Enumeration Complexity of Poor Man’s Propositional Dependence Logic

Abstract

Dependence logics are a modern family of logics of independence and dependence which mimic notions of database theory. In this paper, we aim to initiate the study of enumeration complexity in the field of dependence logics and thereby get a new point of view on enumerating answers of database queries. Consequently, as a first step, we investigate the problem of enumerating all satisfying teams of formulas from a given fragment of propositional dependence logic. We distinguish between restricting the team size by arbitrary functions and the parametrised version where the parameter is the team size. We show that a polynomial delay can be reached for polynomials and otherwise in the parametrised setting we reach FPT delay. However, the constructed enumeration algorithm with polynomial delay requires exponential space. We show that an incremental polynomial delay algorithm exists which uses polynomial space only. Negatively, we show that for the general problem without restricting the team size, an enumeration algorithm running in polynomial space cannot exist.T |= x :⇔ s(x) = 1 ∀s ∈ T, T |= ¬x :⇔ s(x) = 0 ∀s ∈ T, T |= 1 :⇔ true,We say that T satisfies ϕ iff T |= ϕ holds.Note that we have T |= (x ∧ ¬x) iff T = ∅. This observation motivates the definition for T |= 0. Observe that the evaluation in classical propositional logic occurs as the special case of evaluating singletons in team-based propositional logic.Definition 2 (Downward closure) A team-based propositional formula ϕ is called downward closed, if for every team T we have that T |= ϕ ⇒ ∀S ⊆ T : S |= ϕ. An operator • of arity k is called downward closed, if •(ϕ 1 , . . . , ϕ k ) is downward closed for all downward closed formulas ϕ i , i = 1, . . . , k. A class φ of team-based propositional formulas is called downward closed, if all formulas in φ are downward closed.

A Parameterized View on the Complexity of Dependence Logic

“…The parameterized complexity analyses in the propositional setting [23,20,21] have considered the combined complexity of model checking and satisfiability as problems of interest. On the cc-level, the picture there is somewhat different, e.g., team size as a parameter for propositional dependence logic enabled a FPT algorithm while in our setting it has no effect on the complexity (paraNEXP).…”

“…Recently, the propositional variant of dependence logic (PDL) has been investigated regarding its parameterized complexity [23,20]. Moreover, propositional independence and inclusion logic have also been studied from the perspective of parameterized complexity [21].…”

Parameterised complexity of model checking and satisfiability in propositional dependence logic

“…The area of research of parameterised problems is tremendously growing and often provides new insights into the inherent difficulty of the studied problems [12]. However, 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. As a subject of this research, we want to initiate and to further push a study of the parameterised complexity of problems in these logics.…”

A Parameterized View on the Complexity of Dependence Logic

“…Recently, the propositional variant of dependence logic (PDL) has been investigated regarding its parameterized complexity [20,23]. Moreover, propositional independence and inclusion logic have also been studied from the perspective of parameterized complexity [21].…”

“…Related Work. The parameterized complexity analyses in the propositional setting [20,21,23] have considered the combined complexity of model checking and satisfiability as problems of interest. On the cc-level, the picture there is somewhat different, e.g., team size as a parameter for propositional dependence logic enabled a FPT algorithm while in our setting it has no effect on the complexity (paraNEXP).…”