Complexity Results for Modal Dependence Logic

  • Peter Lohmann
  • Heribert Vollmer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)


Modal dependence logic was introduced very recently by Väänänen. It enhances the basic modal language by an operator dep. For propositional variables p 1,...,p n , dep(p 1,...,p n − 1)p n intuitively states that the value of p n only depends on those of p 1,...,p n − 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 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Peter Lohmann
    • 1
  • Heribert Vollmer
    • 1
  1. 1.Institut für Theoretische InformatikLeibniz Universität HannoverHannoverGermany

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