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

Algorithmic correspondence and completeness in modal logic

Abstract 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 equiv- alently 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 consti- tute 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 nega- tive/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 at- tacked with any second-order quantifier elimination algorithm. Our treatment of this ap- proach takes the form of a study of the DLS-algorithm. We partially characterize the for- mulae 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

Deontic Epistemic stit Logic Distinguishing Modes of `Mens Rea\u27

Most juridical systems contain the principle that an act is only unlaw- ful if the agent conducting the act has a `guilty mind\u27 (`mens rea\u27). Dif- ferent law systems distinguish different modes of mens rea. For instance, American law distinguishes between `knowingly\u27 performing a criminal act, `recklessness\u27, `strict liability\u27, etc. I will show we can formalize several of these categories. The formalism I use is a complete stit-logic featuring operators for stit-actions taking effect in `next\u27 states, S5-knowledge op- erators and SDL-type obligation operators. The different modes of `mens rea\u27 correspond to the violation conditions of different types of obligation definable in the logic