Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-16T08:17:16.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Fine hierarchies and Boolean terms

Published online by Cambridge University Press:  12 March 2014

V. L. Selivanov
V. L. Selivanov*
Affiliation:
Department of Mathematics, Pedagogical University, Viluiskaya St. 28, 630126 Novosibirsk, Russia, E-mail: vseliv@ngpi.nsk.su
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider fine hierarchies in recursion theory, descriptive set theory, logic and complexity theory. The main results state that the sets of values of different Boolean terms coincide with the levels of suitable fine hierarchies. This gives new short descriptions of these hierarchies and shows that collections of sets of values of Boolean terms are almost well ordered by inclusion. For the sake of completeness we mention also some earlier results demonstrating the usefulness of fine hierarchies.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 289 - 317
Copyright
Copyright © Association for Symbolic Logic 1995

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

[Ad65]Addison, J., The method of alternating chains, The theory of models (Addison, J. W., Henkin, L. A., and Tarski, A., editors), North Holland, Amsterdam, 1965, Proceedings of the 1963 International Symposium at Berkeley, pp. 1–16.Google Scholar
[BDG88]Balcázar, J. L., Díaz, J., and Gabarró, , Structural complexity 1, EATCS Monographs on Theoretical Computer Science, vol. 11, Springer-Verlag, 1988.CrossRefGoogle Scholar
[BG84]Blass, A. and Gurevich, Y., Equivalence relations, invariants, and normal forms, Logic and machines: Decision problems and complexity, Lecture Notes in Computer Science, vol. 171, Springer-Verlag, 1986, pp. 24–42.CrossRefGoogle Scholar
[Er68]Ershov, Y. L., On a hierarchy of sets II, Algebra and Logic, vol. 7 (1968), pp. 15–47, in Russian.Google Scholar
[Haul4]Hausdorff, F., Grundzüge der Mengenlehre, Leipzig, 1914.Google Scholar
[Hay78]Hay, L., Convex subsets of n2 and bounded truth-table reducibility, Discrete Mathematics, vol. 21 (1978), pp. 31–46.CrossRefGoogle Scholar
[Hi78]Hinman, P., Recursion-theoretic hierarchies, Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[KSW86]Köbler, J., Shöning, U., and Wagner, K. W., The difference and truth-table hierarchies for NP, Preprint 7, Department of Informatics, Koblenz, 1986.Google Scholar
[KM67]Kuratowski, K. and Mostowski, A., Set theory, North Holland, Amsterdam, 1967.Google Scholar
[Lo83]Louveau, A., Some results in the Wadge hierarchy of Borel sets, Cabal seminar '79-81 (Kechris, A. S., Martin, D. A., and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, 1983, pp. 28–55.CrossRefGoogle Scholar
[Mo80]Moschovakis, Y. N., Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[Och55]Ochan, Y. S., Theory of operations on sets, Uspekhi Matematicheskikh Nauk, vol. 65 (1955), pp. 71–128, in Russian.Google Scholar
[Ro67]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[Se82]Selivanov, V. L., On index sets in the Kleene-Mostowski hierarchy, Trudy Matematicheskogo Instituta Sbornik Obzornykh Akademiya Nauk SSSR (1982), pp. 135–158, in Russian, there is an English translation.Google Scholar
[Se83]Selivanov, V. L., Hierarchies of hyperarithmetical sets and functions, Algebra and Logic, vol. 22 (1983), pp. 666–692, in Russian, there is an English translation.CrossRefGoogle Scholar
[Se84]Selivanov, V. L., Index sets in hyperarithmetical hierarchy, Siberian Mathematical Journal, vol. 25 (1984), pp. 164–181, in Russian, there is an English translation.Google Scholar
[Se89]Selivanov, V. L., Fine hierarchies of arithmetical sets and definable index sets, Trudy Matematicheskaga Instituta Akademiya Nauk SSSR, vol. 12 (1989), pp. 165–185, in Russian.Google Scholar
[Se91]Selivanov, V. L., Fine hierarchies and definable index sets, Algebra and Logic, vol. 30 (1991), pp. 705–725, in Russian, there is an English translation.CrossRefGoogle Scholar
[Se91a]Selivanov, V. L., Fine hierarchy of formulas, Algebra and Logic, vol. 30 (1991), pp. 568–582, in Russian, there is an English translation.CrossRefGoogle Scholar
[Se92]Selivanov, V. L., computing degrees of definable classes of sentences, Comtemprory Mathematics, vol. 131 (1992), pp. 657–666.Google Scholar
[St80]Steel, J., Determinateness and the separation property, this Journal, vol. 45 (1980), pp. 143–146.Google Scholar
[VW78]van Wesep, R., Wadge degrees and descriptive set theory, Cabal seminar 76–77 (Kechris, A. S., Martin, D. A., and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 689, Springer-Verlag, 1978, pp. 151–170.CrossRefGoogle Scholar
[We85]Wechsung, G., On the Boolean closure of NP, Proceedings of the 1985 international conference on fundamentals of computation theory, Lecture Notes in Computer Science, vol. 199, Springer-Verlag, 1985, pp. 485–493.Google Scholar