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.

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We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and preserve the syntactic structure of bounded formulae. As a corollary, for the bounded domain case we obtain Buss's result concerning 1 -expressibility in V 2 as a direct consequence of his main result for rst order theories. Using only plain corollaries of the cut elimination theorem, we show that V 2 (BD) prove the same formulae where 8 stand for rst order quanti ers. Combined with the above mentioned result this gives an alternative proof of Buss's characterization of 2 functions. All this readily extends to the case V k (BD) vs. S k+1 (i; k 1).

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

. We define an extension R 0 2 of the bounded arithmetic theory R 0 2 and show that the class of functions \Sigma b 1 -definable in R 0 2 coincides with the computational complexity class TC 0 of functions computable by polynomial size, constant depth threshold circuits. 1 Introduction The theories S i 2 , for i 2 N, of Bounded Arithmetic were introduced by Buss [3]. The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8xt , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj , 9x jtj is called sharply bounded. A formula is called (sharply) bounded if all quantifiers in it are (sharply) bounded. The class of bounded formulae is divided into an hierarchy analogous to the arithmetical hierarchy: The class of sharply bounded formulae is denoted \Sigma b 0 or \Pi b 0 . For i...

We define a property of substructures of models of arithmetic, that of being length-initial , and show that sharply bounded formulae are absolute between a model and its length-initial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as length-initial submodels of some given model. Mathematics Subject Classification: 03F30, 03H15 Introduction First we review the definitions of the theories S i 2 and T i 2 of Bounded Arithmetic introduced by S. Buss [2]: The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8x t , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj, 9x jtj is called sharply bounded. A formula is called sharply bounded if all quantifiers in it are sharply bounded. The class of sh...

Summary. The theory � b 1-CR of Bounded Arithmetic axiomatized by the � b 1-bit-comprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1-definable functions of � b 1-CR are those in uniform TC 0, and the � b 2-definable functions are computable by counterexample computations using TC 0-functions. The latter is used to show that a collapse of stronger theories to � b 1-CR implies that NP is contained in non-uniform TC 0. 1

) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursion-theoretic, proof-theoretic and machine-theoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifier-free, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...

. A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f .

This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k -definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely proof-theoretic and use the method of witness functions and witness oracles.