Probabilistic Disjunctive Normal Forms in Temporal Logic and Automata Theory

2026-03-10

AI Summary

This work addresses the challenge of effectively modeling and reasoning about uncertainty within temporal logic and automata theory. It proposes Probabilistic Disjunctive Normal Form (PDNF), which encodes the probabilities of a variable’s presence, absence, or negation through real-valued weights, and constructs a PDNF vector space to enable algebraic evidence fusion. Innovatively integrating probability distributions with venjunction structures from temporal logic, the approach establishes a Banach space framework that unifies logical semantics with functional-analytic properties. Leveraging exponential parameterization and tools from functional analysis, the study demonstrates that PDNF addition is equivalent to Bayesian evidence fusion and derives probabilistic bounds for identifying outcomes from random samples.

Technology Category

Reasoning under Uncertainty: Relational Probabilistic ModelsKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolution

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.

Problem

• probabilistic disjunctive normal forms
• temporal logic
• uncertainty reasoning
• probability distributions
• logical systems

Innovation

• Probabilistic Disjunctive Normal Forms
• Temporal Logic
• Bayesian Evidence Fusion
• Functional Analysis
• Uncertainty Reasoning