Hybridizing the Discrete Hopfield Neural Network with the Imperialist Competitive Algorithm for Logic Mining: A Neuro-Symbolic Framework for Fertility Classification

ABSTRACT The Boolean Satisfiability Problem (SAT) remains a fundamental challenge in computational theory, particularly when addressing large and complex instances efficiently. Hopfield Neural Networks (HNNs) have been employed as SAT solvers; however, conventional implementations frequently suffer from premature convergence due to local minima. To overcome this limitation, a hybrid neuro-symbolic model designated Hopfield Neural Network-Imperialist Competitive Algorithm-Random k Satisfiability Reverse Analysis (HNN-ICA-RAND k SATRA) was developed by integrating the Imperialist Competitive Algorithm (ICA) into HNN dynamics for Random k-Satisfiability (RAND k SAT) logic representation. The proposed HNN-ICA-RAND k SATRA hybrid model was first validated on simulated datasets comprising 10 to 200 neurons, where it achieved superior performance with a mean classification accuracy of 90.4%, mean Hamming Loss of 0.073, mean Log Loss of 0.259, and mean Average Precision of 0.876 across all network configurations. Peak simulation performance reached 92.5% accuracy at lower network complexities. Following simulation-based validation, the hybrid model was applied to two real-world fertility classification benchmarks: the Medical Fertility Dataset (MFDS) and the Agricultural Soil Fertility Dataset (ASFDS). On the MFDS, HNN-ICA-RAND k SATRA attained 88% accuracy, 86% precision, 90% recall, and an F1 score of 88%. On the ASFDS, the hybrid model achieved 84% accuracy, 83% precision, 84% recall, and an F1 score of 84%. Comparative analyses were conducted against two other hybrid models: Hopfield Neural Network-Ant Colony Optimization-Random k Satisfiability Reverse Analysis (HNN-ACO-RAND k SATRA) and Hopfield Neural Network-Exhaustive Search-Random k Satisfiability Reverse Analysis (HNN-ES-RAND k SATRA). Runtime analysis demonstrated enhanced computational efficiency, with MFDS and ASFDS completed in 45.3s and 40.7s respectively, outperforming the HNN-ACO-RAND k SATRA hybrid (47.8s, 42.2s) and executing over 120% faster than the HNN-ES-RAND k SATRA hybrid (102.5s, 98.6s). Statistical validation through the Wilcoxon Signed-Rank Test and Friedman Test confirmed the significance of these findings. The Wilcoxon test revealed that the HNN-ICA-RAND k SATRA hybrid achieved significantly higher accuracy (p = 0.031), lower Hamming Loss, higher Average Precision, and lower Log Loss compared to HNN-ES-RAND k SATRA, with differences from HNN-ACO-RAND k SATRA being smaller yet consistently favourable. The Friedman Test yielded a Chi-Square value of 14 (p < 0.05) across all metrics, ranking HNN-ICA-RAND k SATRA first, HNN-ACO-RAND k SATRA second, and HNN-ES-RAND k SATRA third among the hybrid models evaluated. These findings demonstrate that the HNN-ICA-RANDkSATRA hybrid delivers an optimal balance between predictive performance and computational efficiency, establishing it as a scalable and adaptable SAT-based logic mining methodology with proven applicability in high-stakes domains including healthcare analytics and agricultural decision-making.