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 show that polynomial time truth-table reducibility via Boolean circuits to SAT is the same as logspace truth-table reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT is equivalent to one round of parallel queries.

The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 /poly .

this article will deal only with the intensional approach. The reader who wants to see the numeralwise representability approach can consult Smorynski [1977] and any number of textbooks such as Mendelson [1987]. The intensional approach is due to Feferman [1960]. An e#ective unification of the two approaches can be given using the fact (independently due to Wilkie and to Nelson [1986]) that I# 0 +# 1 and S 2 are interpretable in Q ; since both I# 0 +# 1 and S 2 admit a relatively straightforward intensional arithmetization of metamathematics (see Wilkie and Paris [1987] and Buss [1986]), this allows strong forms of incompleteness obtained via the intensional approach to apply also to the theory Q ; paragraph 2.1.4 below sketches how the interpretation of S 2 in Q can be used to give an intensional arithmetization in Q . The book of Smullyan [1992] gives a modern, in-depth treatment of Godel's incompleteness theorems

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.

The complexity class of Π p k-polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σ p i-definable functions of T k+1 2 are characterized in terms of Π p k-PLS problems. These Π p k-PLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p k-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σb 1(α)-principle that is conjectured to separate T k 2 (α) and T k+1 2 (α). 1

This paper deals with the weak fragments of arithmetic PV and S and their induction-free fragments PV - and S -1 2 . We improve the bootstrapping of S 2 , which allows us to show that the theory S 2 can be axiomatized by the set of axioms BASIC together with any of the following induction schemas: # 1 -PIND , # 1 -PIND or # 1 -LIND .

This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence results.

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.

this paper. 2 General orderings This section states a couple results about general orderings. By a "general ordering" we mean any order defined by a # 1 -formula; by comparison the results of sections 3 and 4 concern specific natural well-orderings based on ordinal notations