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A finite model-theoretical proof of a property of bounded query classes within PH

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: 1.kolodziejczyk@zodiac.mimuw.edu.pl
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Abstract.

We use finite model theory (in particular, the method of FM-truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove:

Let m ≥ 2. Then:

(A) If there exists k such that NP⊆ Σm TIME(nk)∩ Πm TIME(nk), then for every r there exists kr such that :

(B) If there exists a superpolynomial time-constructible function f such that NTIME(f), then additionally .

This strengthens a result by Mocas [M96] that for any r, .

In addition, we use FM-truth definitions to give a simple sufficient condition for the arity hierarchy to be strict over finite models.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 4 , December 2004 , pp. 1105 - 1116
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[A83]Ajtai, M., sentences on finite structures, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 1–48.CrossRefGoogle Scholar
[BFFT01]Buhrman, H., Fenner, S., Fortnow, L., and Torenvliet, L., Two oracles that force a big crunch, Computational Complexity, vol. 10 (2001), pp. 93–116.CrossRefGoogle Scholar
[BT94]Buhrman, H. and Torenvliet, L., On the cutting edge of relativization: the resource bounded injury method, ICALP proceedings, Springer-Verlag, Berlin, 1994, pp. 263–273.Google Scholar
[BH91]Buss, S. R. and Hay, L., On truth-lable reducibility to SAT, Information and Computation, vol. 91 (1991), pp. 86–102.CrossRefGoogle Scholar
[F74]Fagin, R., Generalized first-order spectra and polynomial-time recognizable sets, Complexity of Computation, SIAM-AMS Proceedings, vol. 7, American Mathematical Society, Providence, R.I., 1974, pp. 43–73.Google Scholar
[F75]Fagin, R., A spectrum hierarchy, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 123–134.CrossRefGoogle Scholar
[FLZ92]Fu, B., Li, H., and Zhong, Y., Some properties of exponential time complexity classes, Proceedings of the Structure in Complexity Theory Conference, IEEE Computer Society Press, 1992, pp. 50–57.Google Scholar
[G97]Gottlob, Georg, Relativized logspace and generalized quantifiers over finite ordered structures, this Journal, vol. 62 (1997), no. 2, pp. 545–574.Google Scholar
[HP93]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
[K04]Kolodziejczyk, L. A., Truth definitions in finite models, this Journal, vol. 69 (2004), pp. 183–200.Google Scholar
[M96]Mocas, S. E., Separating classes in the exponential-time hierarchy from classes in, PH, Theoretical Computer Science, vol. 158 (1996), pp. 221–231.Google Scholar
[MM01]Mostowski, M., On representing concepts in finite models, Mathematical Logic Quarterly, vol. 47 (2001), pp. 513–523.3.0.CO;2-J>CrossRefGoogle Scholar
[Ste93]Stewart, Iain A., Logical characterizations of bounded query classes, I. Logspace oracle machines, Fundamenta Informaticae, vol. 18 (1993), pp. 65–92.Google Scholar
[Sto77]Stockmeyer, L., The polynomial-time hierarchy, Theoretical Computer Science, vol. 3 (1977), pp. 1–22.Google Scholar
[W90]Wagner, K. W., Bounded query classes, SIAM Journal on Computing, vol. 19 (1990), pp. 833–846.CrossRefGoogle Scholar