Unary primitive recursive functions
Daniel E. Severin
Source: J. Symbolic Logic Volume 73, Issue 4
(2008), 1122-1138.
Abstract
In this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396909
Digital Object Identifier: doi:10.2178/jsl/1230396909
Mathematical Reviews number (MathSciNet): MR2467207
Zentralblatt MATH identifier: 1158.03023
References
Paul Axt, Iteration of relative primitive recursion, Mathematische Annalen, vol. 167 (1966), pp. 53--55.
Mathematical Reviews (MathSciNet): MR209147
Digital Object Identifier: doi:10.1007/BF01361215
Zentralblatt MATH: 0192.05002
Nelu Dima, Sudan function is universal for the class of primitive recursive functions, Studii şi Cercetări Matematice, vol. 33 (1981), pp. 59--67.
Mathematical Reviews (MathSciNet): MR616787
Nadejda Georgieva, Another simplification of the recursion scheme, Archiv für Mathematische Logik, vol. 18 (1976), pp. 1--3.
Mathematical Reviews (MathSciNet): MR485271
--------, Classes of one-argument recursive functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 127--130.
Mathematical Reviews (MathSciNet): MR460085
Giorgio Germano and Stefano Mazzanti, Primitive iteration and unary functions, Annals of Pure and Applied Logic, vol. 40 (1988), pp. 217--256.
Mathematical Reviews (MathSciNet): MR973481
Digital Object Identifier: doi:10.1016/0168-0072(88)90032-2
Zentralblatt MATH: 0664.68047
M. D. Gladstone, A reduction of the recursion scheme, Journal of Symbolic Logic, vol. 32 (1967), no. 4, pp. 505--508.
Mathematical Reviews (MathSciNet): MR224460
Digital Object Identifier: doi:10.2307/2270177
JSTOR: links.jstor.org
Zentralblatt MATH: 0157.01702
--------, Simplifications of the recursion scheme, Journal of Symbolic Logic, vol. 36 (1971), no. 4, pp. 653--665.
Mathematical Reviews (MathSciNet): MR305993
Digital Object Identifier: doi:10.2307/2272468
JSTOR: links.jstor.org
Zentralblatt MATH: 0248.02040
Reuben L. Goodstein, Recursive number theory: A development of recursive arithmetic in a logic-free equation calculus, 1957.
Andrzej Grzegorczyk, Some classes of recursive functions, Rozprawy Matematyczne, vol. 4 (1953), pp. 1--44.
Mathematical Reviews (MathSciNet): MR60426
Zentralblatt MATH: 0052.24902
Stefano Mazzanti, Bounded iteration and unary functions, MLQ Mathematical Logic Quarterly, vol. 51 (2005), pp. 89--94.
Mathematical Reviews (MathSciNet): MR2099389
Digital Object Identifier: doi:10.1002/malq.200410010
Zentralblatt MATH: 1077.03024
Jovan Naumović, A classification of the one-argument primitive recursive functions, Archiv für Mathematische Logik, vol. 23 (1983), pp. 161--174.
Mathematical Reviews (MathSciNet): MR725042
Robert W. Ritchie, Classes of recursive functions based on Ackermann's function, Pacific Journal of Mathematics, vol. 15 (1965), no. 3, pp. 1027--1044.
Mathematical Reviews (MathSciNet): MR193013
Project Euclid: euclid.pjm/1102995586
Julia Robinson, General recursive functions, Proceedings of the American Mathematical Society, vol. 1 (1950), no. 6, pp. 703--718.
Mathematical Reviews (MathSciNet): MR38912
Digital Object Identifier: doi:10.2307/2031973
JSTOR: links.jstor.org
Zentralblatt MATH: 0041.15101
--------, A note on primitive recursive functions, Proceedings of the American Mathematical Society, vol. 6 (1955), no. 4, pp. 667--670.
Mathematical Reviews (MathSciNet): MR73536
Digital Object Identifier: doi:10.2307/2033449
JSTOR: links.jstor.org
Zentralblatt MATH: 0067.00204
Raphael M. Robinson, Primitive recursive functions, Bulletin of the American Mathematical Society, vol. 53 (1947), no. 10, pp. 925--942.
Mathematical Reviews (MathSciNet): MR22536
Digital Object Identifier: doi:10.1090/S0002-9904-1947-08911-4
Project Euclid: euclid.bams/1183511140
Zentralblatt MATH: 0034.29102
--------, Primitive recursive functions II, Proceedings of the American Mathematical Society, vol. 6 (1955), no. 4, pp. 663--666.
Mathematical Reviews (MathSciNet): MR73535
Digital Object Identifier: doi:10.2307/2033448
JSTOR: links.jstor.org
Zentralblatt MATH: 0067.00203
István Szalkai, On the algebraic structure of primitive recursive functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 551--556.
Mathematical Reviews (MathSciNet): MR813895
