Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-18T15:11:57.166Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexity of interpolation and related problems in positive calculi

Published online by Cambridge University Press:  12 March 2014

Larisa Maksimova
Larisa Maksimova*
Affiliation:
Institute of Mathematics, Siberian Division of Russian Academy of Sciences, 630090 Novosibirsk, Russia, E-mail: lmaksi@math.nsc.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity. and interpolation problems over the intuitionistic logic Int and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.

In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int+. In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 397 - 408
Copyright
Copyright © Association for Symbolic Logic 2002

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

[I]Chagrov, A. and Zakharyaschev, M., Modal logics, Clarendon Press, Oxford, 1997.CrossRefGoogle Scholar
[2]Craig, W., Three uses of Herbrand-Gentzen theorem in relating model theory, this Journal, vol. 22 (1957), pp. 269–285.Google Scholar
[3]Fisher, M. J. and Ladner, R. E., Prepositional dynamic logic of regular programs, Journal of Computer and Systems Sciences, vol. 18 (1979), pp. 194–211.CrossRefGoogle Scholar
[4]Halpern, J. Y. and Moses, Y., A guide to completeness and complexity for modal logics of knowledge and belief, Artificial Intelligence, vol. 54 (1992), pp. 319–379.CrossRefGoogle Scholar
[5]Hosoi, T. and Ono, H., Intermediate prepositional logics. A survey, Journal of Tsuda College, vol. 5 (1973), pp. 67–82.Google Scholar
[6]Johnson, D. S., A catalog of complexity classes, Handbook of theoretical computer science (van Leeuwen, J., editor), vol. A, Elsevier, 1990, pp. 67–161.Google Scholar
[7]Kreisel, G., Explicit definability in intuitionistic logic, this Journal, vol. 25 (1960), pp. 389–390.Google Scholar
[8]Kuznetsov, A. V., Some properties of the lattice of varieties ofpseudo-boolean algebras, 11th Sovjet algebraic colloquium, Abstracts, Kishinev, 1971, pp. 255–256.Google Scholar
[9]Ladner, R. E., The computational complexity of provability in systems of modal prepositional logics, SIAM Journal of Computing, vol. 6 (1977), no. 3, pp. 467–480.CrossRefGoogle Scholar
[10]Maksimova, L., Intuitionistic logic and implicit definability, Annals of Pure and Applied Logic, vol. 105 (2000), pp. 83–102.CrossRefGoogle Scholar
[11]Maksimova, L. and Voronkov, A., Complexity of some problems in modal and superintuitionistic logics, Logic colloquium '99, Abstracts, Utrecht, 1999, Bulletin of Symbolic Logic, vol. 6 (2000), pp. 118–119.Google Scholar
[12]Maksimova, L. L., Implicit definability and positive logics, to appear in Algebra and Logic.Google Scholar
[13]Maksimova, L. L., Pretabular superintuitionistic logics, Algebra and Logic, vol. 11 (1972), pp. 558–570.CrossRefGoogle Scholar
[14]Maksimova, L. L., Craig's interpolation theorem and amalgamable varieties, Doklady AN SSSR, vol. 237 (1977), pp. 1281–1284.Google Scholar
[15]Maksimova, L. L., Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudoboolean algebras, Algebra and Logic, vol. 16 (1977), pp. 643–681.CrossRefGoogle Scholar
[16]Maksimova, L. L., Decidability of projective Beth's property in varieties of Heyting algebras, Algebra and Logic, vol. 40 (2001), pp. 290–301.CrossRefGoogle Scholar
[17]Miura, S., A remark on the intersection of two logics, Nagoya Mathematics Journal, vol. 26 (1966), pp. 167–171.CrossRefGoogle Scholar
[18]Papadimitriou, C. H., Computational complexity, Addison-Wesley, Reading, Massachusets, 1994.Google Scholar
[19]Pratt, V. R., Models of program logics, Proceedings 20th IEEE symposium on foundations of computer science, 1979, pp. 115–122.Google Scholar
[20]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, Polish Scientific Publishers, Warszawa, 1963.Google Scholar
[21]Spaan, E., Complexity of modal logics, Dissertation, Institute for Logic, Language and Computation, University of Amsterdam, 1992.Google Scholar
[22]Statman, R., Intuitionistic logic is polynomial-space complete, Theoretical Computer Science, vol. 9 (1979), pp. 67–72.CrossRefGoogle Scholar