On the Hierarchy of Intuitionistic Bounded Arithmetic.

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Bibliographic Details
Title:	On the Hierarchy of Intuitionistic Bounded Arithmetic.
Authors:	MONIRI, MORTEZA¹ ezmoniri@gmail.com
Source:	Journal of Logic & Computation. Aug2008, Vol. 18 Issue 4, p625-630. 6p.
Subjects:	Mathematical logic, Combinatory logic, Nonclassical mathematical logic, Computer logic, Constructive mathematics, Polynomials, Algebra
Abstract:	In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory IS12 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 ISi2, collapses. We show that if ISi2 ├ ISi+12, then Si2(PV) ├ Σbi = Πbi and so the polynomial hierarchy collapses to Σpi=Πpi. 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 ISi2 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. [ABSTRACT FROM AUTHOR]
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Abstract:	In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory IS12 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 ISi2, collapses. We show that if ISi2 ├ ISi+12, then Si2(PV) ├ Σbi = Πbi and so the polynomial hierarchy collapses to Σpi=Πpi. 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 ISi2 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. [ABSTRACT FROM AUTHOR]
ISSN:	0955792X