A canonical generalization of OBDD

2026-04-07

AI Summary

This work addresses the limited succinctness of Ordered Binary Decision Diagrams (OBDDs) in representing CNF formulas with bounded treewidth. To overcome this limitation, the paper introduces Tree Decision Diagrams (TDDs), a generalization of OBDDs grounded in vtree-structured deterministic Decomposable Negation Normal Form (d-DNNF). TDDs substantially enhance representational capacity for treewidth-bounded formulas while preserving efficient support for key operations such as model counting, conditioning, and apply. Theoretically, the authors establish that any CNF formula of treewidth $k$ admits a fixed-parameter tractable (FPT)-size TDD representation—a guarantee unattainable by OBDDs—and further uncover a formal connection between TDD size and factor width. An effective bottom-up compilation algorithm is also developed to construct TDDs.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesData Mining & Knowledge Management: Data Compression

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce Tree Decision Diagrams (TDD) as a model for Boolean functions that generalizes OBDD. They can be seen as a restriction of structured d-DNNF; that is, d-DNNF that respect a vtree $T$. We show that TDDs enjoy the same tractability properties as OBDD, such as model counting, enumeration, conditioning, and apply, and are more succinct. In particular, we show that CNF formulas of treewidth $k$ can be represented by TDDs of FPT size, which is known to be impossible for OBDD. We study the complexity of compiling CNF formulas into deterministic TDDs via bottom-up compilation and relate the complexity of this approach with the notion of factor width introduced by Bova and Szeider.

Problem

• Boolean functions
• OBDD
• TDD
• CNF formulas
• treewidth

Innovation

• Tree Decision Diagrams
• OBDD generalization
• structured d-DNNF
• treewidth
• knowledge compilation