Willem Conradie

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 equiva- lents is important for both logical and computational reasons. On the other hand, canon- icity 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 elim- ination: 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 equiva- lents (using a modal version of Ackermanns 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, in- troduced 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.

Algorithmic correspondence for intuitionistic modal mu-calculus

In the present paper, the algorithmic correspondence theory developed in Conradie and Palmigiano [9] is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus. We enhance the algorithm ALBA introduced in Conradie and Palmigiano [9] so as to guarantee its success on the class of recursive mu-inequalities, which we introduce in this paper. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators. We show that, when restricted to the Boolean setting, the recursive mu-inequalities coincide with the “Sahlqvist mu-formulas” defined in van Benthem, Bezhanishvili and Hodkinson [22].

Algebraic modal correspondence: Sahlqvist and beyond

The present paper proposes a new introductory treatment of the very well known Sahlqvist correspondence theory for classical modal logic. The first motivation for the present treatment is {\em pedagogical}: classical Sahlqvist correspondence is presented in a uniform and modular way, and, unlike the existing textbook accounts, extends itself to a class of formulas laying outside the Sahlqvist class proper. The second motivation is {\em methodological}: the present treatment aims at highlighting the {\em algebraic} and {\em order-theoretic} nature of the correspondence mechanism. The exposition remains elementary and does not presuppose any previous knowledge or familiarity with the algebraic approach to logic. However, it provides the underlying motivation and basic intuitions for the recent developments in the Sahlqvist theory of nonclassical logics, which compose the so-called unified correspondence theory.

Canonicity results for mu-calculi: an algorithmic approach

We investigate the canonicity of inequalities of the intuitionistic mu-calculus. The notion of canonicity in the presence of fixed point operators is not entirely straightforward. In the algebraic setting of canonical extensions we examine both the usual notion of canonicity and what we will call tame canonicity. This latter concept has previously been investigated for the classical mu-calculus by Bezhanishvili and Hodkinson. Our approach is in the spirit of Sahlqvist theory. That is, we identify syntactically-defined classes of inequalities, namely the restricted inductive and tame inductive inequalities, which are, respectively, canonical or tame canonical. Our approach is to use an algorithm which processes inequalities with the aim of eliminating propositional variables. The algorithm we introduce is closely related to the algorithms ALBA and mu-ALBA studied by Conradie, Palmigiano, et al. It is based on a calculus of rewrite rules, the soundness of which rests upon the way in which algebras embed into their canonical extensions and the order-theoretic properties of the latter. We show that the algorithm succeeds on every restricted inductive inequality by means of a so-called proper run, and that this is sufficient to guarantee their canonicity. Likewise, we are able to show that the algorithm succeeds on every tame inductive inequality by means of a so-called tame run. In turn, this guarantees their tame canonicity.

Algorithmic correspondence and completeness in modal logic

In (Conradie et al., 2006a) 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 Lyndons monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of SemSQEMA which guarantee canonicity, too.In (Conradie et al., 2006a) 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 Lyndons monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versio...

Algorithmic Correspondence and Completeness in Modal Logic. II. Polyadic and Hybrid Extensions of the Algorithm SQEMA

In Conradie, Goranko, and Vakarelov (2006, Logical Methods in Computer Science, 2) we introduced a new algorithm, , for computing first-order equivalents and proving the canonicity of modal formulae of the basic modal language. Here we extend , first to arbitrary and reversive polyadic modal languages, and then to hybrid polyadic languages too. We present the algorithm, illustrate it with some examples, and prove its correctness with respect to local equivalence of the input and output formulae, its completeness with respect to the polyadic inductive formulae introduced in Goranko and Vakarelov (2001, J. Logic. Comput., 11, 737–754) and Goranko and Vakarelov (2006, Ann. Pure. Appl. Logic, 141, 180–217), and the d-persistence (with respect to descriptive frames) of the formulae on which the algorithm succeeds. These results readily expand to completeness with respect to hybrid inductive polyadic formulae and di-persistence (with respect to discrete frames) in hybrid reversive polyadic languages.

On the strength and scope of DLS

We provide syntactic necessary and sufficient conditions on the formulae reducible by the second-order quantifier elimination algorithm DLS. It is shown that DLS is compete for all modal Sahlqvist and Inductive formulae, and that all modal formulae in a single propositional variable on which DLS succeeds are canonical.

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 Lyndons monotonicity theorem and imposing additional requirement s on the Ackermann rule we obtain restricted versions ofSemSQEMA which guarantee canonicity, too.

Constructive Canonicity for Lattice-Based Fixed Point Logics

We prove the algorithmic canonicity of two classes of

An algebraic look at filtrations in modal logic

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