Scalable Approximate Inference in LIMEN-AI: Gradient-Guided Algorithms for the Neuralized Lukasiewicz Markov Engine

Introduction: The deployment of artificial intelligence in high-stakes domains increasingly demands systems that combine performance with fundamental interpretability, a requirement formalized by regulations such as the EU AI Act. This paper addresses the computational challenge of approximate inference in LIMEN-AI (Łukasiewicz Interpretable Markov Engine for Neuralized AI), a Small Reasoning Model engine that represents knowledge through weighted first-order logic formulas interpreted under Łukasiewicz fuzzy semantics. While this approach ensures human-readable reasoning steps, efficient inference over continuous interpretation spaces in relational settings remains a critical hurdle.

Methods: We develop a family of sampling-based inference algorithms tailored to the energy-based distribution induced by Łukasiewicz Markov Logic. Our approach includes importance sampling with mixture proposals and power sampling variants operating across multiple temperature levels. To address the saturation problem inherent in fuzzy logic—where truth values at boundaries cause vanishing gradients—we introduce ε-regularized operators that preserve informative gradients throughout the interpretation space. We employ the Metropolis-Adjusted Langevin Algorithm (MALA) to exploit the piecewise smooth gradients of the Łukasiewicz energy manifold, ensuring efficient exploration in high-dimensional spaces while maintaining convergence guarantees.

Results: We provide theoretical complexity bounds and extensive quantitative validation on relational domains containing up to 10⁴ ground atoms. Performance is evaluated through Effective Sample Size (ESS) metrics and comparison with analytical ground truth. Our results demonstrate that gradient-guided sampling maintains reliability where uniform baseline approaches collapse, particularly in high-dimensional settings.

Conclusions: The resulting inference routines preserve interpretability while achieving computational tractability, producing structured explanation traces that satisfy EU AI Act transparency requirements. This work bridges the gap between geometric logic and regulatory compliance, enabling auditable decision support in critical applications.