Parameterised complexity of model checking and satisfiability in propositional dependence logic

Abstract

Dependence Logic was introduced by Jouko Väänänen in 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parameterisations with respect to central decision problems. The model checking problem (MC) of PDL is NP-complete (Ebbing and Lohmann, SOFSEM 2012). The subject of this research is to identify a list of parameterisations (formula-size, formula-depth, treewidth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) which classically is known to be NP-complete as well (Lohmann and Vollmer, Studia Logica 2013). There we are presenting a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is FPT. Finally, we introduce a variant of the satisfiability problem, asking for a team of a given size, and show for this problem an almost complete picture.

Parameterised Complexity of Propositional Inclusion and Independence Logic

“…This is followed by proving general results that hold not only for logics considered in this paper but for any team based logic L such that L-atoms can be evaluated in polynomial time. The parameterised complexity of PDL was analysed in [17]. We classify the parameterised complexity of model checking and satisfiability for inclusion and independence logic in Section 4 and 5 recpectively.…”

“…In order to consider specific structural parameters, we need to agree on a representation of an input instance. We follow the conventions given in [17]. Recall that the well-formed L-formulas, for every L ∈ {PDL, PIN C, PIN D}, can be seen as binary trees (the syntax tree) with leaves as atomic subformulas (variables and dependency atoms).…”

“…Recall that the well-formed L-formulas, for every L ∈ {PDL, PIN C, PIN D}, can be seen as binary trees (the syntax tree) with leaves as atomic subformulas (variables and dependency atoms). Similar to PDL [17], we take the syntax structure (defined below) rather than syntax tree as a graph structure in order to consider treewidth as a parameter. We use the same graph representation for each L-formula for L ∈ {PDL, PIN C, PIN D}.…”

“…Introduced by Downey and Fellows [2], parameterised complexity provides a way to further pin down the true source of hardness. In the propositional team semantics setting, the study of parameterised complexity was initiated by Meier and Reinbold [18] in the context of parameterised enumeration problems, and by Mahmood and Meier [17] in the context of classical decision problems, for PDL. The current paper continues the line of research initiated in [17].…”

“…In the propositional team semantics setting, the study of parameterised complexity was initiated by Meier and Reinbold [18] in the context of parameterised enumeration problems, and by Mahmood and Meier [17] in the context of classical decision problems, for PDL. The current paper continues the line of research initiated in [17]. Our focus is on the parameterised complexity of propositional inclusion and independence logic.…”