The Complexity of Satisfiability in Non-Iterated and Iterated Probabilistic Logics

Let L be some extension of classical propositional logic. The noniterated probabilistic logic over L, is the logic PL that is defined by adding non-nested probabilistic operators in the language of L. For example in PL we can express a statement like “the probability of truthfulness of A is at 0.3” where A is a formula of L. The iterated probabilistic logic over L is the logic PPL, where the probabilistic operators may be iterated (nested). For example, in PPL we can express a statement like “this coin is counterfeit with probability 0.6”. In this paper we investigate the influence of probabilistic operators in the complexity of satisfiability in PL and PPL. We obtain complexity bounds, for the aforementioned satisfiability problem, which are parameterized in the complexity of satisfiability of conjunctions of positive and negative formulas that have neither a probabilistic nor a classical operator as a top-connective. As an application of our results we obtain tight complexity bounds for the satisfiability problem in PL and PPL when L is classical propositional logic or justification logic. Acknowledgements. The author is grateful to Antonis Achilleos and Thomas Studer for several useful discussions and suggestions and to the anonymous reviewers for many valuable comments and remarks that helped him improve the quality of this paper substantially. The initial part of this research was conducted when the author was at the Institute of Computer Science (INF) in Bern, Switzerland (financial support by the Swiss National Science Foundation project 153169, Structural Proof Theory and the Logic of Proofs) and the final part when the author was at the Lorraine Research Laboratory in Computer Science and its Applications (LORIA) in Nancy, France (financial support by the European Research Council project 313360, Epistemic Protocol Synthesis).