The Complexity of Non-Iterated Probabilistic Justification Logic

Abstract

The logic PJ is a probabilistic logic defined by adding (non-iterated) probability operators to the basic justification logic J. In this paper we establish upper and lower bounds for the complexity of the derivability problem in the logic PJ. The main result of the paper is that the complexity of the derivability problem in PJ remains the same as the complexity of the derivability problem in the underlying logic J, which is Π p 2 -complete. This implies hat the probability operators do not increase the complexity of the logic, although they arguably enrich the expressiveness of the language.1 J stands for justification, whereas PJ stands for probabilistic justification. 2 the two P's stand for iterations of the probability operator.

Probabilistic justification logic

“…Of course we have to make sure that both kinds of constraints are satisfied in the same PPJ CS,Meas -model. The satisfiability testing for the "justification and classical constraints" will be done using an adaptation of the satisfiability algorithm for the logic J ( [16,17,26,31]), whereas the satisfiability testing for the "probabilistic constraints" will be done using similar ideas as the ones used for the satisfiability testing in the logic PJ [13]. In order to formally present our satisfiability algorithm we will first explain what is meant under "satisfiability testing for justification and classical constraints", then what is formally meant under "satisfiability testing for probabilistic constraints" and finally how both kind of constraints can be satisfied at the same model.…”

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

“…Fagin and Halpern [11] mention (without giving a complete formal proof) that complexity bounds for the satisfiability problem in a modal logic that allows nesting of the probabilistic operators (like in LPP 1 ) can be obtained by employing an algorithm based on a tableau construction as in classical modal logic [13]. In [17] we used the idea of Fagin and Halpern in order to obtain tight bounds for the complexity of satisfiability in PPJ.…”

“…(16) has a solution. Thus whenever we come to an equation like (16) we can nondeterministically chose one of the equivalent set of equations and add it to the constructed linear system. We conclude that we can guess a tree for t and also a linear system in nondeterministic polynomial time.…”

“…Although the expressive power of the formal systems of [10] and [28] is richer than the one of [12], the authors of [10] and [28] were able to use arguments similar to those in [12] to show that the satisfiability problem in their logics is also NP-complete. In [16] we obtained tight bounds for the complexity of the satisfiability problem in non-iterated probabilistic justification logic, using again some results from the theory of linear programming. Fagin and Halpern [11] mention (without giving a complete formal proof) that complexity bounds for the satisfiability problem in a modal logic that allows nesting of the probabilistic operators (like in LPP 1 ) can be obtained by employing an algorithm based on a tableau construction as in classical modal logic [13].…”

“…The goal of this paper is to summarize and generalize the results for the complexity of the satisfiability problem in non-iterated and iterated probabilistic logics. This paper is the extended journal version of [16] and [17] which were presented at "Foundations of Information and Knowledge Systems" in 2016 and the "11th Panhellenic Logic Symposium" in 2017 respectively. The results of [16,17] refer only to probabilistic justification logic, whereas in the present paper we make clear that our results can be applied to an non-iterated and iterated probabilistic logic over any extension of classical propositional logic.…”