A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction

Leander Kurscheidt, Paolo Morettin, Roberto Sebastiani, Andrea Passerini, Antonio Vergari
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:2431-2471, 2025.

Abstract

In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash over obstacles or go off-road. Neural models struggle in the presence of these constraints, especially when they involve intricate algebraic relationships. To address this, we introduce a differentiable probabilistic layer that guarantees the satisfaction of non-convex algebraic constraints over continuous variables. This probabilistic algebraic layer (PAL) can be seamlessly plugged into any neural architecture and trained via maximum likelihood without requiring approximations. PAL defines a distribution over conjunctions and disjunctions of linear inequalities, parametrized by polynomials. This formulation enables efficient and exact renormalization via symbolic integration, which can be amortized across different data points and easily parallelized on a GPU. We showcase PAL and our integration scheme on a number of benchmarks for algebraic constraint integration and on real-world trajectory data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-kurscheidt25a, title = {A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction}, author = {Kurscheidt, Leander and Morettin, Paolo and Sebastiani, Roberto and Passerini, Andrea and Vergari, Antonio}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {2431--2471}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/kurscheidt25a/kurscheidt25a.pdf}, url = {https://proceedings.mlr.press/v286/kurscheidt25a.html}, abstract = {In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash over obstacles or go off-road. Neural models struggle in the presence of these constraints, especially when they involve intricate algebraic relationships. To address this, we introduce a differentiable probabilistic layer that guarantees the satisfaction of non-convex algebraic constraints over continuous variables. This probabilistic algebraic layer (PAL) can be seamlessly plugged into any neural architecture and trained via maximum likelihood without requiring approximations. PAL defines a distribution over conjunctions and disjunctions of linear inequalities, parametrized by polynomials. This formulation enables efficient and exact renormalization via symbolic integration, which can be amortized across different data points and easily parallelized on a GPU. We showcase PAL and our integration scheme on a number of benchmarks for algebraic constraint integration and on real-world trajectory data.} }
Endnote
%0 Conference Paper %T A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction %A Leander Kurscheidt %A Paolo Morettin %A Roberto Sebastiani %A Andrea Passerini %A Antonio Vergari %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-kurscheidt25a %I PMLR %P 2431--2471 %U https://proceedings.mlr.press/v286/kurscheidt25a.html %V 286 %X In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash over obstacles or go off-road. Neural models struggle in the presence of these constraints, especially when they involve intricate algebraic relationships. To address this, we introduce a differentiable probabilistic layer that guarantees the satisfaction of non-convex algebraic constraints over continuous variables. This probabilistic algebraic layer (PAL) can be seamlessly plugged into any neural architecture and trained via maximum likelihood without requiring approximations. PAL defines a distribution over conjunctions and disjunctions of linear inequalities, parametrized by polynomials. This formulation enables efficient and exact renormalization via symbolic integration, which can be amortized across different data points and easily parallelized on a GPU. We showcase PAL and our integration scheme on a number of benchmarks for algebraic constraint integration and on real-world trajectory data.
APA
Kurscheidt, L., Morettin, P., Sebastiani, R., Passerini, A. & Vergari, A.. (2025). A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:2431-2471 Available from https://proceedings.mlr.press/v286/kurscheidt25a.html.

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