Temporal Properties of Conditional Independence in Dynamic Bayesian Networks

2025-11-13

AI Summary

This paper addresses the formal verification of temporal evolution of conditional independence (CI) assertions in dynamic Bayesian networks (DBNs), distinguishing between *structural CI*—determined solely by the graph topology—and *stochastic CI*—dependent on specific probability distributions. We introduce the first verification framework integrating linear temporal logic (LTL) and nondeterministic Büchi automata (NBA) to model and verify CI evolution over time. We establish tight computational complexity bounds: stochastic CI verification is at least as hard as the Skolem problem (implying undecidability in general), whereas structural CI verification is PSPACE-complete and NP∩coNP-hard; however, it becomes tractable for DBNs with bounded treewidth or other favorable structural constraints. Our main contributions are (i) the first formal verification paradigm for temporal CI evolution in DBNs, (ii) a precise characterization of the fundamental distinction between structural and stochastic CI, and (iii) structured sufficient conditions for decidability and tractability of CI verification.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• 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 semantics

Abstract

Dynamic Bayesian networks (DBNs) are compact graphical representations used to model probabilistic systems where interdependent random variables and their distributions evolve over time. In this paper, we study the verification of the evolution of conditional-independence (CI) propositions against temporal logic specifications. To this end, we consider two specification formalisms over CI propositions: linear temporal logic (LTL), and non-deterministic B""uchi automata (NBAs). This problem has two variants. Stochastic CI properties take the given concrete probability distributions into account, while structural CI properties are viewed purely in terms of the graphical structure of the DBN. We show that deciding if a stochastic CI proposition eventually holds is at least as hard as the Skolem problem for linear recurrence sequences, a long-standing open problem in number theory. On the other hand, we show that verifying the evolution of structural CI propositions against LTL and NBA specifications is in PSPACE, and is NP- and coNP-hard. We also identify natural restrictions on the graphical structure of DBNs that make the verification of structural CI properties tractable.

Problem

• Verifying conditional independence evolution in dynamic Bayesian networks
• Analyzing stochastic versus structural CI properties complexity
• Identifying tractable DBN structures for efficient CI verification

Innovation

• Verifying conditional independence using temporal logic
• Analyzing structural properties via graphical DBN representations
• Establishing PSPACE complexity for LTL verification