We show that every model of I \Delta 0 has an end extension to a model of a theory (extending Buss' S 0 2 ) where logspace computable function are formalizable. We also show the existence of an isomorphism between models of I \Delta 0 and models of linear arithmetic LA (i.e., secondorder Presburger arithmetic with finite comprehension for bounded formulas). 0 Introduction. In the last two decades the research on bounded arithmetic has been focussed mainly on the theories of I \Delta 0 and I \Delta 0+\Omega 1 and on their fragments. The interest in these theories is in part motivated by complexity theoretical considerations. It is well known that the provably recursive function of I \Delta 0+\Omega 1 are those computable by algorithms in the polynomial time hierarchy while the provably recursive functions of I \Delta 0 correspond to the linear time hierarchy. Some open problems in bounded arithmetic are known to be equivalent to problems of complexity theory. More precisely, to the s...

We give combinatorial principles GIk, based on k-turn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a non-logical question about multiparty communication protocols.

In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1-formulas does not imply the intuitionistic theory IS1 2 of polynomial induction on Σ b+ 1-formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polynomial induction. We also investigate the relation between various other intuitionistic first-order theories of bounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories.

Classication of Search Problems and Their Denability in Bounded Arithmetic Tsuyoshi Morioka Master of Science Graduate Department of Computer Science University of Toronto 2001 We present a new framework for the study of search problems and their denability in bounded arithmetic. We identify two notions of complexity of search problems: veri- cation complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b i -denability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witness-oracle Turing machines.

The complexity class of Π p k-Polynomial Local Search (PLS) problems with Π p ℓ-goal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the Paris-Wilkie-translation. For ℓ ≤ k, the Σb ℓ+1-definable search problems of T k+1 2 are exactly characterised by Π p k-PLS problems with Πp ℓ-goals. These Π p k-PLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p k-PLS definitions can be Skolemised with simple polynomial time functions. The Skolemised Π p k-PLS definitions give rise to a new ∀Σb1(α) principle conjectured to separate Tk 2(α) from T k+1 2 (α). 1