Random Satisfiability: A Higher-Order Logical Approach in Discrete Hopfield Neural Network

A conventional systematic satisfiability logic suffers from a nonflexible logical structure that leads to a lack of interpretation. To resolve this problem, the advantage of introducing nonsystematic satisfiability logic is important to improve the flexibility of the logical structure. This paper proposes Random 3 Satisfiability (<italic>RAN3SAT</italic>) with three types of logical combinations (<inline-formula> <tex-math notation="LaTeX">$k = 1, 3, k =2, 3$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$k =1$ </tex-math></inline-formula>, 2, 3) to report the behaviors of multiple logical structures. The different types of <italic>RAN3SAT</italic> enforced with Discrete Hopfield Neural Network (DHNN) are included with benchmark searching techniques, such as Exhaustive Search algorithm. Additionally, to strengthen and certify the behavior of the proposed model, we extensively conducted several performance evaluation metrics with a specific number of neurons. In particular, the experimental results revealed that <italic>RAN3SAT</italic> was able to be implemented in DHNN, and each logical combination has its characteristics. Nonetheless, <italic>RAN3SAT</italic> provides more neuron variations in the whole solution space. The proposed model can also be applied in real-world applications such as the logic mining approach since <italic>RAN3SAT</italic> consists of various logic combinations that behave as input language to transform raw data into informative output.


