. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.

We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...

We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV -isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a by-product of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.

The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no one-to-one mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR-9877150. y Department of Computer Science, University o...

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 study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two di#erent space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.

We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constant-depth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss

. We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations. Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special attention to recent attempts on reducing such bounds to some purely complexity results or assumptions. As one of the main motivations for this research we discuss provability of extremely important propositional formulae that express hardness of explicit Boolean functions with respect to various non-uniform computational models. 1. Propositional proofs as feasible proofs of plain statements Interesting and viable logical theories do not appear as result of sheer speculation. Conversely, they attempt to summarize and capture a certain amount of reasoning of a certain style about a certain class of objects that had existed in the math community before the mathematical logics entered the stage....

We consider tautologies formed from a pseudo-random number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-logicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any text-book system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...

We present some recent results on algorithms for satisfiability of k-CNF formulas: fastest probabilistic algorithms. We mention some results in proof complexity that can be used to derive lower bounds on classes of algorithms for satisfiability.