Dynamic Logic of Propositional Assignments: A Well-Behaved Variant of PDL

Abstract

Abstract-We study a version of Propositional Dynamic Logic (PDL) that we call Dynamic Logic of Propositional Assignments (DL-PA). The atomic programs of DL-PA are assignments of propositional variables to true or to false. We show that DL-PA behaves better than PDL, having e.g. compactness and eliminability of the Kleene star. We establish tight complexity results: both satisfiability and model checking are EXPTIME-complete.

A Dynamic Logic Framework for Abstract Argumentation: Adding and Removing Arguments

“…Dynamic Logic of Propositional Assignments DL-PA [1,10] is a variant of Propositional Dynamic Logic PDL [9], with operators for sequential and nondeterministic composition of programs, test and iteration (the Kleene star), but where atomic programs are assignments of truth values to propositional variables. Modal operators associated to programs can express that some property holds after the modification of the current valuation by the program.…”

“…In that case we write π 1 ≡ π 2 . An expression is a formula or a program; equivalence is preserved under replacement of a sub-expression by an equivalent expression [1]. A formula ϕ is DL-PA valid if it is true in all valuations, i.e., if ||ϕ|| = 2 P .…”

“…In these contributions, the argument system is described by a boolean formula and each type of semantics is also represented by a boolean formula; for a given semantics, any interpretation of the propositional variables for which both formulas are true characterizes an extension. In [6], the use of a dynamic logic (DL-PA-Dynamic Logic of Propositional Assignments [1]) furthermore allows us to model updates of the argument system, such as addition or removal of an attack, or modification of extensions. [6] is not the only approach which has tackled the question of the dynamics of argument systems (see [5,15] for instance), but this one provides a single framework which encompasses at the same time the argument system, the logical definition of the change to enforce, and the change operations to perform.…”

A Dynamic View of Active Integrity Constraints

“…Repairing a database can then be done by means of a complex program that combines active integrity constraints. We use a simple yet powerful dialect of dynamic logic: Dynamic Logic of Propositional Assignments, abbreviated DL-PA [HLMT11,BHT13]. The latter is a simple instantiation of Propositional Dynamic Logic PDL [Har84,HKT00]: instead of PDL's abstract atomic programs, its atomic programs are update actions: assignments of propositional variables to either true or false, written p←⊤ and p←⊥.…”

“…The models of DL-PA are considerably simpler than PDL's Kripke models: valuations of classical propositional logic are enough. The assignment p←⊤ inserts p, while the assignment p←⊥ deletes p. It is shown in [HLMT11,BHT13] that every DL-PA formula can be reduced to an equivalent boolean formula. This will allow us to construct repaired databases syntactically.…”