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

Ryoma Onaka, Kengo Nakamura, Masaaki Nishino, Norihito Yasuda

TL;DR

The paper investigates whether tensor trains can serve as a concise, tractable Boolean-function representation within the knowledge compilation map. By treating Boolean functions as tensors and introducing TT_<, the authors prove that TT_< is at least as succinct as OBDD_< and, for some function families, strictly more succinct, while preserving the same polynomial-time operations. The core contributions include a transformation from OBDD_< to TT_< via LSBDD that preserves variable order and a HWB-based separation demonstrating an exponential gap for OBDD_<; TT_< thus offers a unique trade-off not captured by NNFs. These results broaden the applicability of tensor decomposition to knowledge compilation, suggesting practical avenues for compilation and parallel computation beyond traditional NNFs.

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.

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

TL;DR

The paper investigates whether tensor trains can serve as a concise, tractable Boolean-function representation within the knowledge compilation map. By treating Boolean functions as tensors and introducing TT_<, the authors prove that TT_< is at least as succinct as OBDD_< and, for some function families, strictly more succinct, while preserving the same polynomial-time operations. The core contributions include a transformation from OBDD_< to TT_< via LSBDD that preserves variable order and a HWB-based separation demonstrating an exponential gap for OBDD_<; TT_< thus offers a unique trade-off not captured by NNFs. These results broaden the applicability of tensor decomposition to knowledge compilation, suggesting practical avenues for compilation and parallel computation beyond traditional NNFs.

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.

Paper Structure

This paper contains 30 sections, 13 theorems, 12 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. OBDD:
  5. Tensor train:
  6. Knowledge compilation map:
  7. Main Results
  8. Proofs
  9. High-Level Ideas
  10. Proof of Theorem \ref{['thm_succinct']}
  11. Proof that TT$_<$ is at least as succinct as OBDD$_<$:
  12. Proof that OBDD$_<$ is not as succinct as TT$_<$:
  13. Proof Sketch of Theorem \ref{['thm_operation']}
  14. Proof of Theorem \ref{['thm_nonpoly_operations']}
  15. $\wedge$C, $\vee$C:
  16. ...and 15 more sections

Key Result

Lemma 1

The size of the tensor train representation, $\sum_{i=1}^{m}|\mathcal{A}_i|$, of a Boolean function can be lower bounded by $r+m$, where $r$ denotes the rank and $m$ denotes the number of modes.

Figures (1)

  • Figure 1: Example of transforming an OBDD into an equivalent tensor train. We first transform an OBDD into an equivalent level-wise smooth OBDD (LSBDD) and then construct a tensor train from the LSBDD. The OBDD and the LSBDD follow the variable order $x_1 < x_2 < x_3$ and the $\pi$ of the tensor train is defined as $\pi(i) = i$, which follows the order.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • proof
  • Definition 7
  • Theorem 1
  • ...and 22 more