Computability of validity and satisfiability in probability logics over finite and countable models

Abstract

The -logic (which is called E-logic in this paper) of Terwijn is a variant of first-order logic (FOL) with the same syntax in which the models are equipped with probability measures and the quantifier is interpreted as ‘there exists a set A of a measure such that for each , ...’. Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational , respectively -complete and -hard, and ii) for , respectively decidable and -complete. The adjective ‘general’ here means ‘uniformly over all languages’. We extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability and validity with respect to finite models in E-logic are, i) for rational , respectively -complete and -complete, and ii) for , respectively decidable and -complete. Although partial results toward the countable case are also achieved, the computability of E-logic over countable models still remains largely unsolved. In addition, most of the results here and of Kuyper and Terwijn do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research. On the positive side, we derive the decidability of the corresponding problems for monadic relational languages – equality- and function-free languages with finitely-many unary and arbitrarily-many nullary predicates. This result holds for all three of the unrestricted, countable, and finite-model cases. Applications in computational learning theory (CLT), weighted graphs, and artificial neural networks (ANNs) are discussed in the context of these decidability and undecidability results.