Algorithmic correspondence and completeness in modal logic

This thesis takes an algorithmic perspective on the correspondence between modal and hybrid logics on the one hand, and first-order logic on the other. The canonicity of formulae, and by implication the completeness of logics, is simultaneously treated. Modal formulae define second-order conditions on frames which, in some cases, are equivalently reducible to first-order conditions. Modal formulae for which the latter is possible are called elementary. As is well known, it is algorithmically undecidable whether a given modal formula defines a first-order frame condition or not. Hence, any attempt at delineating the class of elementary modal formulae by means of a decidable criterium can only constitute an approximation of this class. Syntactically specified such approximations include the classes of Sahlqvist and inductive formulae. The approximations we consider take the form of algorithms. We develop an algorithm called SQEMA, which computes first-order frame equivalents for modal formulae, by first transforming them into pure formulae in a reversive hybrid language. It is shown that this algorithm subsumes the classes of Sahlqvist and inductive formulae, and that all formulae on which it succeeds are d-persistent (canonical), and hence axiomatize complete normal modal logics. SQEMA is extended to polyadic languages, and it is shown that this extension succeeds on all polyadic inductive formulae. The canonicity result is also transferred. SQEMA is next extended to hybrid languages. Persistence results with respect to discrete general frames are obtained for certain of these extensions. The notion of persistence with respect to strongly descriptive general frames is investigated, and some syntactic sufficient conditions for such persistence are obtained. SQEMA is adapted to guarantee the persistence with respect to strongly descriptive frames of the hybrid formulae on which it succeeds, and hence the completeness of the hybrid logics axiomatized with these formulae. New syntactic classes of elementary and canonical hybrid formulae are obtained. Semantic extensions of SQEMA are obtained by replacing the syntactic criterium of negative/positive polarity, used to determine the applicability of a certain transformation rule, by its semantic correlate — monotonicity. In order to guarantee the canonicity of the formulae on which the thus extended algorithm succeeds, syntactically correct equivalents for monotone formulae are needed. Different version of Lyndon’s monotonicity theorem, which guarantee the existence of these equivalents, are proved. Constructive versions of these theorems are also obtained by means of techniques based on bisimulation quantifiers. Via the standard second-order translation, the modal elementarity problem can be attacked with any second-order quantifier elimination algorithm. Our treatment of this approach takes the form of a study of the DLS-algorithm. We partially characterize the formulae on which DLS succeeds in terms of syntactic criteria. It is shown that DLS succeeds in reducing all Sahlqvist and inductive formulae, and that all modal formulae in a single propositional variable on which it succeeds are canonical.