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(Neural-Symbolic) Machine Learning for Inconsistency Measurement

Sven Weinzierl, Carl Cora

TL;DR

The problem addressed is the computational hardness of quantifying inconsistency in propositional knowledge bases. The authors propose regression- and neural-based models to predict the degrees of $\mathcal{I}_{\mathsf{MI}}$ and $\mathcal{I}_{at}$ from binary-encoded KB representations, enabling constant-time inference after training. A key contribution is the integration of rationality postulates as symbolic constraints, implemented via feature flags and a custom neuro-symbolic loss, which improves prediction accuracy. Empirical results on synthetic data show that MLPs, especially with symbolic constraints, achieve lower MAE than baselines, and runtime analyses demonstrate break-even points where learning-based approaches outperform solver-based computation, highlighting practical impact for continual inconsistency assessment.

Abstract

We present machine-learning-based approaches for determining the \emph{degree} of inconsistency -- which is a numerical value -- for propositional logic knowledge bases. Specifically, we present regression- and neural-based models that learn to predict the values that the inconsistency measures $\incmi$ and $\incat$ would assign to propositional logic knowledge bases. Our main motivation is that computing these values conventionally can be hard complexity-wise. As an important addition, we use specific postulates, that is, properties, of the underlying inconsistency measures to infer symbolic rules, which we combine with the learning-based models in the form of constraints. We perform various experiments and show that a) predicting the degree values is feasible in many situations, and b) including the symbolic constraints deduced from the rationality postulates increases the prediction quality.

(Neural-Symbolic) Machine Learning for Inconsistency Measurement

TL;DR

The problem addressed is the computational hardness of quantifying inconsistency in propositional knowledge bases. The authors propose regression- and neural-based models to predict the degrees of and from binary-encoded KB representations, enabling constant-time inference after training. A key contribution is the integration of rationality postulates as symbolic constraints, implemented via feature flags and a custom neuro-symbolic loss, which improves prediction accuracy. Empirical results on synthetic data show that MLPs, especially with symbolic constraints, achieve lower MAE than baselines, and runtime analyses demonstrate break-even points where learning-based approaches outperform solver-based computation, highlighting practical impact for continual inconsistency assessment.

Abstract

We present machine-learning-based approaches for determining the \emph{degree} of inconsistency -- which is a numerical value -- for propositional logic knowledge bases. Specifically, we present regression- and neural-based models that learn to predict the values that the inconsistency measures and would assign to propositional logic knowledge bases. Our main motivation is that computing these values conventionally can be hard complexity-wise. As an important addition, we use specific postulates, that is, properties, of the underlying inconsistency measures to infer symbolic rules, which we combine with the learning-based models in the form of constraints. We perform various experiments and show that a) predicting the degree values is feasible in many situations, and b) including the symbolic constraints deduced from the rationality postulates increases the prediction quality.

Paper Structure

This paper contains 13 sections, 7 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setup
  4. Machine Learning Models
  5. Experimental Setup
  6. Data Set Generation
  7. Symbolic Rules/Heuristics
  8. Training Configuration
  9. Results
  10. Experiment Results
  11. Discussion
  12. Scalability
  13. Related Work and Conclusion

Figures (4)

  • Figure 1: Exemplary instances (with minimal inconsistent subsets), constructed over a rule base and a seq. of case-dependent inputs.
  • Figure 2: SHAP plot for an exemplary instance of the data set with max. 3 atoms and max. 5 formulas (KB=$\{\neg b, \neg c, \neg b\vee\neg c\}$) and the MLP with flags and constraints. The SHAP values represent the importance of the features of this instance on the predicted MI measure.
  • Figure 3: Runtimes (s) of the Tweety-library ("solver") and the MLP approaches from this work for producing inconsistency values for all data sets (AT measure). The x-axis shows the individual data sets, for example, "3-5" means max. 3 atoms, max. 5 formulas.
  • Figure 4: MAE of a data set (max. 6 atoms, max. 10 formulas) for the MI measure ($\mathcal{I}_{\mathsf{MI}}$) with varying training set sizes from 1K to 9K instances (average over ten folds).

Theorems & Definitions (5)

  • Definition 1: Knowledge Base (KB)
  • Definition 2: Minimal Inconsistent Subset
  • Example 1
  • Definition 3: Considered Measures
  • Example 2