Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA

Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equivalents is important for both logical and computational reasons. On the other hand, canonicity of modal formulae is important, too, because it implies frame-completeness of logics axiomatized with canonical formulae. Computing a first-order equivalent of a modal formula amounts to elimination of second-order quantifiers. Two algorithms have been developed for second-order quantifier elimination: SCAN, based on constraint resolution, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing first-order equivalents (using a modal version of Ackermann's lemma) and, moreover, for proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on modal formulae, thus avoiding Skolemization and the subsequent problem of unskolemization. We present the core algorithm and illustrate it with some examples. We then prove its correctness and the canonicity of all formulae on which the algorithm succeeds. We show that it succeeds not only on all Sahlqvist formulae, but also on the larger class of inductive formulae, introduced in our earlier papers. Thus, we develop a purely algorithmic approach to proving canonical completeness in modal logic and, in particular, establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer Scienc

Regression in Modal Logic

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic

Temporal Vagueness, Coordination and Communication

How is it that people manage to communicate even when they implicitly differ on the meaning of the terms they use? Take an innocent-sounding expression such as tomorrow morning. What counts as morning? There is a surprising amount of variation across different people.

The {Markgraf Karl} Refutation Procedure

The goal of the \em MKRP project} is the development of a theorem prover which \u000Acan be used as an inference engine in various applications, in particular it \u000Ashould be capable of proving significant mathematical theorems. Our first \u000Aimplementation, the {\em Markgraf Karl Refutation Procedure (MKRP) realizes \u000Asome of the ideas we have developed to this end. It is a general purpose \u000Aresolution based deduction system that exploits the representation of formulae \u000Aas a graph (clause graph). The main features are its well tailored selection \u000Acomponents, heuristics and control mechanisms for guiding the search for a \u000Aproof. mechanisms for guiding the search for a proof. This paper gives an \u000Aoverview of the system. It summarizes and evaluates our experience with the \u000Asystem in particular, and the logics we use as well as the clause graph \u000Aapproach: as 1990 marks the fifteenth birthday of the system, the time may have \u000Acome to ask: ``Was it worth the effort?'

Towards the {MEDLAR} Framework

This is an outline description of work seeking an integrated framework for mechanising nonclassical logics. The particular logics and calculi we are concerned with are structured from the point of view of applications. As a first example for testing our prototype of the general framework, a generalised interpretation of modal logics is presented. Next, we introduce the methodology of \it Labelled Deductive Systems\/, demonstrating why this approach for a general framework is adequate to integrate various logical systems via a unified methodology. Finally, the need for different operational methods of solving problems in formal logic are briefly discussed in the context of an ambitious example suggested from MEDLAR case studies