Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:51:47.742Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZATIONS OF THE RECURSION THEOREM

Published online by Cambridge University Press:  21 December 2018

SEBASTIAAN A. TERWIJN
SEBASTIAAN A. TERWIJN*
Affiliation:
DEPARTMENT OF MATHEMATICS RADBOUDUNIVERSITY NIJMEGEN P.O. BOX 9010, 6500 GL NIJMEGEN, THE NETHERLANDSE-mail: terwijn@math.ru.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider two generalizations of the recursion theorem, namely Visser’s ADN theorem and Arslanov’s completeness criterion, and we prove a joint generalization of these theorems.

Keywords

03D2503D2803B40recursion theoremADN theoremArslanov completeness criterion
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1683 - 1690
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arslanov, M. M., On some generalizations of the fixed point theorem. Soviet Mathematics (Izvestiya VUZ. Matematika), vol. 25 (1981), no. 5, pp. 1–10. (English translation).Google Scholar
Arslanov, M. M., Nadirov, R. F., and Solov’ev, V. D., Completeness criteria for recursively enumerable sets and some general theorems on fixed points. Soviet Mathematics(Izvestiya VUZ. Matematika), vol. 21 (1977), no. 4, pp. 1–4. (English translation).Google Scholar
Barendregt, H., Representing ‘undefined’ in lambda calculus. Journal of Functional Programming, vol. 2 (1992), no. 3, pp. 367–374.CrossRefGoogle Scholar
Bernardi, C. and Sorbi, A., Classifying positive equivalence relations, this Journal, vol. 48 (1983), no. 3, pp. 529–538.Google Scholar
Downey, R. G. and Hirschfeldt, D. R., Algorithmic Randomness and Complexity, Springer-Verlag, New York, 2010.CrossRefGoogle Scholar
Ershov, Y. L., Theorie der Numerierungen I. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289–388.CrossRefGoogle Scholar
Jockusch, C. G. Jr., Lerman, M., Soare, R. I., and Solovay, R. M., Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion. this Journal, vol. 54 (1989), no. 4, pp. 1288–1323.Google Scholar
Jockusch, C. G. Jr. and Soare, R. I., Degrees of members of ${\rm{\Pi }}_1^0$ classes . Pacific Journal of Mathematics, vol. 40 (1972), pp. 605–616.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Soare, R. I., ${\rm{\Pi }}_1^0$ classes and degrees of theories . Transactions of the American Mathematical Society , vol. 173 (1972), pp. 33–56.Google Scholar
Kjos-Hanssen, B., Merkle, W., and Stephan, F., Kolmogorov complexity and the recursion theorem. Transactions of the American Mathematical Society, vol. 363 (2011), pp. 5465–5480.CrossRefGoogle Scholar
Kleene, S. C., On notation for ordinal numbers, this Journal, vol. 3 (1938), pp. 150–155.Google Scholar
Kučera, A., An alternative, priority-free solution to Post’s problem, Mathematical Foundations of Computer Science 1986 (Gruska, J., Rovan, B., and Wiedermann, J., editors), Lecture Notes in Computer Science, vol. 233, Springer, 1986, pp. 493–500.CrossRefGoogle Scholar
Montagna, F. and Sobi, A., Universal recursion theoretic properties of r.e. preordered structures. this Journal, vol. 50 (1985), no. 2, pp. 397–406.Google Scholar
Moschovakis, Y. N., Kleene’s amazing second recursion theorem. Bulletin of Symbolic Logic, vol. 16 (2010), no. 2, pp. 189–239.CrossRefGoogle Scholar
Odifreddi, P., Classical recursion theory, vol. 1, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam, 1989.Google Scholar
Owings, J. C., Diagonalization and the recursion theorem. Notre Dame Journal of Formal Logic, vol. 14 (1973), pp. 95–99.CrossRefGoogle Scholar
Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.Google Scholar
Terwijn, S. A., The noneffectivity of Arslanov’s completeness criterion and related theorems, preprint, 2018, arXiv:1804.01522.Google Scholar
Visser, A., Numerations, λ-calculus, and arithmetic, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (Seldin, J. P. and Hindley, J. R. editors), Academic Press, New York, 1980, pp. 259–284.Google Scholar