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Journal of Logic and Computation Advance Access published online on February 6, 2008
Journal of Logic and Computation, doi:10.1093/logcom/exn003
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

On the Hierarchy of Intuitionistic Bounded Arithmetic

Morteza Moniri

Department of Mathematics, Shahid Beheshti University, Evin, Tehran, Iran.

E-mail: m-moniri{at}cc.sbu.ac.ir, ezmoniri{at}gmail.com

Received 22 January 2007.


  Abstract

In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory Formula defined and studied by Cook and Urquhart as the first level. We prove level by level equivalence between the two hierarchies (for the first level, the fact was first proved by Cook and Urquhart using realizability and functional interpretation and later by Buss by an elementary method). Next we investigate the question of whether the hierarchy, denoted Formula, collapses. We show that if Formula, then Formula and so the polynomial hierarchy collapses to Formula. Our proof for this is independent from earlier works on relating the collapse of the hierarchy of classical bounded arithmetic and the collapse of the polynomial hierarchy. We give an elementary model theoretic proof using only the basic properties of the theories Formula and we do not use results which belong to Cook and Urquhart and also Harnik that characterize the definable functions of these theories with long witnessing proofs.

Keywords: Bounded arithmetic; intuitionistic logic; polynomial hierarchy; kripke model


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