Tableau Systems for Deontic Action Logics Based on Finite Boolean Algebras, and Their Complexity

Abstract

We introduce a family of tableau calculi for deontic action logics based on finite boolean algebras (or DAL for short), these logics provide deontic operators (e.g., obligation, permission, prohibition) which are applied to a finite number of actions (the vocabulary of the logic); furthermore, in these formalisms, actions can be combined by means of boolean operators, this provides an expressive algebra of actions. We define a tableau calculus for the basic logic and then we extend this calculus to cope with extant variations of this formalism; we prove the soundness and completeness of these proof systems. In addition, we investigate the computational complexity of the satisfiability problem for DAL and its extensions; we show this problem is NP-complete when the number of actions considered is fixed, and it is Σ2p-Hard (in Stockmeyer’s polynomial hierarchy) when the number of actions is taken as an extra parameter. The tableau systems introduced here can be implemented in PSPACE, this seems reasonable taking into consideration the computational complexity of the logics.