Modal dependence logic (MDL) was introduced recently by Väänänen. 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 NPcomplete 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 while for some fragments, e.g., the fragment with only ♦, the complexity drops to P.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 with both classical as well as dependence disjunction. This is the second arXiv version of this paper. It extends the first version by the investigation of the classical disjunction. A shortened variant of the first arXiv version was presented at SOFSEM 2012 [EL12].
We study the extension of dependence logic D by a majority quantifier M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, D(M) captures the complexity class counting hierarchy.
Abstract. In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D * -Horn and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore, we show that the open D * -Horn-formulae correspond to the negative fragment of SO∃-Horn.
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