Abstract not found

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds ...

We show that there is a pair of disjoint NP-sets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admitting an NP -definition of primes about which it can prove that no number satisfying the definition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit feasible interpolation theorem unless the RSA cryptosystem is not secure, and that an extension of EF by tautologies p (p primes), formalizing that p is not composite, as additional axioms does not admit feasible interpolation theorem unless factoring and the discrete logarithm are in P=poly . The NP 6= coNP conjecture is equivalent to the statement that no propositional proof system (as defined in [6]) admits polynomial size proofs of all tautologies. However, only for few proof systems occur...

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...

. We consider some problems about pairs of disjoint NP sets.

We propose a framework for proving lower bounds to the size of EF - proofs (equivalently, to the number of proof-steps in F-proofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P -proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...

We obtain two results about the proof complexity of deep inference: 1) deep-inference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference proof systems that exhibit an exponential speed-up over analytic Gentzen proof systems that they polynomially simulate.

We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs.

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