Complexity of Model Checking for Modal Dependence Logic

Modal dependence logic (MDL) was introduced recently by Väänänen. It enhances the basic modal language by an operator =(·). For propositional variables p1, . . . , pn the atomic formula =(p1, . . . , pn−1, pn) intuitively states that the value of pn is determined solely by those of p1, . . . , pn−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].