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...

An exponential lower bound for the size of tree-like Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between tree-like resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13].

We prove an exponential lower bound for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution from (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Putnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...

It is known that random k-SAT instances with at least cn clauses where c = ck is a suitable constant are unsatisfiable (with high probability). We consider the problem to certify efficiently the unsatisfiability of such formulas. A backtracking based algorithm of Beame et al. shows that k-SAT instances with at least n clauses can be certified unsatisfiable in polynomial time. We employ spectral methods to improve on this bound: For even k 4 we present a polynomial time algorithm which certifies random k-SAT instances with at least clauses as unsatisfiable (with high probability). For odd k we focus on 3-SAT instances and obtain an ecient algorithm for formulas with at least n clauses, where " > 0 is an arbitrary constant.

Abstract. We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain. Contents

In the thesis we have investigated the complexity of proofs in several propositional proof systems. Our main motivation has been to contribute to the line of research started by Cook and Reckhow to obtain how much knowledge as possible about the complexity of different proof systems to show that there is no super proof system. We also have been motivated by more applied questions concerning the automatic generation of Theorems. The results presented in the thesis were obtained in the papers [15, 14, 16] Regarding the complexity of proofs we prove both lower and upper bounds for the size of the proofs in several proof systems (resolution and some of its restrictions, Cutting Planes, Polynomial Calculus, Frege systems). Our results give better or new separations between such proof systems. On the other hand our work also concerns with automated theorem proving questions in resolution and Polynomial Calculus. Some of our results imply that restricting the search space to seek for resolut...

Reengineering Unification and t-Entailment for Mantra in C++ Tania Kharma The objective of this thesis lies in two directions. In one hand, understanding the concepts of unification and decidable inference mechanism, used in building the logic formalism of Mantra and implemented in its previous version in Common Lisp in 1991. On the other hand, learning about the object oriented paradigm and applying it in the design and implementation through the use of design patterns, the OMT notations, and C++ as the implementation language. Mantra is a shell for hybrid knowledge representation and hybrid inferences. It supports three different formalisms: logic, frames, and semantic networks. In representing any kind of knowledge through a knowledge base system, one is faced with choosing among a broad repertoire of formalisms. Mantra provides a combination of knowledge representation formalisms, so the user can decide which representation is convenient for each piece of knowledge, and is not lim...

Abstract: Recent work has shown the value of using propositional SAT solvers, as opposed to pure BDD solvers, for solving many real-world Boolean Satisfiability problems including Bounded Model Checking problems (BMC). We propose a SAT solver paradigm which combines the use of BDDs and search methods to support efficient implementation of complex search heuristics and effective use of early (preprocessor) learning. We implement many of these ideas in software called SBSAT. We show that SBSAT solves many of the benchmarks tested competitively or substantially faster than state-of-the-art SAT solvers. SBSAT differs from standard propositional SAT solvers by working directly with non-CNF propositional input; its input format is BDDs. This allows some BDD-style processing to be used as a preprocessing tool. After preprocessing, the BDDs are transformed into state machines (different state machines than the ones used in the original model checking problem) and a good deal of lookahead information is precomputed and memoized. This provides for fast implementation of a new form of lookahead, called local-function-complete lookahead (contrasting with the depth-first lookahead of

: We provide two different model theoretic characterizations of a fragment of first-order logic which we call d-Horn formulas. This fragment is dual to the well known Horn fragment and has the same complexity for proving unsatisfiability. The method used in the characterization (syntactic translation functions between formulas which are mimicked by translation functions between models) might be applied to characterize other first-order restrictions. This paper is related to the work of Henschen and Wos (1974), but we study semantic translation together with syntactic renaming functions. We comment shortly on a number of applications of d-Horn formulas, one of which is the characterization of Context Free Grammars through a Horn [ d-Horn first-order theory. Keywords: Horn formulas, d-Horn formulas, renaming functions, model theoretic characterizations. 0.1 Introduction The Horn restriction of first-order logic (FO) is a relevant fragment, specially for Computer Science. For instance,...

this papers is as follows: In section 1.2, the syntax and the semantic of a basic feature type system is presented which corresponds essentially to a Horn system where first-order terms have been replaced by feature terms. In section 1.3, the corresponding sequent calculus is defined and arguments for its soundness and completeness are given. Then the handling of the disjunction operator is added, and opportunities for optimizations are discussed.