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y = 2x VS. y = 3x

Published online by Cambridge University Press:  12 March 2014

Alexei Stolboushkin  and
Damian Niwiński
Alexei Stolboushkin
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA, E-mail: [email protected]
Damian Niwiński
Affiliation:
Institute of Informatics, University of Warsaw, 02 097 Warsaw, Poland, E-mail: [email protected]
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Abstract

We show that no formula of first order logic using linear ordering and the logical relation y = 2x can define the property that the size of a finite model is divisible by 3. This answers a long-standing question which may be of relevance to certain open problems in circuit complexity.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 661 - 672
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Ajtai, M., formulae on finite structure, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 1–48.CrossRefGoogle Scholar
[2]Barrington, D. A., Bounded-width polynomial-size branching programs recognize exactly those languages in NC1, Journal of Computer and Systems Sciences, vol. 38 (1989), no. 1, pp. 150–164.CrossRefGoogle Scholar
[3]Barrington, D. A., Immerman, N., and Straubing, H., On uniformity within NC1, Structure in complexity theory: Third annual conference (Washington), IEEE Computer Society Press, 1988, pp. 47–59, the Journal version in: Journal of Computer and Systems Sciences, vol. 41 (1990), pp. 274–306.Google Scholar
[4]Barrington, D. A. M., Compton, K. J., Thérien, D., and Straubing, H., Regular languages in NC1, Journal of Computer and Systems Sciences, vol. 44 (1989), pp. 478–499.CrossRefGoogle Scholar
[5]Büchi, J., Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlagen Math., vol. 6 (1960), pp. 66–92.CrossRefGoogle Scholar
[6]Compton, K. and Straubing, H., Characterization of regular languages in low level complexity classes, Bulletin of the EATCS (1992), pp. 134–142.Google Scholar
[7]Ehrenfeucht, A., An application of games to the completeness problem for formalized theories, Fundamenta Mathematicae, vol. 49 (1961), pp. 129–141.CrossRefGoogle Scholar
[8]Fagin, R., Generalized first-order spectra and polynomial time recognizable sets, Complexity of computation (Karp, R. M., editor), SIAM-AMS Proceedings, vol. 7, 1974, pp. 43–73.Google Scholar
[9]Fraïssé, R., Sur quelques classifications des systèmes de relations, Publ. Sci. Univ.Alger. Ser. A, vol. 1 (1954), pp. 35–182.Google Scholar
[10]Furst, M., Saxe, J. B., and Sipser, M., Parity, circuits and the polynomial time hierarchy, Mathematical Systems Theory, vol. 17 (1984), pp. 13–27.CrossRefGoogle Scholar
[11]Gurevich, Yu. and Lewis, H. R., A logic for constant-depth circuits, Information and Control, vol. 61 (1984), no. 1, pp. 65–74.CrossRefGoogle Scholar
[12]Immerman, N., Languages which capture complexity classes, SIAM Journal on Computing, vol. 16 (1987), no. 4, pp. 760–778.CrossRefGoogle Scholar
[13]Immerman, N., Descriptive and computational complexity, Computational complexity theory (Hartmanis, J., editor), Proceedings of Symposia in Applied Mathematics, vol. 38, American Mathematical Society, 1989, pp. 75–91.CrossRefGoogle Scholar
[14]McNaughton, R. and Papert, S., Counter-free automata, MIT Press, Cambridge, 1971.Google Scholar
[15]Nurmonen, J., Counting modulo quantifiers on finite linearly ordered trees, Proceedings of the 11th LICS, 1996, to appear.Google Scholar
[16]Péladeau, P., Logically defined subsets of Nk, Proceedings of the MFCS, LNCS, vol. 379, 1989, pp. 397–407.Google Scholar
[17]Straubing, H., Thérien, D., and Thomas, W., Regular languages defined with generalized quantifiers, Proceedings of the 15th ICALP, 1989, pp. 561–575.Google Scholar
[18]Williams, J. W. J., Algorithm 232: Heapsort, Comm. ACM, vol. 7 (1964), no. 6, pp. 347–348.Google Scholar