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On Scaling Neurosymbolic Programming through Guided Logical Inference

Thomas Jean-Michel Valentin, Luisa Sophie Werner, Pierre Genevès, Nabil Layaïda

TL;DR

The paper tackles the scalability bottleneck in probabilistic neurosymbolic learning caused by reductions to PWMC and the explosion of logical provenance. It introduces DPNL, an exact, provenance-free inference framework that uses an oracle-guided ProbDPLL decomposition to steer inference, along with ApproxDPNL, which adds ε- and (ε,δ)-guaranteed approximations. The authors prove termination and correctness of DPNL and show that the oracle-based approach can dramatically prune the search space, enabling exact reasoning on substantially larger problems than prior methods. Empirical results on MNIST-N-SUM demonstrate that DPNL scales to N=4 digits per summand, outperforming existing exact systems and offering reliable approximations with bounded error via ApproxDPNL, thereby significantly advancing scalable neurosymbolic inference.

Abstract

Probabilistic neurosymbolic learning seeks to integrate neural networks with symbolic programming. Many state-of-the-art systems rely on a reduction to the Probabilistic Weighted Model Counting Problem (PWMC), which requires computing a Boolean formula called the logical provenance.However, PWMC is \\#P-hard, and the number of clauses in the logical provenance formula can grow exponentially, creating a major bottleneck that significantly limits the applicability of PNL solutions in practice.We propose a new approach centered around an exact algorithm DPNL, that enables bypassing the computation of the logical provenance.The DPNL approach relies on the principles of an oracle and a recursive DPLL-like decomposition in order to guide and speed up logical inference.Furthermore, we show that this approach can be adapted for approximate reasoning with $ε$ or $(ε, δ)$ guarantees, called ApproxDPNL.Experiments show significant performance gains.DPNL enables scaling exact inference further, resulting in more accurate models.Further, ApproxDPNL shows potential for advancing the scalability of neurosymbolic programming by incorporating approximations even further, while simultaneously ensuring guarantees for the reasoning process.

On Scaling Neurosymbolic Programming through Guided Logical Inference

TL;DR

The paper tackles the scalability bottleneck in probabilistic neurosymbolic learning caused by reductions to PWMC and the explosion of logical provenance. It introduces DPNL, an exact, provenance-free inference framework that uses an oracle-guided ProbDPLL decomposition to steer inference, along with ApproxDPNL, which adds ε- and (ε,δ)-guaranteed approximations. The authors prove termination and correctness of DPNL and show that the oracle-based approach can dramatically prune the search space, enabling exact reasoning on substantially larger problems than prior methods. Empirical results on MNIST-N-SUM demonstrate that DPNL scales to N=4 digits per summand, outperforming existing exact systems and offering reliable approximations with bounded error via ApproxDPNL, thereby significantly advancing scalable neurosymbolic inference.

Abstract

Probabilistic neurosymbolic learning seeks to integrate neural networks with symbolic programming. Many state-of-the-art systems rely on a reduction to the Probabilistic Weighted Model Counting Problem (PWMC), which requires computing a Boolean formula called the logical provenance.However, PWMC is \\#P-hard, and the number of clauses in the logical provenance formula can grow exponentially, creating a major bottleneck that significantly limits the applicability of PNL solutions in practice.We propose a new approach centered around an exact algorithm DPNL, that enables bypassing the computation of the logical provenance.The DPNL approach relies on the principles of an oracle and a recursive DPLL-like decomposition in order to guide and speed up logical inference.Furthermore, we show that this approach can be adapted for approximate reasoning with or guarantees, called ApproxDPNL.Experiments show significant performance gains.DPNL enables scaling exact inference further, resulting in more accurate models.Further, ApproxDPNL shows potential for advancing the scalability of neurosymbolic programming by incorporating approximations even further, while simultaneously ensuring guarantees for the reasoning process.

Paper Structure

This paper contains 67 sections, 6 theorems, 21 equations, 1 figure, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Problem statement and definitions
  4. Notations.
  5. Probabilistic Learning Problem.
  6. Probabilistic Neurosymbolic Learning System.
  7. Major Challenge.
  8. Probabilistic Weighted Model Counting (PWMC).
  9. A novel architecture for PNL systems
  10. ProbDPLL
  11. The ProbDPLL Algorithm.
  12. Correctness and termination.
  13. Application to logic programs.
  14. Key observations.
  15. The DPNL Algorithm
  16. ...and 52 more sections

Key Result

Theorem 1

For every $\text{ProbDPLL}(G,\sigma)$ terminates and returns $\text{PWMC}_G^\sigma$.

Figures (1)

  • Figure 1: Trace of $\text{ProbDPLL}(G,\sigma)$ where $G = (A \lor B \lor \neg C) \land (A \lor B \lor C) \land (B \lor \neg A) \land (\neg B \lor C) \land (\neg B \lor \neg C)$.

Theorems & Definitions (18)

  • Example 1: MNIST Classification
  • Example 2: MNIST-SUM Task
  • Example 3: Inverting $S$ for MNIST-N-SUM
  • Example 4: Logic for MNIST
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 8 more