I. INTRODUCTION
In the first two decades of the 21 st century, artificial intelligence (AI) has provided the impetus in building the algorithmic structure for the development of constraint satisfaction and Boolean satisfiability. For the computational complexity, logic, and AI, the satisfiability problem (SAT) has become a problem of major interest. The concept of imposing an artificial neural network (ANN) in AI is one step ahead in understanding our human actual brain's learning and memory task. ANN is a computational model that tries to simulate the structure and functional aspects of biological neural networks. An essential objective of ANN is to store practical knowledge and make it available for all. The uses of ANN have been exploited in many areas, such as diagnosing autism spectrum disorder [1], predicting animals category [2], and forecasting the stock market index [3]. A variant of ANN The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou. was introduced by Hopfield and Tank [4] in 1985, named Hopfield Neural Network (HNN). A feature of HNN is that it has single and multiple associative memory systems with no hidden layers [5]. HNN comprises two distinct structures: continuous and discrete [6]. Discrete Hopfield Neural Network (DHNN) is depicted as a suitable ANN platform of linearized interconnected neuron states to analyze discrete entries. It is worth mentioning that DHNN has a variety of applications such as resources management [7], location detector [8], and students' performance evaluation [9]. Despite the recent and fast improvement in DHNN, there has been no recent development of demonstrating the output of DHNN in the form of the symbolic rules.
SAT is the decision problem of whether there exists a satisfying assignment for a given formula [10]. Based on the language of logic programming, Wan Abdullah [11] introduced the fundamental concept of logical rules in ANN. Likewise, Sathasivam [12] extended the work by [11] and presented a new concept of SAT named Horn Satisfiability (Horn SAT).
In [12], the Sathasivam relaxation method for optimizing the final state of the neuron was introduced. Utilizing previously published works, ANN researchers conducted research on SAT integrated with ANN.
Interestingly, Mansor et al. [13] extended a systematic SAT logical function expressed in Conjunctive Normal Form (CNF) which is known as 3 Satisfiability (3SAT) with DHNN. Note that the 3SAT logical rule contains strictly three literals in each independent clause. The important concept of pattern satisfiability, 3SAT, and DHNN was discussed here. Subsequently, Mansor et al. extended the work by initiating a study of [14] that creates another variant of 3SAT imposed in DHNN, known as Maximum 3 Satisfiability (MAX3SAT). Here, the result of the proposed logical rule was negative due to nonredundant literals existing in MAX3SAT. The proposed work was capable of optimally exhibiting the behavior of MAX3SAT during the testing phase of DHNN. Research moved in a new direction when Kasihmuddin et al. [15] suggested another systematic, logical rule named 2SAT.
In their study, a hybrid approach that employs DHNN in optimizing the 2SAT logical rule was initiated. Note that DHNN was used to minimize logical inconsistency in interpreting the logic clauses. The emergence of 2SAT work motivated a new concept in systematic, logical rule whose outcome is negative, which is Maximum 2 Satisfiability (MAX2SAT). Also, DHNN can exhibit the behavior of MAX2SAT in all utilized performance metrics. Another meaningful work in utilizing systematic SAT was done by Zamri et al. [16], whereby the 3SAT logical rule was implemented in DHNN with a modified Imperialistic Competitive Algorithm (ICA). The proposed model was further incorporated with the reverse analysis method and able to analyze real data sets with 3SAT as a neuron representation in DHNN. This study is significant as it highlighted the potential of systematic SAT to represent real data sets. Another fascinating direction of 2SAT was proposed by Kasihmuddin et al. [17]. This work inaugurated a mutation operator in the retrieval phase of DHNN. Therefore, the suggested study shows an optimizer in the testing phase can help systematically find other solutions in other search spaces. Thus, by observing previous studies, a systematic approach was the main focus of many ANN researchers. One may pose the question, what happens if DHNN integrates nonsystematic logical rules?
The Random k Satisfiability (RANkSAT) is a variant of SAT where RANkSAT plays an essential role for studying the typical case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity spin-glasses [18]. As for the RANkSAT structure, each clause depends on the same number of literals k, and a clause is uniformly and randomly selected with a given number of literals per clause [19]. Thus, RANkSAT has a significant impact on both theoretical computer science and as well as statistical studies. Meanwhile, RANkSAT was rigorously studied in statistical physics. Lemoy et al. [20] introduced variable-focused local search algorithms for SAT problems. This work focused on random literals in unsatisfied clauses. Variants are considered where literals are selected uniformly and randomly. The result of Sathasivam et al. [21] introduced a new phenomenon of 2SAT, which is expressed as Random 2 Satisfiability (RAN2SAT) that consists of the first and second-order of SAT logical structure. [21] states the key goal is to ensure the cost function of RAN2SAT tends to zero, which indicates the logical rule is satisfied. Those mentioned above discussed the use of RANkSAT to represent their cases. The study of choosing RANkSAT is not only to consider k literals per clause but also to integrate as a logical rule in ANN. The work by [21] explains how RANkSAT is extended for nonsystematic logical rule by emphasizing random structure with higher orders of k and reporting the behavior of more logical combination.
Recently, the authors proposed the study of RAN2SAT with DHNN [21]. However, the proposed logic was restricted up k ≤ 2. In our study, the investigation of RANkSAT with DHNN for various orders of k is reported. Higher-order k with different logical combinations provides more variability and holds more literals in representing neuron states in DHNN. The behavior of RANkSAT with higher-order k has never been reported in terms of performance evaluation metrics and similarity index. Thus, the dynamics of SAT logic for Random 3 Satisfiability (RAN3SAT) in providing symbolic instruction to DHNN was reported in our new study.
The main contributions of this paper are as follows: • To formulate various Random 3 Satisfiability logical structures such as Random 3,1 Satisfiability, Random 3,2 Satisfiability, and Random 3,2,1 Satisfiability as a symbolic instruction in Discrete Hopfield Neural Network.
• To investigate the quality of solution of Random 3 Satisfiability in Discrete Hopfield Neural Network in terms of minimization of the cost function, synaptic weight management, energy profile, and final neuron variations.
• To assess numerically the impact of various Random 3 Satisfiability in Discrete Hopfield Neural Network by using different performance measures, it is shown that in the specific number of neurons, 100% accurate synaptic weight can be achieved to fulfil global minimum solution and also for a particular number of neurons the simulation achieved 100% local minimum ratio. That is the logical rule of RAN3SAT that can be embedded in DHNN for getting a higher global minimum ratio.
The paper is arranged as follows: the overall structure of RAN3SAT and the implementation of RAN3SAT in DHNN resulting in the proposed model. Introduction and the proposed random 3 satisfiability are discussed in Section 1 and Section 2, respectively. Random 3 Satisfiability in Discrete Hopfield Neural Network (DHNN-RAN3SAT) and the experimental settings involved in our study are described in Sections 3 and 4. The results and discussion of the proposed method are depicted in Section 5. The conclusion and future 50832 VOLUME 9, 2021 work of our study are presented in the last section of this paper.

