A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction

• URL: http://arxiv.org/abs/2503.19466v3
• Date: Mon, 16 Jun 2025 10:05:22 GMT
• Title: A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction
• Authors: Leander Kurscheidt, Paolo Morettin, Roberto Sebastiani, Andrea Passerini, Antonio Vergari,
• Abstract summary: In safety-critical applications, satisfaction of constraints linear continuous environments is crucial, e.g., an autonomous agent should never crash or go off-road.<n>We introduce different approximations that guarantee the satisfaction of non-junctionable constraints.<n>This formulation enables efficient and exact renormalization via symbolic integration.
• Score: 13.245011236407166
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash into 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, parameterized 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.

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