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A Smooth Transition Between Induction and Deduction: Fast Abductive Learning Based on Probabilistic Symbol Perception

Lin-Han Jia, Si-Yu Han, Lan-Zhe Guo, Zhi Zhou, Zhao-Long Li, Yu-Feng Li, Zhi-Hua Zhou

TL;DR

This work tackles the efficiency bottleneck in abductive learning caused by the transition between numerical induction and symbolic deduction. It introduces Probabilistic Symbol Perception (PSP), a two-step framework that first leverages probabilistic perceptual signals and sequence models to identify which symbols should be treated as variables, and then converts the resulting continuous probabilities into a set of discrete Boolean sequences via a fast max-heap–based search. The approach reduces the number of KB accesses needed for logical reasoning ($T_{ac}$) while preserving correctness, and it accumulates experiential knowledge to improve future perception. Empirical results on DBA, RBA, and HMS under incomplete and complete knowledge-base conditions show PSP outperforms gradient-free baselines in efficiency and improves KB–model consistency and rule generation speed, highlighting its practical value for neuro-symbolic learning with limited reasoning budgets.

Abstract

Abductive learning (ABL) that integrates strengths of machine learning and logical reasoning to improve the learning generalization, has been recently shown effective. However, its efficiency is affected by the transition between numerical induction and symbolical deduction, leading to high computational costs in the worst-case scenario. Efforts on this issue remain to be limited. In this paper, we identified three reasons why previous optimization algorithms for ABL were not effective: insufficient utilization of prediction, symbol relationships, and accumulated experience in successful abductive processes, resulting in redundant calculations to the knowledge base. To address these challenges, we introduce an optimization algorithm named as Probabilistic Symbol Perception (PSP), which makes a smooth transition between induction and deduction and keeps the correctness of ABL unchanged. We leverage probability as a bridge and present an efficient data structure, achieving the transfer from a continuous probability sequence to discrete Boolean sequences with low computational complexity. Experiments demonstrate the promising results.

A Smooth Transition Between Induction and Deduction: Fast Abductive Learning Based on Probabilistic Symbol Perception

TL;DR

This work tackles the efficiency bottleneck in abductive learning caused by the transition between numerical induction and symbolic deduction. It introduces Probabilistic Symbol Perception (PSP), a two-step framework that first leverages probabilistic perceptual signals and sequence models to identify which symbols should be treated as variables, and then converts the resulting continuous probabilities into a set of discrete Boolean sequences via a fast max-heap–based search. The approach reduces the number of KB accesses needed for logical reasoning () while preserving correctness, and it accumulates experiential knowledge to improve future perception. Empirical results on DBA, RBA, and HMS under incomplete and complete knowledge-base conditions show PSP outperforms gradient-free baselines in efficiency and improves KB–model consistency and rule generation speed, highlighting its practical value for neuro-symbolic learning with limited reasoning budgets.

Abstract

Abductive learning (ABL) that integrates strengths of machine learning and logical reasoning to improve the learning generalization, has been recently shown effective. However, its efficiency is affected by the transition between numerical induction and symbolical deduction, leading to high computational costs in the worst-case scenario. Efforts on this issue remain to be limited. In this paper, we identified three reasons why previous optimization algorithms for ABL were not effective: insufficient utilization of prediction, symbol relationships, and accumulated experience in successful abductive processes, resulting in redundant calculations to the knowledge base. To address these challenges, we introduce an optimization algorithm named as Probabilistic Symbol Perception (PSP), which makes a smooth transition between induction and deduction and keeps the correctness of ABL unchanged. We leverage probability as a bridge and present an efficient data structure, achieving the transfer from a continuous probability sequence to discrete Boolean sequences with low computational complexity. Experiments demonstrate the promising results.

Paper Structure

This paper contains 28 sections, 1 theorem, 4 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Integration of Perception and Reasoning
  4. Derivative-Free Optimization
  5. Problem Setting and Formulation
  6. The Machine Learning Module
  7. The Logical Reasoning Module
  8. The Optimization Problem
  9. Time Consumption
  10. Methodology
  11. From Sample Perception to Symbolic Perception
  12. Utilization of Perceptual Information
  13. Utilization of Symbol Structure
  14. Accumulation of Experience
  15. From Symbolic Perception to Symbolic Reasoning
  16. ...and 13 more sections

Key Result

Theorem 1

Let $a > 1$. After identifying the probabilities of the top $a-1$ sequences, it holds that at least one sequence among the top $a-1$ solutions does not require any positions to transition from flipped to initial states in order to obtain the $a$-th top sequence. Additionally, only one position needs

Figures (13)

  • Figure 1: The process diagram of ABL reveals a one-to-many relationship between the inductive learning and logical reasoning modules.
  • Figure 2: The schematic diagram of ABL transitioning from sample to symbolic perception.
  • Figure 3: Ranking positions based on overall probability loss.
  • Figure 4: The schematic diagram of the search algorithm for finding the $T_{ac}$ discrete Boolean sequences $B$ with the highest probability based on continuous probability sequence $PB$.
  • Figure 5: The variation of accuracy on the DBA dataset with the incomplete knowledge base.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof