Abstract

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].

Citations