Annals of Pure and Applied Logic 100 (1-3):189-245 (1999)

Abstract
The bounded arithmetic theories R2i, S2i, and T2i are closely connected with complexity theory. This paper is motivated by the questions: what are the Σi+1b-definable multifunctions of R2i? and when is one theory conservative over another? To answer these questions we consider theories , and where induction is restricted to prenex formulas. We also define which has induction up to the 0 or 1-ary L2-terms in the set τ. We show and and for . We show that the -multifunctions of are FPΣip and that those of are FPΣip. For -definability we get FPΣi+k+1p for all these theories. Write for the set of terms 2min,t) where ℓ is a finite product of terms in τ and tL2. We prove and we show - INDτ provided τO2. This gives a proof theoretic proof that S2iΔi+1b-LIND and -LLIND solving an open problem. For τO2, we define using weak replacement axioms and show . We show if or if or if where τ′ has an unbounded term then PH=B. We separate PΣip from PΣip for behaved ℓ and deduce theory separations. We lastly introduce a notion of a model separating two theories and derive some consequences
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DOI 10.1016/s0168-0072(99)00008-1
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References found in this work BETA

Existence and Feasibility in Arithmetic.Rohit Parikh - 1971 - Journal of Symbolic Logic 36 (3):494-508.
Bounded Arithmetic and the Polynomial Hierarchy.Jan Krajíček, Pavel Pudlák & Gaisi Takeuti - 1991 - Annals of Pure and Applied Logic 52 (1-2):143-153.

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Citations of this work BETA

Multifunction Algebras and the Provability of PH↓.Chris Pollett - 2000 - Annals of Pure and Applied Logic 104 (1-3):279-303.
Dynamic Ordinal Analysis.Arnold Beckmann - 2003 - Archive for Mathematical Logic 42 (4):303-334.
Bootstrapping, Part I.Sedki Boughattas & J. -P. Ressayre - 2010 - Annals of Pure and Applied Logic 161 (4):511-533.

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