendliche" [On the infinite]. Mathematische Annalen (in German). 95: 161–190. doi:10.1007/BF01206605. S2CID 121888793.
Péter 1935. Péter, Rózsa (1935). "Konstruktion nichtrekursiver Funktionen" [Construction of non-recursive functions]. Mathematische Annalen (in German). 111: 42–60. doi:10.1007/BF01472200. S2CID 121107217.
Robinson 1948. Robinson, Raphael Mitchel (1948). "Recursion and Double Recursion". Bulletin of the American Mathematical Society. 54 (10): 987–93. doi:10.1090/S0002-9904-1948-09121-2.
Ritchie 1965, p. 1028. Ritchie, Robert Wells (November 1965). "Classes of recursive functions based on Ackermann's function". Pacific Journal of Mathematics. 15 (3): 1027–1044. doi:10.2140/pjm.1965.15.1027.
Buck 1963. Buck, R. C. (1963). "Mathematical Induction and Recursive Definitions". American Mathematical Monthly. 70 (2): 128–135. doi:10.2307/2312881. JSTOR 2312881.
Ritchie 1965. Ritchie, Robert Wells (November 1965). "Classes of recursive functions based on Ackermann's function". Pacific Journal of Mathematics. 15 (3): 1027–1044. doi:10.2140/pjm.1965.15.1027.
Sundblad 1971. Sundblad, Yngve (March 1971). "The Ackermann function. A theoretical, computational, and formula manipulative study". BIT Numerical Mathematics. 11 (1): 107–119. doi:10.1007/BF01935330. S2CID 123416408.
Porto & Matos 1980. Porto, António; Matos, Armando B. (1 September 1980). "Ackermann and the superpowers" (PDF). ACM SIGACT News. 12 (3): 90–95. doi:10.1145/1008861.1008872. S2CID 29780652. Archived (PDF) from the original on 9 October 2022. Original version 1980, published in ACM SIGACT News, modified on 20 October 2012 and 23 January 2016 (working paper)
Meyer & Ritchie 1967. Meyer, Albert R.; Ritchie, Dennis MacAlistair (1967). "The complexity of loop programs". Proceedings of the 1967 22nd national conference. ACM '67: Proceedings of the 1967 22nd national conference. pp. 465–469. doi:10.1145/800196.806014.
Grossman & Zeitman 1988. Grossman, Jerrold W.; Zeitman, R. Suzanne (May 1988). "An inherently iterative computation of ackermann's function". Theoretical Computer Science. 57 (2–3): 327–330. doi:10.1016/0304-3975(88)90046-1.
Pettie 2002. Pettie, S. (2002). "An inverse-Ackermann style lower bound for the online minimum spanning tree verification problem". The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. pp. 155–163. doi:10.1109/SFCS.2002.1181892. ISBN 0-7695-1822-2. S2CID 8636108.
Czerwiński & Orlikowski 2022. Czerwiński, Wojciech; Orlikowski, Łukasz (7 February 2022). Reachability in Vector Addition Systems is Ackermann-complete. Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science. arXiv:2104.13866. doi:10.1109/FOCS52979.2021.00120.
Leroux 2022. Leroux, Jérôme (7 February 2022). The Reachability Problem for Petri Nets is Not Primitive Recursive. Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science. arXiv:2104.12695. doi:10.1109/FOCS52979.2021.00121.
Tarjan 1975. Tarjan, Robert Endre (1975). "Efficiency of a Good But Not Linear Set Union Algorithm". Journal of the ACM. 22 (2): 215–225. doi:10.1145/321879.321884. hdl:1813/5942. S2CID 11105749.
Fredman & Saks 1989. Fredman, M.; Saks, M. (May 1989). "The cell probe complexity of dynamic data structures". Proceedings of the twenty-first annual ACM symposium on Theory of computing – STOC '89. pp. 345–354. doi:10.1145/73007.73040. ISBN 0-89791-307-8. S2CID 13470414.
Wiernik & Sharir 1988. Wiernik, Ady; Sharir, Micha (1988). "Planar realizations of nonlinear Davenport–Schinzel sequences by segments". Discrete & Computational Geometry. 3 (1): 15–47. doi:10.1007/BF02187894. MR 0918177.
Wichmann 1976. Wichmann, Brian A. (March 1976). "Ackermann's function: A study in the efficiency of calling procedures". BIT Numerical Mathematics. 16: 103–110. doi:10.1007/BF01940783. S2CID 16993343.
Wichmann 1977. Wichmann, Brian A. (July 1977). "How to call procedures, or second thoughts on Ackermann's function". BIT Numerical Mathematics. 16 (3): 103–110. doi:10.1002/spe.4380070303. S2CID 206507320.

