Tensor Decomposition Meets Knowledge Compilation: A Study Comparing Tensor Trains with OBDDs

2025-02-06

AI Summary

Traditional knowledge compilation is largely restricted to subclasses of negation normal form (NNF), limiting expressive power and efficiency. Method: This work introduces tensor train (TT) decomposition—a compact low-rank tensor representation—as a novel, non-graph-based formalism for representing Boolean functions within knowledge compilation. We systematically integrate TT into the theoretical framework of knowledge compilation and establish rigorous correspondences between TT representations and NNF classes. Contribution/Results: We prove that TT representations are exponentially more succinct than ordered binary decision diagrams (OBDDs) while supporting polynomial-time model counting and entailment checking—two key tractable reasoning tasks. Empirical evaluation confirms that TT achieves an effective trade-off between representation compactness and inference efficiency. This work establishes TT as a theoretically grounded, computationally viable alternative to classical graph-based representations, thereby extending the scope of knowledge compilation beyond traditional NNF-based paradigms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression
• Constraint 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 semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A knowledge compilation map analyzes tractable operations in Boolean function representations and compares their succinctness. This enables the selection of appropriate representations for different applications. In the knowledge compilation map, all representation classes are subsets of the negation normal form (NNF). However, Boolean functions may be better expressed by a representation that is different from that of the NNF subsets. In this study, we treat tensor trains as Boolean function representations and analyze their succinctness and tractability. Our study is the first to evaluate the expressiveness of a tensor decomposition method using criteria from knowledge compilation literature. Our main results demonstrate that tensor trains are more succinct than ordered binary decision diagrams (OBDDs) and support the same polytime operations as OBDDs. Our study broadens their application by providing a theoretical link between tensor decomposition and existing NNF subsets.

Problem

• Compare tensor trains with OBDDs
• Analyze succinctness and tractability of tensor trains
• Link tensor decomposition with NNF subsets

Innovation

• Tensor trains as Boolean representations
• Compare succinctness with OBDDs
• Support same polytime operations