We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of second-order bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...

We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV -isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a by-product of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.

In this paper we study the pairs (U; V ) of disjoint NP-sets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NP-pair which is complete for the class of disjoint NP-pairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.

We investigate the possibility to characterize (multi)functions that are \Sigma b i -definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are \Sigma b 1 - and \Sigma b 2 -definable in the theories S 1 2 and T 1 2 . 2. New characterizations of (multi)functions that are \Sigma b 2 - and \Sigma b 3 - definable in the theory T 2 2 . 3. A new non-conservation result: the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)- conservative over the theory S 2 2 (ff). To prove that the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)-conservative over the theory S 2 2 (ff), we present two examples of a \Sigma b 1 (ff)-principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 ; f; g) formalizing that no function f is a bijection between a 2 and a with the inverse...

We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

This paper is motivated by the questions: what are the \Sigma

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