The DPLL procedure is the basis of some of the most successful propositional satisfiability solvers to date. Although originally devised as a proofprocedure for first-order logic, it has been used almost exclusively for propositional logic so far because of its highly inefficient treatment of quantifiers, based on instantiation into ground formulas. The recent FDPLL calculus by Baumgartner was the first successful attempt to lift the procedure to the first-order level without resorting to ground instantiations. FDPLL lifts to the first-order case the core of the DPLL procedure, the splitting rule, but ignores other aspects of the procedure that, although not necessary for completeness, are crucial for its effectiveness in practice. In this paper, we present a new calculus loosely based on FDPLL that lifts these aspects as well. In addition to being a more faithful litfing of the DPLL procedure, the new calculus contains a more systematic treatment of universal literals, one of FDPLL's optimizations, and so has the potential of leading to much faster implementations.

The Davis-Putnam-Logemann-Loveland procedure (DPLL) was introduced in the early

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Abstract. This paper presents a technique for automated theorem proving with free variable tableaux that does not require backtracking. Most existing automated proof procedures using free variable tableaux require iterative deepening and backtracking over applied instantiations to guarantee completeness. If the correct instantiation is hard to find, this can lead to a significant amount of duplicated work. Incremental Closure is a way of organizing the search for closing instantiations that avoids this inefficiency. 1

this paper every formula is equivalent to a formula in negation normal form

This paper describes two hyperresolution-based decision procedures for a subfragment of the guarded fragment. The rst procedure is a polynomial space decision procedure which eectively corresponds to polynomial space tableaux-based algorithms without blocking. The second procedure is a minimal model generation procedure which constructs all and only minimal Herbrand models for guarded formulae. This procedure is based on hyperresolution, complement splitting and a model constraint propagation rule. Both procedures have concrete application domains and are relevant for all multi-modal and description logics that can be embedded into the guarded fragment.

ost with b. The two occurrences are variable independent. In a calculus with universal variables this is easily recognized, but there are cases where a variable is not universal, but still independent of many other occurrences of the same variable. Now, let us reverse the order of rule application such that the -inference is below the -inference. If the calculus introduces a new free variable with every -inference, then the derivations are variable-pure [13]. This is exemplified in (b), where the above variable independence is revealed due to the di#erence in inference order. In order to have goal-directed search and keep a tight relation to matrix systems [15, 13, 7], it is desirable to have invariance under order of rule application, which is not a property enjoyed by variable-pure calculi. (The leaf sequents in (a) di#er from those in (b).) To obtain this invariance, one can employ a way of reusing free variables. If the di#erent occurrences of the same -formula introduce the sa

Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of first-order (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of first-order logic; recent

We overview the development of first-order automated reasoning systems starting from their early years. Based on the analysis of current and potential applications of such systems, we also try to predict new trends in first-order automated reasoning. Our presentation will be centered around two main motives: efficiency and usefulness for existing and future potential applications. This paper expresses the views of the author on past, present, and future of theorem proving in first-order logic gained during ten years of working on the development, implementation, and applications of the theorem prover Vampire, see [Riazanov and Voronkov, 2002a]. It reflects our recent experience with applications of Vampire in verification, proof assistants, theorem proving, and semantic Web, as well as the analysis of future potential applications. 1 Theorem Proving in First-Order Logic The idea of automatic theorem proving has a long history both in mathematics and computer science. For a long time, it was believed by many that hard theorems in mathematics can be proved in a completely automatic way, using the ability of computers to perform fast combinatorial calculations. The very first experiments in automated theorem proving have shown that the purely combinatorial methods of proving firstorder theorems are too week even for proving theorems regarded as relatively easy by mathematicians. Provability in first-order logic is a very hard combinatorial problem. First-order logic is undecidable, which means that there is no terminating procedure checking provability of formulas. There are decidable classes of first-order formulas but formulas of these classes do not often arise in applications. Due to undecidability, very short formulas may turn out to be extremely complex, while very long ones rather easy. Sometimes first-order provers find proofs consisting of several thousand steps in a few seconds, but sometimes it takes hours to find a ten-step proof. The theory of first-order reasoning is centered around the completeness theorems while in practice completeness is often not an issue due to the intrinsic * Partially supported by a grant from EPSRC.

Abstract. We present randoCoP, a theorem prover for classical firstorder logic, which integrates randomized search techniques into the connection prover leanCoP 2.0. By randomly reordering the axioms of the problem and the literals within its clausal form, the incomplete search variants of leanCoP 2.0 can be improved significantly. We introduce details of the implementation and present comprehensive practical results by comparing the performance of randoCoP with leanCoP and other theorem provers on the TPTP library and problems involving large theories. 1