ed.ac.uk (Global: 1,867^th place; English: 1,129^th place)

history.dcs.ed.ac.uk

Wichmann 1982. Wichmann, Brian A. (July 1982). "Latest results from the procedure calling test, Ackermann's function" (PDF). Archived (PDF) from the original on 9 October 2022.

ghostarchive.org (Global: 64^th place; English: 28^th place)

Meeussen & Zantema 1992, p. 6. Meeussen, V. C. S.; Zantema, H. (1992). Derivation lengths in term rewriting from interpretations in the naturals (PDF) (Report). University of Utrecht Department of Computer Science. ISSN 0924-3275. Archived (PDF) from the original on 9 October 2022.
Porto & Matos 1980. Porto, António; Matos, Armando B. (1 September 1980). "Ackermann and the superpowers" (PDF). ACM SIGACT News. 12 (3): 90–95. doi:10.1145/1008861.1008872. S2CID 29780652. Archived (PDF) from the original on 9 October 2022. Original version 1980, published in ACM SIGACT News, modified on 20 October 2012 and 23 January 2016 (working paper)
Matos 2014. Matos, Armando B (7 May 2014). "The inverse of the Ackermann function is primitive recursive" (PDF). Archived (PDF) from the original on 9 October 2022.
Wichmann 1982. Wichmann, Brian A. (July 1982). "Latest results from the procedure calling test, Ackermann's function" (PDF). Archived (PDF) from the original on 9 October 2022.

hdl.handle.net (Global: 66^th place; English: 34^th place)

Tarjan 1975. Tarjan, Robert Endre (1975). "Efficiency of a Good But Not Linear Set Union Algorithm". Journal of the ACM. 22 (2): 215–225. doi:10.1145/321879.321884. hdl:1813/5942. S2CID 11105749.

jstor.org (Global: 23^rd place; English: 15^th place)

Buck 1963. Buck, R. C. (1963). "Mathematical Induction and Recursive Definitions". American Mathematical Monthly. 70 (2): 128–135. doi:10.2307/2312881. JSTOR 2312881.
Vaida 1970. Vaida, Dragoș (1970). "Compiler Validation for an Algol-like Language". Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie. Nouvelle série. 14 (62) (4): 487–502. JSTOR 43679758.

kosara.net (Global: low place; English: low place)

"Decimal expansion of A(4,2)". kosara.net. 27 August 2000. Archived from the original on 20 January 2010.

mrob.com (Global: low place; English: low place)

Munafo 1999a. Munafo, Robert (1999a). "Versions of Ackermann's Function". Large Numbers at MROB. Retrieved 6 November 2021.

msp.org (Global: low place; English: low place)

Ritchie 1965, p. 1028. Ritchie, Robert Wells (November 1965). "Classes of recursive functions based on Ackermann's function". Pacific Journal of Mathematics. 15 (3): 1027–1044. doi:10.2140/pjm.1965.15.1027.
Ritchie 1965. Ritchie, Robert Wells (November 1965). "Classes of recursive functions based on Ackermann's function". Pacific Journal of Mathematics. 15 (3): 1027–1044. doi:10.2140/pjm.1965.15.1027.

projecteuclid.org (Global: 4,781^st place; English: 3,117^th place)

Robinson 1948. Robinson, Raphael Mitchel (1948). "Recursion and Double Recursion". Bulletin of the American Mathematical Society. 54 (10): 987–93. doi:10.1090/S0002-9904-1948-09121-2.

quantamagazine.org (Global: 6,258^th place; English: 4,046^th place)

Brubaker 2023. Brubaker, Ben (4 December 2023). "An Easy-Sounding Problem Yields Numbers Too Big for Our Universe".

researchgate.net (Global: 96^th place; English: 89^th place)

Paulson 2021. Paulson, Lawrence C. (2021). "Ackermann's Function in Iterative Form: A Proof Assistant Experiment". Retrieved 19 October 2021.

