On the Hierarchy of Intuitionistic Bounded Arithmetic

摘要

In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory IS_2~1 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 IS'2, collapses. We show that if IS_2~i∣― IS_2~(i+1), then S_2~i(PV)1-Σ_i~b = ∏_i~b and so the polynomial hierarchy collapses to Σ_t~p = ∏_i~p. 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 IS'-, 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.