Complexity Results for Modal Dependence Logic

Abstract

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator =(). For propositional variables p 1 , . . . , p n , =(p 1 , . . . , p n−1 , p n ) intuitively states that the value of p n is determined by 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 ∧, , ♦ (i. e., disallowing disjunction), remains NEXP-TIME-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 NEXP-TIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction.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.A short version of this was presented at CSL 2010 [LV10].

Dependence Logic: A survey of some recent work

“…In (Grädel, 2012), it was also shown that, for the fragment of D with constancy atoms only, the model checking problem is PSPACE-complete. For modal versions of dependence logic, the model checking problem has been studied in (Ebbing and Lohmann, 2012) and , and the satisfiability problem in (Lohmann and Vollmer, 2013) and (Sevenster, 2009).…”

“…The semantics of MDL is defined with respect to sets of worlds in a frame (Kripke structure). Modal dependence logic has been further studied in (Sevenster, 2009;Lohmann, 2012;Ebbing and Lohmann, 2012), and (Lohmann and Vollmer, 2013). Also, the extension of MDL by the intuitionistic implication has been studied in .…”

Modal dependence logics: axiomatizations and model-theoretic properties

“…where each α i is a classical formula. This normal form is a generalization of a similar normal form for InqL defined in [5,6], and similar disjunctive normal forms for modal dependence logics without intuitionistic implication were discussed in the literature with a slightly different presentation (see e.g., [12,19]). Let us now introduce our disjunctive normal form for MT 0 as a recursively defined translation τ(φ) for every formula φ of MT 0 :…”

“…Basic model-theoretic properties of the logics were studied in e.g., [26], a van Benthem Theorem for the logics was proved in [16], and the frame definability of the logics was studied in [24,25]. The expressive power and the relevant computation complexity problems of the logics were investigated extensively in e.g., [7,8,9,12,19,20,26]. In this paper, we study two problems that received less attention in the literature, namely the axiomatization problem and the comparison between team semantics and the single-world semantics.…”

Enumeration Complexity of Poor Man’s Propositional Dependence Logic

“…From a computational complexity perspective, DL is well understood: most of the possible operator fragments have been classified [16,17]. However, it turned out that model checking and satisfiability for propositional dependence logic PDL are already NP-complete [10,24]. As a result, tractable enumeration of solutions in the full logic is impossible (unless P and NP coincide) and we focus on a fragment of PDL which we will call, for historical reasons, the Poor Man's fragment [18].…”