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On primitive recursive permutations and their inverses1

Published online by Cambridge University Press:  12 March 2014

Frank B. Cannonito  and
Mark Finkelstein
Frank B. Cannonito
Affiliation:
University of California, Irvine
Mark Finkelstein
Affiliation:
University of California, Irvine
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Extract

It has been known for some time that there is a primitive recursive permutation of the nonnegative integers whose inverse is recursive but not primitive recursive. For example one has this result apparently for the first time in Kuznecov [1] and implicitly in Kent [2] or J. Robinson [3], who shows that every singularly recursive function ƒ is representable as

where A, B, C are primitive recursive and B is a permutation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 4 , February 1970 , pp. 634 - 638
Copyright
Copyright © Association for Symbolic Logic 1970

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Footnotes

1

The first author was supported by Grant No. AF-AFOSR 1321-67 and the second by NSF GP-7500.

References

[1] Kuznecov, A. V., On primitive recursive functions of large oscillation, Doklady Akademii Nauk SSSR 71 (1950), pp. 233–236 (Russian).Google Scholar
[2] Kent, C. F., Algebraic structure of some groups of recursive permutations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1960.Google Scholar
[3] Robinson, J., General recursive functions, Proceedings of the American Mathematical Society, vol. 1 (1950), pp. 703–718.CrossRefGoogle Scholar
[4] Cannonito, F. B., Hierarchies of computable groups and the word problem, this Journal , vol. 31 (1966), pp. 376–392.Google Scholar
[5] Rogers, Hartley Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[6] Uspenskií, V. A., Lectures on computable functions, Fizmatgiz, Moscow, 1960; French transl., Actualités Scientifiques et Industrielles, no. 1317, Hermann, Paris, 1966; English transl., Hindustan, Delhi (to appear).Google Scholar
[7] Kleene, S. C., Introduction to metamathematics, Van Nostrand, Princeton, N.J., 1952.Google Scholar
[8] Kleene, S. C., Mathematical logic, Wiley, New York, 1967.Google Scholar
[9] Davis, M., Computability and unsolvability, McGraw-Hill, New York, 1957.Google Scholar
[10] Hermes, H., Aufzählbarkeit, Entscheidbarkeit, Berechenbarkeit, Springer-Verlag, Berlin, 1961.CrossRefGoogle Scholar
[11] Ritchie, R. W., Classes of predictably computable functions, Transactions of the American Mathematical Society, vol. 106 (1963), pp. 139–173.CrossRefGoogle Scholar