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



Journal of Symbolic Logic

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

Leszek Aleksander Kołodziejczyk and Neil Thapen

Source: J. Symbolic Logic Volume 73, Issue 2 (2008), 578-592.

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 S21 exists in which NP is not in the second level of the linear hierarchy; and that a model of S21 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 S21 the failure of WPHP (for Σb1 definable relations) implies that the strict version of PH does not collapse to a finite level.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208359061
Digital Object Identifier: doi:10.2178/jsl/1208359061
Mathematical Reviews number (MathSciNet): MR2414466

References

S. Buss, Bounded arithmetic, Bibliopolis, 1986.
Mathematical Reviews (MathSciNet): MR880863
S. Cook and N. Thapen, The strength of replacement in weak arithmetic, ACM Transactions on Computational Logic, vol. 7 (2006), no. 4.
Mathematical Reviews (MathSciNet): MR2264422
E. Jeřábek, On independence of variants of the weak pigeonhole principle, Journal of Logic and Computation, vol. 17 (2007), no. 3, pp. 587--604.
Mathematical Reviews (MathSciNet): MR2334518
Digital Object Identifier: doi:10.1093/logcom/exm017
J. Krajíček, Bounded arithmetic, propositional logic, and complexity theory, Cambridge University Press, 1995.
Mathematical Reviews (MathSciNet): MR1366417
--------, On the weak pigeonhole principle, Fundamenta Mathematicae, vol. 170 (2001), pp. 123--140.
Mathematical Reviews (MathSciNet): MR1881373
J. Krajíček and P. Pudlák, Some consequences of cryptographical conjectures for $\text\upshape S^1_2$ and $\text\upshape EF$, Information and Computation, vol. 140 (1998), pp. 82--89.
Mathematical Reviews (MathSciNet): MR1492845
Digital Object Identifier: doi:10.1006/inco.1997.2674
A. Maciel, T. Pitassi, and A. R. Woods, A new proof of the weak pigeonhole principle, Journal of Computer and System Sciences, vol. 64 (2002), pp. 843--872.
Mathematical Reviews (MathSciNet): MR1912305
Digital Object Identifier: doi:10.1006/jcss.2002.1830
J. B. Paris and A. J. Wilkie, Counting problems in bounded arithmetic, Methods in mathematical logic, Lecture Notes in Mathematics, vol. 1130, Springer--Verlag, 1985, pp. 317--340.
Mathematical Reviews (MathSciNet): MR799046
Zentralblatt MATH: 0572.03034
Digital Object Identifier: doi:10.1007/BFb0075316
J. B. Paris, A. J. Wilkie, and A. R. Woods, Provability of the pigeonhole principle and the existence of infinitely many primes, Journal of Symbolic Logic, vol. 53 (1988), pp. 1235--1244.
Mathematical Reviews (MathSciNet): MR973114
Digital Object Identifier: doi:10.2307/2274618
Project Euclid: euclid.jsl/1183742795
C. Pollett, Multifunction algebras and the provability of $PH\downarrow$, Annals of Pure and Applied Logic, vol. 104 (2000), pp. 279--303.
Mathematical Reviews (MathSciNet): MR1778942
Digital Object Identifier: doi:10.1016/S0168-0072(00)00015-4
J. P. Ressayre, A conservation result for systems of bounded arithmetic, unpublished manuscript, 1986.
N. Thapen, A model-theoretic characterization of the weak pigeonhole principle, Annals of Pure and Applied Logic, vol. 118 (2002), pp. 175--195.
Mathematical Reviews (MathSciNet): MR1934122
Digital Object Identifier: doi:10.1016/S0168-0072(02)00038-6
--------, Structures interpretable in models of bounded arithmetic, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 247--266.
Mathematical Reviews (MathSciNet): MR2169685
Digital Object Identifier: doi:10.1016/j.apal.2005.04.005
D. Zambella, Notes on polynomially bounded arithmetic, Journal of Symbolic Logic, vol. 61 (1996), pp. 942--966.
Mathematical Reviews (MathSciNet): MR1412519
Digital Object Identifier: doi:10.2307/2275794
Project Euclid: euclid.jsl/1183745086

2008 © Association for Symbolic Logic