Complexity of Model Checking for Modal Dependence Logic

TL;DR

This paper investigates the computational complexity of model checking in modal dependence logic (MDL), establishing NP-completeness across various fragments and extensions, and providing a comprehensive classification of these complexities.

Contribution

It provides the first detailed complexity classification of model checking for MDL and its fragments, including bounded and extended versions with classical disjunction.

Findings

Model checking for MDL is NP-complete.

Several MDL fragments without certain connectives remain NP-complete.

Adding classical disjunction does not reduce complexity, often maintaining NP-completeness.

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.