It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size tree-resolution proofs or is "di#cult" i.e requires exponential size tree-resolution proofs. It is shown that the class of tautologies which are hard (for tree-resolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is

An algorithm for satisfiability testing in the propositional calculus with a worst case running time that grows at a rate less than 2 (.25+ε) L is described, where L can be either the length of the input expression or the number of occurrences of literals (i.e., leaves) in it. This represents a new upper bound on the complexity of non-clausal satisfiability testing. The performance is achieved by using lemmas concerning assignments and pruning that preserve satisfiability, together with choosing a "good " variable upon which to recur. For expressions in conjunctive normal form, it is shown that an upper bound is 2.128 L.

. ZRes is a propositional prover based on the original procedure of Davis and Putnam, as opposed to its modied version of Davis, Logeman and Loveland, on which most of the current ecient SAT provers are based. On some highly structured SAT instances, such as the well known Pigeon Hole and Urquhart problems, both proved hard for resolution, ZRes performs very well and surpasses all classical SAT provers by an order of magnitude. 1 The DP and DLL algorithms Stimulated by hardware progress, many more and more ecient SAT solvers have been designed during the last decade. It is striking that most of the complete solvers are based on the procedure of Davis, Logeman and Loveland (DLL for short) presented in 1962 [11]. The DLL procedure may roughly be described as a backtrack procedure that searches for a model. Each step amounts to the extension of a partial interpretation by choosing an assignment for a selected variable. The success of this procedure is mainly due to its space comp...

We investigate algorithms deciding propositional tautologies for DNF and coNP--complete subclasses given by restrictions on the number of occurrences of literals. Especially polynomial use of resolution for reductions in combination with a new combinatorial principle called "Generalized Sign Principle" is studied. Upper bounds on time complexity are given with exponential part 2 ff\Delta(F ) where the measure (F ) for a clause set F either is the number n(F ) of variables, the number `(F ) of literal occurrences or the number k(F ) of clauses. ff is called a "power coefficient" for the class of formulas under consideration w.r.t. measure . Power coefficients are derived with the help of a method estimating the size of trees, which is also used to find "good" branching rules. Under natural conditions power coefficients ff; fi; fl for n; k; ` respectively fulfill ff fi fl. We obtain the following power coefficients. - 0:1112 for DNF w.r.t. ` - 0:3334 for DNF w.r.t. k These result...

An algorithm for the Satisfiability problem is presented and its probabilistic behavior is analysed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamically assigns values to literals appearing in a given instance until a satisfying assignment is found or the algorithm "gives up" without determining whether or not a solution exists. It is shown that if n clauses are constructed independently from r boolean variables where the probability that a variable appears in a clause as a positive literal is p and as a negative literal is p then almost all randomly generated instances of Satisfiability are solved in polynomial time if p ! :4 ln(n)=r or p ? ln(n)=r or p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r 1\Gammaffl ! 1 for any ffl ? 0. It is also shown that if p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r = 1 then almost ...

This thesis explores the relative complexity of proofs produced by the automatic theorem proving procedures of analytic tableaux, linear resolution, the connection method, tree resolution and the Davis-Putnam procedure. It is shown that tree resolution simulates the improved tableau procedure and that SL-resolution and the connection method are equivalent to restrictions of the improved tableau method. The theorem by Tseitin that the Davis-Putnam Procedure cannot be simulated by tree resolution is given an explicit and simplified proof. The hard examples for tree resolution are contradictions constructed from simple Tseitin graphs.

We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Zn q (or Zn) for any given integer q 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Zp. The result is obtained through learning linear systems over an arbitrary eld. In general a counting function mayhavea composite modulus. We prove that, for any given integer q 2, over the domain Z n 2, the class of read-once disjunctions of Boolean-weighted counting functions with modulus q is polynomial time learnable with only one equivalence query, and the class of disjunctions of log log n Booleanweighted counting functions with modulus q is polynomial time learnable. Finally, we present an algorithm for learning graph-based counting functions. 1

this paper we develop an approach which allows to produce explicitly a system of polynomials of degree 6 and to prove a linear lower bound on the degree of its boolean Nullstellensatz refutation. This approach borrows an idea from [13] to reduce the issue of Nullstellensatz refutations to Thue systems. First, we introduce and study (see section 1) boolean multiplicative Thue systems (basically, they consist of binomials necessarily containing among them the polynomials X

We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the average-case complexity of NP-complete problems. Gurevich showed that a problem with a flat distribution is not DistNP complete (for deterministic reductions), unless DEXPTime<F NaN> 6= NEXPTime. We express the known results concerning fixed size and fixed density distributions for CNF in the framework of average-case complexity and show that all these distributions are flat. We introduce the family of symmetric distributions, which generalizes those mentioned before, and show that bounded symmetric distributions on ordered tuples of clauses (CNFTuples) and on k-CNF (sets of k-literal-clauses), are flat. This eliminates all these distributions as candidates for 'provably hard' (i.e. DistNP complete) distributions for SAT, if one considers only deterministic reductions. Given the (presumed) naturalness and generality o...

: The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed [31, 35, 34, 32]. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the Universal SAT problem. In this paper, we prove that, when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio fi ! 1, Newton's method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1 \Gamma fi=m) for the Universal SAT problem with m variables. An algorithm based on the coordinate descent method for the...