II. THE PROPOSED RANDOM 3 SATISFIABILITY
RAN3SAT is a new type of nonsystematic SAT logical structure. It can be expressed in an arbitrary number of literals and clauses [21]. The main three components of RAN3SAT consist of a set of x variables where A 1 , A 2 , A 3 , . . . , A x with a set of y clauses where Z y . The values for each literal are in the form of {1, −1} which represents either true or false [7]. In our study, each clause contains x distinct literals chosen in random bases where the ratio of the probability in the positive form to negative form is 1:1 or 1:2 or 2:1 for k ≤ 3. According to the theory of propositional calculus, SAT can be expressed in CNF. The general formulation RAN3SAT (δ k RAN 3SAT ) for k = 1, 2, 3 is shown in (1) - (4). Note that, δ 1,3

RAN 3SAT
comprises of the first and third order of logic, δ 2,3 RAN 3SAT forms by second, and third-order of logic, and δ 1,2,3

RAN 3SAT
composed of the first, second, and third-order of logic respectively.
whereby u, v, and w is the total number of first, second, and third-order of logic in each clause in δ k RAN 3SAT respectively. Note that, u > 0, v > 0, w > 0, and Z (k) i consists of clauses with various orders of δ k RAN 3SAT . Conjointly, the selection Z (k) i is set at random as we want to investigate the behavior of δ k RAN 3SAT . Following (4), the literals (positive or negative) is set at random where and δ 1,2,3 RAN 3SAT with a different order of logical structure is formulated in (5) -(7) as follows: Following the above equations, if any of the δ k RAN 3SAT is satisfiable, an example δ 1,3 RAN 3SAT is said to be satisfiable when δ 1,3 RAN 3SAT = 1 that provides truth values. On the contrary, if δ 1,3 RAN 3SAT = −1, δ 1,3 RAN 3SAT is unsatisfiable, which gives false values. The key reason why δ k RAN 3SAT is proposed is because the logic can provide more variability of structure, avoid repetitive neuron states, and retrieve solution in other solution space. Furthermore, the basic structure δ k RAN 3SAT is not limited compared to conventional kSAT. Table 1 shows a comparison of existing logical structure, 2 Satisfiability (δ 2SAT ), 3 Satisfiability (δ 3SAT ), and Random 2 Satisfiability (δ RAN 2SAT ) with δ k RAN 3SAT . The previous work by [21] only considers up to twodimensional decision systems which consider k = 1, 2. In contrast, we utilize k = 1, 3, k = 2, 3 and k = 1, 2, 3 where we create a more logical combination of high dimensional decision system. The novelty δ k RAN 3SAT is vital to guarantee satisfiable logic with a different type of SAT since the study of δ RAN 2SAT [21] proved their logic function is satisfied. There are no recent works that investigate the higher order of RANkSAT k > 2 in an ANN. Therefore, this study focuses on δ k RAN 3SAT a symbolic form representation which will be integrated with the network as Discrete Hopfield Neural Network (DHNN).

III. RAN3SAT IN DISCRETE HOPFIELD NEURAL NETWORK
Discrete Hopfield Neural Network (DHNN) is a recurrent neural network with no hidden layer and was initiated by John Hopfield [4]. The recurrent feature is when the input is fed back as the output. DHNN is an extended structure of Elman Neural Network [24]. Researchers utilized DHNN as it is equipped with associative memory to solve constraint problems [25]. For example, the study by [26] utilized DHNN integrated with 2SAT as a platform in solving the optimization Bezier Curves model. In this study, the memory storage or Content Addressable Memory (CAM) stores synaptic weights up to 2 n (matrix form) binary vectors [27]. Generally, the calculation in DHNN is executed by assortments of interconnected activated neurons [28]. According to [29], DHNN contains essential properties, including parallel execution for intensive optimization problems. The units of DHNN are represented in bipolar values {1, −1}, and we utilized the asynchronous neuron adaptation by Theorem 1. Theorem 1 defined DHNN worked in asynchronous mode with its condition.  Theorem 1: All networks described by (8) in randomly asynchronous mode will fall into a network gap with a probability of one when it starts at any initial state in search space [30].
From (8), W ij is the synaptic weight from unit i to j. S j Depicts the state of the unit j and ρ i is the threshold of the unit i. The weight between neurons i and j represent the strength of connections between two neurons. Also, the neuron connections are approached W ij as W (9) -(10) that accommodates all types of logic combinations δ k RAN 3SAT is as follows.
Here S A 1 is the neuron state where A 1 ∈ {1, −1}. Note that the formulation of probability for consistent interpretation is presented in (11). Referring to Table 2, the probability (11) of consistent interpretation for δ k RAN 3SAT independent clauses is listed. Note that β Z k i is the number of Z k i clauses. We can ensure that the minimum value of Q δ k RAN 3SAT which indicates δ k RAN 3SAT is successfully minimized and obtain optimal values of W ij .
The local field of DHNN is given by (12). S i (t) is denoted as the final state of neurons whereby W is the W ij for third, second, and first-order, respectively. The dynamics of the testing phase in DHNN utilized an activation function of Hyperbolic Tangent Activation Function (HTAF) to enable the convergence of final neuron states by avoiding neuron oscillation [31]. The local field of our proposed model is formulated in (12) and (13) as follows: whereby (12) is the general formulation of the local field for δ k RAN 3SAT and (13) is a piecewise function of the generated final neuron states according to the value of (12). A study by [17] claimed that Wan Abdullah (WA) method is compatible with DHNN in finding optimal W ij by comparing directly the coefficients of (9) and Lyapunov energy function (14). The Lyapunov energy function of DHNN is formulated as (14). is calculated by (15).
Note that, represents the number of 3 literals, 2 literals, and 1 literal existing in δ k RAN 3SAT . Alternatively, H min it can be predetermined in the training phase from the Lyapunov energy function of each independent Z (k) i in δ k RAN 3SAT , which is guaranteed constant [7]. However, new data will be incorporated during the retrieval phase of HNN.
Finally, the quality of final neuron states can be examined through (16) that differentiates the global and local minima solution. Note that, when (16) is satisfied, the final neuron states will achieve global minima solution; otherwise, it is trapped into local minima solution. τ Does the user predetermine the tolerance value?
In this paper, the implementation of δ k RAN 3SAT DHNN will be abbreviated as DHNN-RAN3SAT. [21] utilized RANkSAT only for k = 1, 2 whereas in this paper, we will discuss the higher order of k and make the variation of logic by introducing multiple logical combinations. Fig. 1 shows the generalized architecture of DHNN-RAN3SAT. In Fig. 1, each main block represents different orders of k (first, second, and third), respectively. Within each main block, pink, orange, and blue colored lines illustrate the connection of each neuron W jj . However, for brown and black colored lines exhibit W k respectively. The red-colored normal line represents the combination of the first and third order, the dashed green line represents the second and third-order, and the dotted purple line depicts all the combinations of orders k.

