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

Published online by Cambridge University Press:  12 March 2014

Leszek Aleksander Kołodziejczyk  and
Neil Thapen
Leszek Aleksander Kołodziejczyk
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, E-mail: lak@mimuw.edu.pl
Neil Thapen
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic Žitná 25, CZ-115 67 Praha 1, Czech Republic, E-mail: thapen@math.cas.cz
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 578 - 592
Copyright
Copyright © Association for Symbolic Logic 2008

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