Abstract not found

A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A four-valued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The four-valued logic is a well-known one, due to Belnap, and is the simplest example of Ginsberg's bilattice notion. An e#cient implementation based on semantic tableaux is sketched; it reduces to SLD resolution when negations are not involved. The resulting system can give reasonable answers to queries that involve both negation and free variables. Also it gives the same results as Prolog when there are no negations. Finally, an implementation in Prolog is given. 1 Introduction The most common treatment of negation in logic programming is negation-as-failure. This leads to problems that are now familiar: meanings of programs become di#cult to specify; program operators need not reach fix...

Even though higher-order calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higher-order logic that use higher-order unification as the key inference procedure. These calculi differ in the treatment of the substitutional properties of equivalences. The first calculus is equivalent in deductive power to the machineoriented higher-order refutation calculi known from the literature, whereas the second is complete with respect to Henkin's general models.

Abstract. In this paper we present an extensional higher-order resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goal-directed) inference rules is of practical applicability, as an implentation of the calculus in the Leo-System shows. Furthermore, we prove the long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1

The first part of this paper introduces an extension for a variant of Huet's higher-order resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed -calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connect higher-order pre-unification with the general refutation process in an appropriate way to establish full extensionality for the whole system. The general idea of the calculus is discussed on different examples. The second part introduces the Leo system which implements the discussed extensional higher-order resolution calculus. This part mainly focus on the embedding of the new extensionality rules into the refutation process and the treatment of higher-order unification. 1 Introduction Many mathematical problems can be expressed shortly and elegantly in higher order logic whereas they often lead to unnatural and inflated formulations in first-order logic, e.g., when coding them into axio...

. Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. Keywords: Partial functions, many-valued logic, sorted logic, tableau. 1 Introduction Many practical applications of deduction systems in mathematics and computer science rely on the correct and efficient treatment of partial functions. For this purpose...

Abstract. We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic — in our case a sequent calculus for classical type theory — is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically. 1

This techreport presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69] , and a calculus ERUE adapting FO RUE-Resolution [Dig79] to HO logic (based on Church's simply typed -calculus). EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable difference-reducing character the HO paramodulation approach looses its pure term-rewriting character. On the other hand examples demonstrate that the extensionality rules harmonise rather well with the difference-reducing HO RUE-resolution idea.

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