Algorithmic correspondence and completeness in modal logic. V. Recursive extensions of SQEMA

In (CON 06b) we introduced the algorithmSQEMA for computing first-order equiv- alents and proving canonicity of modal formulae, and thus es tablished a very general cor- respondence and canonical completeness result. SQEMA is based on transformation rules, the most important of which employs a modal version of a resul t 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. I this paper we develop several exten- sions of SQEMA where that syntactic condition is replaced by a semantic one , viz. downward monotonicity. For the first, and most general, extensionSemSQEMA we prove correctness for a large class of modal formulae containing an extension o the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a spe cial modal version of Lyndon's monotonicity theorem and imposing additional requirement s on the Ackermann rule we obtain restricted versions ofSemSQEMA which guarantee canonicity, too.

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