Synthesizing semantics for extensions of propositional logic

Given a Hilbert style specification of a propositional extension of standard propositional logic, it is shown how the basic model theoretic semantics can be obtained from the axioms by syntactic transformations. The transformations are designed in such a way that they eliminate certain derived theorems from the Hilbert axiomatization by turning them into tautologies. The following transformations are considered. Elimination of the reflexivity and transitivity of a binary consequence relation yields the basic possible worlds framework. Elimination of the congruence properties of the connectives yields weak neighbourhood semantics. Elimination of certain monotonicity properties yields a stronger neighbourhood semantics. Elimination of certain closure properties yields relational possible worlds semantics for the connectives. If propositional logic is the basis of the specification, the translated Hilbert axioms can be simplified by eliminating the formula variables with a quantifier elimination algorithm. This way we obtain the frame conditions for the semantic structures. All transformations work for arbitrary n-place connectives. The steps can be fully automated by means of PL1 theorem provers and quantifier elimination algorithms. The meta theory guarantees soundness and completeness of all transformation steps. As a by--product, translations into multi--modal logic are developed

Preprints of Proceedings of GWAI-92

This is a preprint of the proceedings of the German Workshop on Artificial Intelligence (GWAI) 1992. The final version will appear in the Lecture Notes in Artificial Intelligence

Towards automating duality

Dualities between different theories occur frequently in mathematics and logic --- between syntax and semantics of a logic, between structures and power structures, between relations and relational algebras, to name just a few. In this paper we show for the case of structures and power structures how corresponding properties of the two related structures can be computed fully automatically by means of quantifier elimination algorithms and predicate logic theorem provers. We illustrate the method with a large number of examples and we give enough technical hints to enable the reader who has access to the {\sc Otter} theorem prover to experiment herself

A multi-dimensional terminological knowledge representation language - preliminary version

An extension of the concept description language {$\cal ALC$} used in {\sc kl-one}-like terminological reasoning is presented. The extension includes multi-modal operators that can either stand for the usual role quantifications or for modalities such as belief, time etc. The modal operators can be used at all levels of the concept terms, and they can be used to modify both concepts and roles. This is an instance of a new kind of combination of modal logics where the modal operators of one logic may operate directly on the operators of the other logic. This preliminary version is no longer available and has been replaced by MPI-I-95-2-005

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

Modal satisfiability via SMT solving

Modal logics extend classical propositional logic, and they are robustly decidable. Whereas most existing decision procedures for modal logics are based on tableau constructions, we propose a framework for obtaining decision procedures by adding instantiation rules to standard SAT and SMT solvers. Soundness, completeness, and termination of the procedures can be proved in a uniform and elementary way for the basic modal logic and some extensions.Fil: Areces, Carlos Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Areces, Carlos Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.Fil: Fontaine, Pascal. Université de Lorraine; Francia.Fil: Fontaine, Pascal. National Institute for Research in Digital Science and Technology; Francia.Fil: Merz, Stephan. Université de Lorraine; Francia.Fil: Merz, Stephan. National Institute for Research in Digital Science and Technology; Francia.Ciencias de la Computació

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.