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 develop a method for establishing the independence of some Zf(a)-formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)-conservative over S'2(a). We characterize the Z^-definable functions of T2 as being precisely the functions definable as projections of polynomial local search (PLS) problems. Although it is still an open problem whether bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1-conservative over T'2 [3], but it is not V2!f+2-conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1-conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomial-time hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute results: S'2 + (a) is V2f+,(a)-conservative but not VZf+2(a)-conservative over T'2(a), and T'2(a) is not VZf+i(c*)-conservative over S'2(a). Here a represents a new uninterpreted predicate symbol adjoined to the language of arithmetic which may be used in induction formulas; from a computer science perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show the unprovability of other 2f(a)-formulas from S'2(a). Our methods are analogous in spirit to the proof strategy of [8]: prove a witnessing theorem to show that provability of a Zf+1(a)-formula A in S'2(a) implies that it is witnessed by a function of certain complexity and then employ techniques of boolean complexity to construct an oracle a such that the formula A cannot be witnessed by a function of the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas [6].

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.

Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove

. We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations. Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special attention to recent attempts on reducing such bounds to some purely complexity results or assumptions. As one of the main motivations for this research we discuss provability of extremely important propositional formulae that express hardness of explicit Boolean functions with respect to various non-uniform computational models. 1. Propositional proofs as feasible proofs of plain statements Interesting and viable logical theories do not appear as result of sheer speculation. Conversely, they attempt to summarize and capture a certain amount of reasoning of a certain style about a certain class of objects that had existed in the math community before the mathematical logics entered the stage....

Abstract. We study the proof–theoretic strength and effective content denote Ram-of the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X ′ ′ ≤T 0 (n). Let I�n and B�n denote the �n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models is conservative of arithmetic enables us to show that RCA0 + I �2 + RT2 2 over RCA0 + I �2 for �1 1 statements and that RCA0 + I �3 + RT2 < ∞ is �1 1-conservative over RCA0 + I �3. It follows that RCA0 + RT2 2 does not imply B �3. In contrast, J. Hirst showed that RCA0 + RT2 < ∞ does imply B �3, and we include a proof of a slightly strengthened version of this result. It follows that RT2 < ∞ is strictly stronger than RT2 2 over RC A0. 1.

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.

Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents

. Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a quasitautology. This is an important problem in logic, specifically in the complexity of proofs. In computer science, specifically in automated theorem proving, one hopes for an algorithm which avoids the guesses of existential substitution axioms involved in proving a theorem. Herbrand's theorem forms the very basis of automated theorem proving where for a given number n we would like to have an algorithm which finds the terms in the n disjunctions of matrices solely from the shape of the matrix. The main result of this paper is that both problems have negative solutions. 1 Introduction By the theorem of Herbrand we have for a quantifier-free OE: j= 9 x OE( x) iff j= OE( a 1 ) OE( ...

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.