Complexity of Model Checking for Modal Dependence Logic

Abstract

Modal dependence logic (MDL) was introduced recently by Väänänen. 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 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].

Dependence Logic: A survey of some recent work

“…In (Grädel, ), it was also shown that, for the fragment of 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, ) and (Ebbing et al, ), and the satisfiability problem in (Lohmann and Vollmer, ) and (Sevenster, ).…”

“…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, ; Lohmann, ; Ebbing and Lohmann, ), and (Lohmann and Vollmer, ). Also, the extension of MDL by the intuitionistic implication has been studied in (Ebbing et al, ). Extensions of dependence logic by generalized quantifiers have been defined and studied in (Engström, ; Engström and Kontinen, ), and (Durand et al, ). A dynamic version of dependence logic was introduced in (Galliani, ) and (Galliani, ) in which formulas are interpreted as transitions between information states.…”

“…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 .…”

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].…”

The expressive power of modal logic with inclusion atoms

“…In [12], Lohmann and Vollmer proved complexity results for syntactic fragments of MDL. Ebbing and Lohmann [4] showed that the model checking problem for MDL is NP-complete. Also, in recent years research on complexity questions has been active; see e.g.…”