Topic
Modal operator
About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.
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16 Jul 2011TL;DR: This paper considers the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and shows they are PSPACE-complete.
Abstract: It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.
30 citations
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TL;DR: It is shown that model checking for MDL formulae over Kripke structures is NP -complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives.
Abstract: Modal dependence logic (MDL) was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n the atomic formula =(p_1,...,p_(n-1),p_n) intuitively states that the value of p_n is determined solely by those of p_1,...,p_(n-1). We show that model checking for MDL formulae over Kripke structures is NP-complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives. It turns out that several fragments, e.g., the one without modalities or the one without propositional connectives, remain NP-complete. We also consider the restriction of MDL where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the model checking problem for this bounded MDL is still NP-complete. We additionally extend MDL by allowing classical disjunction - introduced by Sevenster - besides dependence disjunction and show that classical disjunction is always at least as computationally bad as bounded arity dependence atoms and in some cases even worse, e.g., the fragment with nothing but the two disjunctions is NP-complete. Furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as bounded MDL.
30 citations
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TL;DR: Very general, though partly non-constructive, methods that cover all previous examples, and extend to an infinite family of modal logics are provided.
30 citations
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09 Nov 2016TL;DR: This paper provides a sound and complete axiomatic system for THT together with several results on interdefinability of modal operators.
Abstract: Temporal Here and There (THT) constitutes the logical foundations of Temporal Equilibrium Logic Nevertheless, it has never been studied in detail since results about axiomatisation and interdefinability of modal operators remained unknown In this paper we provide a sound and complete axiomatic system for THT together with several results on interdefinability of modal operators
30 citations
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TL;DR: Discrete dualities are presented for Heyting algebras with various modal operators, for Heytson-Brouwer algebraes with an external negation, and for symmetric HeytingAlgebrAs.
Abstract: Discrete dualities are presented for Heyting algebras with various modal operators, for Heyting algebras with an external negation, for symmetric Heyting algebras, and for Heyting-Brouwer algebras.
29 citations