SQEMA
Algorithmic correspondence and completeness in modal logic. IV. Semantic extensions of SQEMA. In [{it W. E. Conradie, V. F. Goranko} and {it D. Vakarelov}, Log. Methods Comput. Sci. 2, No. 1, Paper 5 (2006; Zbl 1126.03018)] we introduced the algorithm SQEMA for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. SQEMA is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of SQEMA where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension SemSQEMA we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon’s monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of SemSQEMA which guarantee canonicity, too.
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References in zbMATH (referenced in 23 articles , 1 standard article )
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- Conradie, Willem; Goranko, Valentin; Vakarelov, Dimiter: Algorithmic correspondence and completeness in modal logic. III. Extensions of the algorithm SQEMA with substitutions (2009)
- Kikot, Stanislav: An extension of Kracht’s theorem to generalized Sahlqvist formulas (2009)
- Conradie, Willem; Goranko, Valentin: Algorithmic correspondence and completeness in modal logic. IV. Semantic extensions of SQEMA. (2008)
- Schmidt, Renate A.: Improved second-order quantifier elimination in modal logic (2008)
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- Conradie, Willem; Goranko, Valentin; Vakarelov, Dimiter: Algorithmic correspondence and completeness in modal logic. I: The core algorithm SQEMA (2006)
- Conradie, Willem; Goranko, Valentin; Vakarelov, Dimiter: Algorithmic correspondence and completeness in modal logic. I. the core algorithm SQEMA (2006)