Structure and Definability in General Bounded Arithmetic Theories

The bounded arithmetic theories Ri 2, S i 2, and T i 2 are closely connected with complexity theory. This paper is motivated by the questions: what are the Σi+1-definable multifunctions of R i 2? and when is one theory conservative over another? To answer these questions we consider theories R̂i 2, Ŝ i 2, and T̂ i 2 where induction is restricted to prenex formulas. We also define T̂ i,τ 2 which has induction up to the 0 or 1-ary L2-terms in the set τ . We show Ŝi 2 = S i 2 and T̂ i 2 = T i 2 and for i > 1, R̂ i 2 1B(Σ̂bi ) R i 2. We show that the Σ̂i+1-multifunctions of T̂ i,τ 2 are FP Σpi (wit, |τ |) and that those of R̂i 2 are FPΣ p i (wit, log log). For Σ̂i+k+2-definability we get FP Σ i+k+1(wit, 1) for all these theories. Write 2τ̇ for the set of terms 2min(`(x),|t(x)|) where ` is a finite product of terms in τ and t ∈ L2. We prove T̂ i,2 τ̇