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.
Highlights
The logics of dependence and independence are a recent innovation studying the notion of dependencies occurring in several areas of research: computer science, logic, statistics, game theory, linguistics, philosophy, biology, physics, and social choice theory [27]
We study a sub-logic of the modal variant which is called propositional dependence logic (PDL) [30, 51]
Proof Given an instance G where G = (V, E) is a graph. We map this input to an instance (T, Φ), 2 where T is a team, and Φ is a PDL-formula with 2 split-junctions
Summary
The logics of dependence and independence are a recent innovation studying the notion of dependencies occurring in several areas of research: computer science, logic, statistics, game theory, linguistics, philosophy, biology, physics, and social choice theory [27]. The area of dependence logic is rather blank with respect to this direction of research, only Meier and Reinbold [41] investigated the (parameterised) enumeration complexity of a fragment of PDL recently. The MC problem for dependence logic, for example, is analogous to determining whether a relation in the database satisfies a functional dependency. The problem is defined as, given a database T and a positive integer k whether there is a non-trivial functional dependency of size (dep-arity in our notion) at most k that is satisfied by T. These authors prove that this problem is W[2]-complete.
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