We describe a novel logic, called HiLog, and show that it provides a more suitable basis for logic programming than does traditional predicate logic. HiLog has a higher-order syntax and allows arbitrary terms to appear in places where predicates, functions and atomic formulas occur in predicate calculus. But its semantics is first-order and admits a sound and complete proof procedure. Applications of HiLog are discussed, including DCG grammars, higher-order and modular logic programming, and deductive databases.

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We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational theories. The method can also be applied to Horn clauses with equality, in which case it corresponds to positive unit resolution plus oriented paramodulation, with unrestricted simplification.

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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

{hustadt, schmidt} topi-sb.mpg.de This paper investigates the evaluation method of decision procedures for multi-modal logic proposed by Giunchiglia and Sebastiani as an adaptation from the evaluation method of Mitchell et al of decision procedures for propositional logic. We compare three different theorem proving approaches, namely the Davis-Putnam-based procedure KSAT, the tableaux-based system KTUS and a translation approach combined with first-order resolution. Our results do not support the claims of Giunchiglia and Sebastiani concerning the computational superiority of KSAT over KRIS, and an easy-hard-easy pattern for randomly generated modal formulae. 1

t represents the sort restrictions on the variables. There are two extra inference rules which transform the sort constraint into solved form: Sort resolution and empty sort. These rules 1 The name is the result of a lunch break, FLOTTER means "faster", in German. 2 Synergetic Prover Augmenting Superposition with Sorts, SPASS means "fun", in German. implement a specific strategy of the sorted unification algorithm [15] on the sort constraint literals. In addition to these inference rules, SPASS includes a splitting rule. The splitting rule is a variant of the usual fi-rule of tableau. If SPASS splits a clause into two different cases, the two parts will not share variables, i.e. these parts can independently be refuted. For SPASS we implemented powerful reduction rules: tautology deletion, subsumption, condensing, an efficient variant of contextual rewriting,

We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

An equational approach to the synthesis of functional and logic program is taken. In this context, the synthesis task involves nding executable equations such that the given speci cation holds in their standard model. Hence, to synthesize such programs, induction is necessary.We formulate procedures for inductiveproof,aswell as for program synthesis, using the framework of \ordered rewriting". We also propose heuristics for generalizing from a sequence of equational consequences. These heuristics handle cases where the deductive process alone is inadequate for coming up with a program. 1.