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

Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects

In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives

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

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