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# Algorithmic Correspondence and Completeness in Modal Logic. III. Extensions of the Algorithm SQEMA with Substitutions.

Fundamenta Informaticae (Impact Factor: 0.4). 01/2009; 92:307-343. DOI: 10.3233/Fi-2009-77
Source: DBLP

ABSTRACT In earlier papers we have introduced an algorithm, SQEMA, for computing first-order equivalents and proving canonicity of modal formulae. However, SQEMA is not complete with respect to the so called complex Sahlqvist formulae. In this paper we, first, introduce the class of complex inductive formulae, which extends both the class of complex Sahlqvist formulae and the class of polyadic inductive formulae, and second, extend SQEMA to SQEMAsub by allowing suitable substitutions in the process of transformation. We prove the correctness of SQEMAsub with respect to local equivalence of the input and output formulae and d-persistence of formulae on which the algorithm succeeds, and show that SQEMAsub is complete with respect to the class of complex inductive formulae.

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ABSTRACT: The previously introduced algorithm SQEMA computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend SQEMA with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points FOμ. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid μ-calculus. In particular, we prove that the recursive extension of SQEMA succeeds on the class of ‘recursive formulae’. We also show that a certain version of this algorithm guarantees the canonicity of the formulae on which it succeeds.
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ABSTRACT: One of the nice features of modal languages is that sometimes they can talk about abstract properties of the corresponding semantic structures. For instance the truth of the modal formula \square pÞ p\square p\Rightarrow p in the Kripke frame (W,R) is equivalent to the reflexivity of the relation R. Using a terminology from modal logic [13], we say that the condition of reflexivity – ( ∀ x)(xRx), is a first-order equivalent of the modal formula \square pÞ p\square p\Rightarrow p, or, that the formula \square pÞ p\square p\Rightarrow p is first-order definable by the condition ( ∀ x)(xRx). More over, adding the formula \square pÞ p\square p\Rightarrow p to the axioms of the minimal modal logic K we obtain a complete logic with respect to the class of reflexive frames and the completeness proof can be done by the well known in modal logic canonical method (such formulas are called canonical). Let us note that definability and completeness are some of the good features in the applications of modal logic, and hence it is important to have algorithmic methods for establishing such properties. In our talk we will describe several algorithmic approaches to this problem.
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