The complexity of satisfiability in non-iterated and iterated probabilistic logics

Abstract

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 least 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. keywords: probabilistic logic, computational complexity, satisfiability, justification logicAcknowledgements. 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.

Multi-agent Logics for Reasoning About Higher-Order Upper and Lower Probabilities

“…Second, we will investigate complexity of satisfiability problem for the logic ILUPP. Such a method is already developed in [16] for probabilistic logics with iterations of standard probability operators. Note that the fact that ILUPP is a generalization of the logic LPP1 already leads to a complexity bound for ILUPP.…”

“…Note that the fact that ILUPP is a generalization of the logic LPP1 already leads to a complexity bound for ILUPP. Namely, it was shown in [16] that the satisfiability problem for the logic LPP1 is PSPACE-complete, thus a lower complexity bound for ILUPP is PSPACE.…”

Probabilistic justification logic

“…As already mentioned, PJ [15] is the precursor of PPJ without iterations of probability operators. Kokkinis [14] shows that the satisfiability problem in PJ has the same complexity as the corresponding problem in the underlying justification logic and that the satisfiability problem in PPJ is PSPACE-complete, which is the usual suspect in modal logics.…”