Answer Set Programming and S4

We develop some ideas in order to obtain a nonmonotonic reasoning system based on the modal logic S4. As a consequence we show how to express the well known answer set semantics using a restricted fragment of modal formulas. Moreover, by considering the full set of modal formulas, we obtain an interesting generalization of answer sets for logic programs with modal connectives. We also depict, by the use of examples, possible applications of this inference system. It is also possible to replace the modal logic S4 with any other modal logic to obtain similar nonmonotonic systems. We even consider the use of multimodal logics in order to model the knowledge and beliefs of agents in a scenario where their ability to reason about each other’s knowledge is relevant. Our results clearly state interesting links between answer sets, modal logics and multi-agent systems

Nonmonotonic Reasoning is Sometimes Simpler!

We establish the complexity of decision problems associated with the nonmonotonic modal logic S4. We prove that the problem of existence of an S4-expansion for a given set A of premises is \Sigma P 2 -complete. Similarly, we show that for a given formula ' and a set A of premises, it is \Sigma P 2 - complete to decide whether ' belongs to at least one S4-expansion for A, and it is \Pi P 2 -complete to decide whether ' belongs to all S4-expansions for A. This refutes a conjecture of Gottlob that these problems are PSPACE-complete. An interesting aspect of these results is that reasoning (testing satisfiability and provability) in the monotonic modal logic S4 is PSPACE-complete. To the best of our knowledge, the nonmonotonic logic S4 is the first example of a nonmonotonic formalism which is computationally easier than the monotonic logic that underlies it (assuming PSPACE does not collapse to \Sigma P 2 ). 1 Introduction First nonmonotonic logics were proposed in late 70s and e..