The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.

Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.

The training phase of the random 3-satisfiability problem in discrete Hopfield neural networks aims to identify more satisfying clauses, enhancing synaptic weight for energy function computation and minimizing network energy. The primary challenge lies in designing synaptic weights that can be dynamically optimized to adapt to various formula structures while ensuring complete clause satisfaction and avoiding local optima. This enables an optimal balance between clause satisfaction and network convergence performance. To address this challenge, this paper first proposes a method for determining synaptic weights during the training phase based on the logical relationships between clauses and variables, simplifying the computation process and improving solution efficiency. Second, the hybrid firefly algorithm is employed during the training phase to optimize the number of satisfied clauses. This is achieved through a balance of global and local search mechanisms and a diversity maintenance strategy, facilitating the identification of more satisfied clauses, thereby leading to the generation of high-quality synaptic weights. Consequently, during the retrieval phase of the network, local field updates are executed based on these synaptic weights to find the optimal neuron states, thereby minimizing the energy function and improving global convergence performance. To evaluate the effectiveness of the hybrid firefly algorithm and the simplification of synaptic weight computation, we employed a comprehensive performance evaluation framework composed of maximum fitness, fitness ratio, entropy-adjusted diversity metrics, weight error, global minima ratio, energy error, similarity indices, and runtime. Experimental results indicate that the proposed model outperforms both traditional discrete Hopfield neural network random 3-satisfiability models and those that combine election algorithms with discrete Hopfield neural network random 3-satisfiability models across multiple performance metrics.