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].

1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp(O(log 2 n)) but such that every depth d refutation must have the size at least exp(n\Omega\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...

The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no one-to-one mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR-9877150. y Department of Computer Science, University o...

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 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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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total. It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective.

) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 91093-0112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we dene certain second-order theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV -isomorphism. 1 Introduction A phenomenon that is commonly observed in Complexity Theory is that proofs of results about counting complexity classes (#P , Mod p P etc.) can often be scaled down to yield results about small depth circuit classes with the corresponding counting gates. For example, Toda's result [17] that every problem in the Polynomial Hierarchy can be solved in polynomial time with an oracle for #P correspond...