Lohmann P., Vollmer H. (2010) Complexity Results for Modal Dependence Logic. In: Dawar A., Veith H. (eds) Computer Science Logic. CSL 2010. Lecture Notes in Computer Science, vol 6247. Springer, Berlin, Heidelberg
Modal dependence logic was introduced very recently by Väänänen. It enhances the basic modal language by an operator dep. For propositional variables p1,...,pn, dep(p1,...,pn − 1)pn intuitively states that the value of pn only depends on those of p1,...,pn − 1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time.
In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ∧, □, \(\Diamond\) (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy.
In this way we completely classifiy the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.
dependence logic satisfiability problem computational complexity poor man’s logic