Probably Half True: Probabilistic Satisfiability over Łukasiewicz Infinitely-Valued Logic

We study probabilistic-logic reasoning in a context that allows for “partial truths”, focusing on computational and algorithmic properties of non-classical Łukasiewicz Infinitely-valued Probabilistic Logic. In particular, we study the satisfiability of joint probabilistic assignments, which we call LIPSAT. Although the search space is initially infinite, we provide linear algebraic methods that guarantee polynomial size witnesses, placing LIPSAT complexity in the NP-complete class. An exact satisfiability decision algorithm is presented which employs, as a subroutine, the decision problem for Łukasiewicz Infinitely-valued (non probabilistic) logic, that is also an NP-complete problem. We develop implementations of the algorithms described and discuss the empirical presence of a phase transition behavior for those implementations.