We introduce some Π1–expressible combinatorial principles which may be treated as axioms for some bounded arithmetic theories. The principles, denoted Sk(Σ b n, length log k) and Sk(Σ b n, depth log k) (where ‘Sk ’ stands for ‘Skolem’), are related to the consistency of Σ b n–induction: for instance, they provide models for Σ b n–induction. However, the consistency is expressed indirectly, via the existence of evaluations for sequences of terms. The evaluations do not have to satisfy Σ b n– induction, but must determine the truth value of Σ b n statements. Our principles have the property that Sk(Σ b n, depth log k) proves Sk(Σ b n+1, length logk). Additionally, Sk(Σ b n, length log k−2) proves Sk(Σ b n+1, length logk). Thus, some provability is involved where conservativity is known in the case of Σ b n induction on an initial segment and induction for higher Σ b m classes on smaller segments. 1

Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way. 1

We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordinals as universal measures for implicit computational complexity. I.e., we describe the connections between generalised dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories. In particular, through the determination of generalised dynamic ordinals we re-obtain well-known independence results for relativised bounded arithmetic theories.

The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss ’ systems S2. We show if IEi(exp) ⊢ A with a proof P of expindrank(P) ≤ n + 1 where all ( ∀ ≤: right) or ( ∃ ≤: left) have bounding terms not containing function symbols then S i 2 ⊇ IEi,2 ⊢ A n. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) � I∆0(exp). Here I∆0(exp) is a conservative extension of I∆0+exp obtained by adding to I∆0 a symbol for 2 x to the language as well as defining axioms for it.

. We define a hierarchy of bounded arithmetic L i 2 for i 1 which are modifications of TAC of Clote and Takeuti. Provably total functions of L i 2 are characterized in terms of circuit complexity. As a consequence non-conservation results are obtained for L i 2 and L i+2 2 by applying a lower bound for st-connectivity. A weaker non-conservation result is also obtained between AC 0 CA and L i 2 using the KPT witnessing theorem. 1 Introduction Theories of bounded arithmetic are, in general, axiomatized by defining axioms for symbols in the language together with restricted forms of well-known axiom schemata. S. Buss introduced the theory S 1 2 which has a weak induction scheme called polynomial induction (PIND) restricted to NP formulae. Much weaker induction was used to define theories for boolean circuit classes. B. Allen [1] and P. Clote and G. Takeuti [5] independently defined theories whose provably total functions correspond to the class NC. Both of them used essenti...