The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its tree-like variant. The following consequences of these relations reveal width as a crucial "resource" of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the width-size relations naturally suggest a simple dynamic programming procedure for automated theorem proving - one which simply searches for small width proofs. This relation guarantees that the running time (and thus the size of the produced proof) is at most quasi-polynomial in the smallest tree-like proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.

We prove that any Resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of ), (for some global constant ffl ? 0). One corollary is that a certain propositional formulation of the statement NP 6ae P=poly does not have short Resolution proofs.

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiable formula of F finds a resolution proof of F , and the runtime of our algorithm is nontrivial as a function of the size of the shortest resolution proof of F . Next we investigate a class of backtrack search algorithms, commonly known as Davis-Putnam procedures and provide the first average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is 2 Q(n(n/m) 1/(k-2) ) with probability 1 - o(1). Finally, we give new lower bounds on res(F), the size of the smallest resolution refutation ...

We study the complexity of proving unsatisfiability for random k-CNF formulas with clause density D = m=n where m is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithms that, in particular, for k = 3 produce refutations almost certainly in time 2 O(n=D) . This is polynomial when m n 2 =logn. We show that our upper bounds are tight for certain natural classes of Davis-Putnam algorithms. We show further that random 3-CNF formulas of clause density D almost certainly have no resolution refutation of size smaller than 2 W(n=D 4+e ) , which implies the same lower bound on any Davis-Putnam algorithm. We also give a much simpler argument based on a novel form of self-reduction that yields a slightly weaker 2 W(n=D 5+e ) lower bound. 1 Introduction The random k-CNF model has been widely studied for several good reasons. First, it is an intrinsically natural model, analogous to the random graph model, that shed...

This paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof system: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of [BW99] for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship reffered to in [BW99] as size-width trade-off. We moreover obtain the optimality of the size-width trade-off for the widely used restrictions of resolution: Regular, Davis-Putnam, Negative, Positive and Linear. As for the second system, we show that the translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than \Omega\Gammaan/ n). A consequence of this is that the simulation of resolution by PC of [CEI97] cannot be improved to better than quasipolynomial in the case we start with small resolution proofs. We conjecture that the simu...

We show that neither Resolution nor tree-like Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xed-parameter tractable by randomized algorithms with one-sided error.

We present the best known separation between tree-like and general resolution, improving on the recent exp(n ) separation of [BEGJ98].

We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.

We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows to relate the space needed in a resolution proof of a formula to other well studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula, and as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n \Gamma c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general Pigeonhole Principle with n holes and more than n pigeons, need space n + 1 independently of the number of pigeons. Since a matching space upper bound of n + 1 for these formulas exist, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations. 3 1

. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixed-clause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh...