Approximate Fuzzy Inference for Neuralized Markov Logic Networks

This paper develops approximate inference algorithms for Neuralized Markov Logic Networks (NMLNs) interpreted under Łukasiewicz fuzzy logic, addressing the computational challenges of reasoning over rich relational structures with continuous truth degrees. We formalize inference tasks as expectations over energy-based distributions and propose importance and power sampling schemes tailored to the piecewise-linear gradient structure of Łukasiewicz semantics. Theoretical analysis establishes unbiased estimators, variance bounds, and computational complexity guarantees. Empirical validation on synthetic benchmarks and real-world legal contract analysis demonstrates that the proposed methods achieve effective sample sizes exceeding 80% with tractable computational cost, while maintaining interpretability through structured explanation traces. The framework may support regulatory compliance requirements through transparent reasoning mechanisms for high-stakes applications where both accuracy and explainability are essential.