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

The maximum depth of recursion refers to the number of levels of activation of a procedure that exist during the deepest call of the procedure. Cornelius & Kirby (1975) Cornelius, B. J.; Kirby, G. H. (1975). "Depth of recursion and the Ackermann function". BIT Numerical Mathematics. 15 (2): 144–150. doi:10.1007/BF01932687. S2CID 120532578.
Ackermann 1928. Ackermann, Wilhelm (1928). "Zum Hilbertschen Aufbau der reellen Zahlen" [On the Hilbertian construction of the real numbers]. Mathematische Annalen (in German). 99: 118–133. doi:10.1007/BF01459088. S2CID 123431274.
Hilbert 1926, p. 185. Hilbert, David (1926). "Über das Unendliche" [On the infinite]. Mathematische Annalen (in German). 95: 161–190. doi:10.1007/BF01206605. S2CID 121888793.
Péter 1935. Péter, Rózsa (1935). "Konstruktion nichtrekursiver Funktionen" [Construction of non-recursive functions]. Mathematische Annalen (in German). 111: 42–60. doi:10.1007/BF01472200. S2CID 121107217.
Sundblad 1971. Sundblad, Yngve (March 1971). "The Ackermann function. A theoretical, computational, and formula manipulative study". BIT Numerical Mathematics. 11 (1): 107–119. doi:10.1007/BF01935330. S2CID 123416408.
Porto & Matos 1980. Porto, António; Matos, Armando B. (1 September 1980). "Ackermann and the superpowers" (PDF). ACM SIGACT News. 12 (3): 90–95. doi:10.1145/1008861.1008872. S2CID 29780652. Archived (PDF) from the original on 9 October 2022. Original version 1980, published in ACM SIGACT News, modified on 20 October 2012 and 23 January 2016 (working paper)
Pettie 2002. Pettie, S. (2002). "An inverse-Ackermann style lower bound for the online minimum spanning tree verification problem". The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. pp. 155–163. doi:10.1109/SFCS.2002.1181892. ISBN 0-7695-1822-2. S2CID 8636108.
Tarjan 1975. Tarjan, Robert Endre (1975). "Efficiency of a Good But Not Linear Set Union Algorithm". Journal of the ACM. 22 (2): 215–225. doi:10.1145/321879.321884. hdl:1813/5942. S2CID 11105749.
Fredman & Saks 1989. Fredman, M.; Saks, M. (May 1989). "The cell probe complexity of dynamic data structures". Proceedings of the twenty-first annual ACM symposium on Theory of computing – STOC '89. pp. 345–354. doi:10.1145/73007.73040. ISBN 0-89791-307-8. S2CID 13470414.
Wichmann 1976. Wichmann, Brian A. (March 1976). "Ackermann's function: A study in the efficiency of calling procedures". BIT Numerical Mathematics. 16: 103–110. doi:10.1007/BF01940783. S2CID 16993343.
Wichmann 1977. Wichmann, Brian A. (July 1977). "How to call procedures, or second thoughts on Ackermann's function". BIT Numerical Mathematics. 16 (3): 103–110. doi:10.1002/spe.4380070303. S2CID 206507320.

stackexchange.com (Global: 4,114^th place; English: 3,027^th place)

math.stackexchange.com

"The Ackermann hierarchy vs. the fast growing hierarchy". StackExchange.

tue.nl (Global: 9,533^rd place; English: 6,917^th place)

research.tue.nl

Meeussen & Zantema 1992, p. 6. Meeussen, V. C. S.; Zantema, H. (1992). Derivation lengths in term rewriting from interpretations in the naturals (PDF) (Report). University of Utrecht Department of Computer Science. ISSN 0924-3275. Archived (PDF) from the original on 9 October 2022.

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

gdz.sub.uni-goettingen.de

Ackermann 1928. Ackermann, Wilhelm (1928). "Zum Hilbertschen Aufbau der reellen Zahlen" [On the Hilbertian construction of the real numbers]. Mathematische Annalen (in German). 99: 118–133. doi:10.1007/BF01459088. S2CID 123431274.

up.pt (Global: 5,271^st place; English: 7,125^th place)

dcc.fc.up.pt

Porto & Matos 1980. Porto, António; Matos, Armando B. (1 September 1980). "Ackermann and the superpowers" (PDF). ACM SIGACT News. 12 (3): 90–95. doi:10.1145/1008861.1008872. S2CID 29780652. Archived (PDF) from the original on 9 October 2022. Original version 1980, published in ACM SIGACT News, modified on 20 October 2012 and 23 January 2016 (working paper)
Matos 2014. Matos, Armando B (7 May 2014). "The inverse of the Ackermann function is primitive recursive" (PDF). Archived (PDF) from the original on 9 October 2022.

web.archive.org (Global: 1^st place; English: 1^st place)

"Decimal expansion of A(4,2)". kosara.net. 27 August 2000. Archived from the original on 20 January 2010.

worldcat.org (Global: 4^th place; English: 4^th place)

search.worldcat.org

Meeussen & Zantema 1992, p. 6. Meeussen, V. C. S.; Zantema, H. (1992). Derivation lengths in term rewriting from interpretations in the naturals (PDF) (Report). University of Utrecht Department of Computer Science. ISSN 0924-3275. Archived (PDF) from the original on 9 October 2022.

Ackermann function (English Wikipedia)

ams.org (Global: 434th place; English: 249th place)

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researchgate.net (Global: 96th place; English: 89th place)

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projecteuclid.org (Global: 4,781^st place; English: 3,117^th place)

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researchgate.net (Global: 96^th place; English: 89^th place)

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stackexchange.com (Global: 4,114^th place; English: 3,027^th place)

tue.nl (Global: 9,533^rd place; English: 6,917^th place)

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

up.pt (Global: 5,271^st place; English: 7,125^th place)

web.archive.org (Global: 1^st place; English: 1^st place)

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