Probabilistic Disjunctive Normal Forms in Temporal Logic and Automata Theory

Abstract

This article introduces probabilistic disjunctive normal forms (PDNFs) as a framework for representing and reasoning about uncertainty in logical systems. Unlike classical DNFs, PDNFs assign real-valued weights to variables, encoding probabilistic information about their presence, absence, or negation. Then we construct a vector space of PDNFs that allows algebraic evidence combination. PDNFs are interpreted as probability distributions over venjunctions (temporal logic constructs) and as integrable functions over partitioned intervals, where the integrals determine variable probabilities. This dual perspective allows for a Banach space structure and the application of functional analysis. We demonstrate that, under exponential parametrisation, PDNF addition aligns with Bayesian evidence fusion and derive bounds for outcome identification from random samples. The formalism thus bridges logic, numerical methods, and continuous probability.