Journal of Logic and Computation Advance Access originally published online on February 6, 2008
Journal of Logic and Computation 2008 18(4):625-630; doi:10.1093/logcom/exn003
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Original Articles |
On the Hierarchy of Intuitionistic Bounded Arithmetic
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.
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Abstract |
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In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory 
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 
, collapses. We show that if 
, then 
and so the polynomial hierarchy collapses to 
. 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 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