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The polynomial and linear hierarchies in models where the weak pigeonhole principle fails

  • Leszek Aleksander Kołodziejczyk (a1) and Neil Thapen (a2)

Abstract

We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of exists in which NP is not in the second level of the linear hierarchy; and that a model of exists in which the polynomial hierarchy collapses to the linear hierarchy.

Our methods are model-theoretic. We use the assumption about factoring to get a model in which the weak pigeonhole principle fails in a certain way, and then work with this failure to obtain our results.

As a corollary of one of the proofs, we also show that in the failure of WPHP (for definable relations) implies that the strict version of PH does not collapse to a finite level.

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[Bus86]Buss, S., Bounded arithmetic, Bibliopolis, 1986.
[CT06]Cook, S. and Thapen, N., The strength of replacement in weak arithmetic, ACM Transactions on Computational Logic, vol. 7 (2006), no. 4.
[Jer07] E. Jeřábek, On independence of variants of the weak pigeonhole principle, Journal of Logic and Computation, vol. 17 (2007), no. 3, pp. 587604.
[Kra95] J. Krajíček, Bounded arithmetic, prepositional logic, and complexity theory, Cambridge University Press, 1995.
[Kra01] J. Krajíček, On the weak pigeonhole principle, Fundamenta Mathematicae, vol. 170 (2001), pp. 123140.
[KP98] J. Krajíček and Pudlák, P., Some consequences of cryptographical conjectures for and EF, Information and Computation, vol. 140 (1998), pp. 8289.
[MPW02]Maciel, A., Pitassi, T., and Woods, A. R., A new proof of the weak pigeonhole principle, Journal of Computer and System Sciences, vol. 64 (2002), pp. 843872.
[PW85]Paris, J. B. and Wilkie, A. J., Counting problems in bounded arithmetic. Methods in mathematical logic, Lecture Notes in Mathematics, vol. 1130, Springer-Verlag, 1985, pp. 317340.
[PWW88]Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 12351244.
[Po1OO]Pollett, C., Multifunction algebras and the provability of PH ↓, Annals of Pure and Applied Logic, vol. 104 (2000), pp. 279303.
[Res]Ressayre, J. P., A conservation result for systems of bounded arithmetic, unpublished manuscript, 1986.
[Th02]Thapen, N., A model-theoretic characterization of the weak pigeonhole principle, Annals of Pure and Applied Logic, vol. 118 (2002), pp. 175195.
[Th05]Thapen, N., Structures interpretable in models of bounded arithmetic. Annals of Pure and Applied Logic, vol. 136 (2005), pp. 247266.
[Zam96]Zambella, D., Notes on polynomially bounded arithmetic, this Journal, vol. 61 (1996), pp. 942966.

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The polynomial and linear hierarchies in models where the weak pigeonhole principle fails

  • Leszek Aleksander Kołodziejczyk (a1) and Neil Thapen (a2)

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