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An application of boolean complexity to separation problems in bounded arithmetic
 Proc. London Math. Society
, 1994
"... 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 ( ..."
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Cited by 54 (15 self)
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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+1conservative over T'2 [3], but it is not V2!f+2conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomialtime 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].
Lower Bounds to the Size of ConstantDepth Propositional Proofs
, 1994
"... 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 ..."
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Cited by 53 (6 self)
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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...
Relating the Bounded Arithmetic and Polynomial Time Hierarchies
 Annals of Pure and Applied Logic
, 1994
"... 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 / ..."
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Cited by 27 (1 self)
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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 .
An Application of Boolean Complexity to
"... We develop a method for establishing the independence of some V,!()formulas from S(). In particular, we show that T() is not VV,!()conservative over S(). ..."
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We develop a method for establishing the independence of some V,!()formulas from S(). In particular, we show that T() is not VV,!()conservative over S().