Parameterized Modal Satisfiability

Abstract

We investigate the parameterized computational complexity of the satisfiability problem for modal logic and attempt to pinpoint relevant structural parameters which cause the problem's combinatorial explosion, beyond the number of propositional variables v. To this end we study the modality depth, a natural measure which has appeared in the literature, and show that, even though modal satisfiability parameterized by v and the modality depth is FPT, the running time's dependence on the parameters is a tower of exponentials (unless P = NP). To overcome this limitation we propose possible alternative parameters, namely diamond dimension and modal width. We show fixed-parameter tractability results using these measures where the exponential dependence on the parameters is much milder (doubly and singly exponential respectively) than in the case of modality depth thus leading to FPT algorithms for modal satisfiability with much more reasonable running times. We also give lower bound arguments which prove that our algorithms cannot be improved significantly unless the Exponential Time Hypothesis fails.Keywords Modal logic · Parameterized complexity · Satisfiability problems A preliminary version of this paper has appeared in ICALP 2010.A. Achilleos · M. Lampis ( ) · V. Mitsou

Does Treewidth Help in Modal Satisfiability?

“…Similar pattern is observed in graded modal logics [22]. With treewidth and modal depth as parameters, our results indicate similar behaviour in the world of parameterized complexity -satisfiability in transitive models is W [1]-hard, while satisfiability in Euclidean and transitive models is Fpt, even with treewidth as the only parameter. However, more work is needed in this direction.…”

“…, k} by [k]. We use standard notation about parameterized complexity like Fpt algorithms, Fpt reductions and W [1]-hardness from [13]. We will also use notation and definitions of relational structures and their tree decompositions from [13]: a relational vocabulary τ is a set of relation symbols.…”

“…The concept of level is similar to the concept of distance defined in [26]. The process of checking satisfiability we describe in section 3 can be considered a variant of the level-based bottom-up algorithm given in [26], which is also implicitly used in [1,Theorem 5]. It requires more work and combination of other ideas to prove that this process can be formalized in MSO logic.…”

“…Restricted to propositional CNF formulae (which are modal formulae with modal depth 0), this graph is precisely the incidence graph associated with propositional CNF formulae (see [30] for details). We prove that 1. with the treewidth of the graph and the modal depth of the formula as parameters, satisfiability in general models is Fpt, 2. with treewidth and modal depth as parameters, satisfiability in transitive models is W [1]-hard and 3. with treewidth as the parameter, satisfiability in models that are Euclidean 1 and any combination of reflexive, symmetric and transitive is Fpt.…”

“…In [25], Nguyen shows that satisfiability of many modal logics reduce to Ptime under the restriction of Horn fragment and bounded modal depth. In [1], Achilleos et. al.…”