In this paper we give an overview of resolution methods for extended propositional modal logics. We adopt the standard translation approach and consider different resolution refinements which provide decision procedures for the resulting clause sets. Our procedures are based on ordered resolution and selection-based resolution. The logics that we cover are multi-modal logics defined over relations closed under intersection, union, converse and possibly complementation.

. This paper gives a new treatment of Maslov's class K in the framework of resolution. More specifically, we show that K and the class DK consisting of disjunction of formulae in K can be decided by a resolution refinement based on liftable orderings. We also discuss relationships to other solvable and unsolvable classes. 1 Introduction Maslov's class K [13] is one of the most important solvable fragments of firstorder logic. It contains a variety of classical solvable fragments including the Monadic class, the initially extended Skolem class, the Godel class, and the twovariable fragment of first-order logic FO 2 [4]. It also encompasses a range of non-classical logics, like a number of extended modal logics, many description logics used in the field of knowledge representation [11, 4, chap. 7], and some reducts of representable relational algebras. For this reason practical decision procedures for the class K are of general interest. According to Maslov [13] the inverse method pro...

Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

1> [ S: In general, if we have two clauses C 1 and C 2 and C 1 ` C 2 ; there is no point in using the clause C 2 since for every clause that can be derived using C 2 there is a subset that can be derived using C 1 : We say that C 1 subsumes C 2 if C 1 is a subset of C 2 : When C 1 subsumes C 2 ; we can forget C 2 : In the case of predicate logic, however, the situation is more complicated. Before giving the definition, let us consider an example. Let \Sigma be the clause set fC 1 ; : : : ; C 5 g, where C 1 = fp(X;<F13.