621 research outputs found

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

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    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

    Cut-free Calculi and Relational Semantics for Temporal STIT Logics

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    We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm , Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames

    Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information

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    We consider a simple modal logic whose non-modal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 and S5, such logics are useful, as shown in previous work by Baltag, Coecke and the first author, for encoding and reasoning about information and misinformation in multi-agent systems. For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of "nested" or "tree-sequent" calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.Comment: This paper is the full version of the article that is to appear in the ENTCS proceedings of the 25th conference on the Mathematical Foundations of Programming Semantics (MFPS), April 2009, University of Oxfor

    Changing a semantics: opportunism or courage?

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    The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

    Women's participation in leisure cycling. A Scottish perspective

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