Discrete Hopfield Neural Network is widely used in solving various optimization problems and logic mining. Boolean algebras are used to govern the Discrete Hopfield Neural Network to produce final neuron states that possess a global minimum energy solution. Non-systematic satisfiability logic is popular due to the flexibility that it provides to the logical structure compared to systematic satisfiability. Hence, this study proposed a non-systematic majority logic named Major 3 Satisfiability logic that will be embedded in the Discrete Hopfield Neural Network. The model will be integrated with an evolutionary algorithm which is the multi-objective Election Algorithm in the training phase to increase the optimality of the learning process of the model. Higher content addressable memory is proposed rather than one to extend the measure of this work capability. The model will be compared with different order logical combinations $k=3,2$, $k=3,2,1$ and $k=3,1$. The performance of those logical combinations will be measured by Mean Absolute Error, Global Minimum Energy, Total Neuron Variation, Jaccard Similarity Index and Gower and Legendre Similarity Index. The results show that $k=3,2$ has the best overall performance due to its advantage of having the highest chances for the clauses to be satisfied and the absence of the first-order logic. Since it is also a non-systematic logical structure, it gains the highest diversity value during the learning phase.

The diversification of retrieved final neuron states through non-systematic satisfiability logical representation is pivotal to ensuring the optimality and functionality of Discrete Hopfield Neural Networks (DHNN) under varying neuron complexities. However, DHNN learning frameworks are predominantly designed for single-objective optimization, which often leads to repetitive neuron state patterns, overfitting, and limited storage capacity, particularly under increasing neuron complexity. These limitations indicate the need for a learning mechanism that can simultaneously enhance solution optimality and diversity. Motivated by this gap, we proposed a multi-objective DHNN framework based on a non-systematic Major 2 Satisfiability (MAJ2SAT) logical representation integrated with a Hybrid Exhaustive Search (HES) learning algorithm enhanced by an intelligent mutation operator. The proposed framework jointly optimized neuron fitness and neuron state diversity, enabling the systematic generation of high-quality and diversified neuron states. Unlike conventional exhaustive or heuristic search methods, the intelligent mutation operator selectively modified neuron states associated with unsatisfied clauses, thereby improving exploration efficiency while preserving solution feasibility. To further enhance the learning capability of DHNN, the proposed model introduced the concept of power strings, which facilitated the construction of multiple content addressable memories and effectively expanded the storage capacity of the network. Extensive experiments conducted on simulated datasets demonstrated that the proposed approach consistently I outperforms several state-of-the-art learning algorithms across problem sizes. The results showed reduced learning error in neuron diversity, increased total neuron variation, and a higher global minimum attainment ratio under varying clause configurations. Overall, the proposed multi-objective DHNN–MAJ2HES framework establishes a robust and scalable learning paradigm that enhances solution quality and diversity, with strong potential for extension to other logic-based neural optimization problems.

The current systematic logical rules in the Discrete Hopfield Neural Network encounter significant challenges, including repetitive final neuron states that lead to the issue of overfitting. Furthermore, the systematic logical rules neglect the impact on the appearance of negative literals within the logical structure, and most recent efforts have primarily focused on improving the learning capabilities of the network, which could potentially limit its overall efficiency. To tackle the limitation, we introduced a Negative Based Higher Order Systematic Logic to the network, imposing restriction on the appearance of negative literals within the clauses. Additionally, a Hybrid Black Hole Algorithm was proposed in the retrieval phase to optimize the final neuron states. This ensured that the optimized states achieved maximum diversity and reach global minima solutions with the lowest similarity index, thereby enhancing the overall performance of the network. The results illustrated that the proposed model can achieve up to 10,000 diversified and global solutions with an average similarity index of 0.09. The findings indicated that the optimized final neuron states are in optimal configurations. Based on the findings, the development of the new systematic SAT and the implementation of the Hybrid Black Hole algorithm to optimize the retrieval capabilities of DHNN to achieve multi-objective functions result in updated final neuron states with high diversity, high attainment of global minima solutions, and produces states with a low similarity index. Consequently, this proposed model could be extended for logic mining applications to tackle classification tasks. The optimized final neuron states will enhance the retrieval of high-quality induced logic, which is effective for classification and knowledge extraction.