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Truth definitions in finite models

Published online by Cambridge University Press:  12 March 2014

Leszek Aleksander Kołodziejczyk
Leszek Aleksander Kołodziejczyk*
Affiliation:
Institute of Philosophy, Warsaw University, Krakowskie Przedmieście 3, 00-047 Warsaw, Poland, E-mail: [email protected]
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Abstract

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski's classical notion of truth definition.

We compare FM-truth definitions with Vardi's concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic does not depend on the syntax of , as long as it is decidable. It follows that for a natural there exist FM-truth definitions whose evaluation is much easier than the combined complexly of would suggest.

We apply the general theory to give a complexity-theoretical characterization of the logics for which the classes (prenex classes of higher order logics) define FM-truth. For any d ≥ 2, m ≥ 1 we construct a family of syntactically defined fragments of which satisfy this characterization. We also use the classes to give a refinement of known results on the complexity classes captured by .

We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary σ, of a fixed number k such that model checking for all first order sentences over σ can be done in deterministic time nk.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 1 , March 2004 , pp. 183 - 200
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1]Balcázar, J. L., Díaz, J., and Gabarró, J., Structural complexity I, second, revised ed., Springer-Verlag, 1995.CrossRefGoogle Scholar
[2]Bennett, J., On spectra, Ph. D. Thesis, Princeton University, 1962.Google Scholar
[3]Börger, E., Decision problems in predicate logic, Logic colloquium '82 (Lolli, G.et al., editors), North-Holland, 1984, pp. 263–301.Google Scholar
[4]Christen, C. A., Spektren und Klassen Elementarer Funktionen, Ph. D. Thesis, ETH Zürich, 1974.Google Scholar
[5]Eiter, T., Gottlob, G., and Gurevich, Y., Normal forms for second-order logic over finite structures, and classification of NP optimization problems, Annals of Pure and Applied Logic, vol. 78 (1996), pp. 111–125.CrossRefGoogle Scholar
[6]Fagin, R., Generalized first-order spectra and polynomial-time recognizable sets, Complexity of Computation, S1AM-AMS Proceedings, vol. 7 (1974), pp. 43–73.Google Scholar
[7]Gottlob, G., Relativized Logspace and generalized quantifiers over finite ordered structures, this Journal, vol. 62 (1997), pp. 545–574.Google Scholar
[8]Gottlob, G., Leone, N., and Veith, H., Succinctness as a source of complexity in logical formalisms, Annals of Pure and Applied Logic, vol. 97 (1999), pp. 231–260.Google Scholar
[9]Hella, L. and Torres, J. M. Turull, Expressing database queries with higher order logics, Technical Report 5/2003, Information Systems Department, Massey University, 2003.Google Scholar
[10]Immerman, N., Descriptive complexity, Springer-Verlag, 1999.CrossRefGoogle Scholar
[11]Leivant, D., Descriptive characterizations of computational complexity, Journal of Computer and System Sciences, vol. 39 (1987), pp. 51–83.CrossRefGoogle Scholar
[12]Leivant, D., Higher order logic, Handbook of logic in artificial intelligence and logic programming (Gabbay, D. M.et al., editors), vol. 2, Oxford University Press, 1994, pp. 228–321.Google Scholar
[13]Makowsky, J. A. and Pnueli, Y. B., Arity and alternation in second order logic, Annals of Pure and Applied Logic, vol. 78 (1996), pp. 189–202.Google Scholar
[14]Mostowski, M., On representing concepts infinite models, Mathematical Logic Quarterly, vol. 47 (2001). pp. 513–523.3.0.CO;2-J>CrossRefGoogle Scholar
[15]Mostowski, M., On representing semantics infinite models, to appear in Proceedings of Contributed Papers, LMPhSc 99.Google Scholar
[16]Stockmeyer, L., The polynomial-time hierarchy, Theoretical Computer Science, vol. 3 (1977). pp. 1–22.CrossRefGoogle Scholar
[17]Tarski, A., Pojȩcie prawdy w językach nauk dedukcyjnych, Warszawskie Towarzystwo Naukowe, Warszawa, 1933, English translation of the German version: The concept of truth in formalized languages, in A. Tarski. Logic, semantics, metamathematics, Oxford University Press. 1956. pp. 152–278.Google Scholar
[18]Vardi, M., The complexity of relational query languages, Proceedings of the 14th ACM Symposium on Theory of Computing, 1982, pp. 137–146.Google Scholar
[19]Zdanowski, K., Arithmetics in finite hut potentially infinite worlds, Ph. D. Thesis, Warsaw University, in preparation.Google Scholar