IV. EXPERIMENTAL SETTING OF DHNN-RAN3SAT
We will present the methodology of DHNN-RAN3SAT, algorithm outlines, and performance evaluation metrics in this section. All involved parameters are defined in Table 3 with values chosen based on the standard settings listed. We report the simulation of DHNN-RAN3SAT δ k RAN 3SAT for different orders of k. For all combinations δ k RAN 3SAT , a restricted training environment of DHNN-RAN3SAT is applied to avoid overfitting of maximum fitness achieved for all neuron states. From Table 3, note that ψ = i . A standard set up by [17] emphasizes that the value ρ is set at zero to guarantee the energy of DHNN decreases uniformly. Aligned with the work by [32], HTAF is considered as one of the most stable activation functions to be implemented in DHNN. In our study, the Sathasivam relaxation rate was utilized to ''pause'' the simulation to avoid the neurons oscillating. A suitable rate was chosen 2 ≤ R ≤ 4 [12]. To verify the effectiveness of the proposed approach, the experiment is conducted based on three subsections: the training and testing phase, energy analysis, and similarity index. Table 4 presents the list of parameters involved in all performance evaluation metrics. Each subsection signifies a different purpose, as listed below: • Training phase -To achieve optimal weight management via an effective training phase.
• Testing phase -To evaluate the quality of the solution produced by DHNN-RAN3SAT.
• Energy analysis -To analyze the difference of energy retrieved by DHNN-RAN3SAT.
• Similarity index -To investigate the quality of final neuron states retrieved by DHNN-RAN3SAT with benchmark neuron states.

A. TRAINING PHASE AND TESTING PHASE SETTING
The proposed performance metrics in this subsection are the root mean square error (RMSE), mean absolute error (MAE), the sum of squared error (SSE), and mean absolute percentage error, % (MAPE). By referring to the study by [6] and [16], all chosen performance evaluation metrics are relevant to be implemented in this study. Note that (17) -(24) are performance metrics formulated based on the training and testing phase. Zero value of errors indicates optimal training and testing phase. We are using 100 neurons string for training iterations as well as the same number for training samples.
The training iterations and the number of training samples of neural networks with the different numbers of neurons are not always the same. Since the different number of neurons will create a different number of iterations. Table 5 shows the range of optimal value of errors for each performance metric.

B. ENERGY ANALYSIS SETTING
Similar to previous performance metrics, this subsection is essential to evaluate the minimization of energy achieved by DHNN-RAN3SAT. The energy profile can be determined by using (25) - (28). Note that Table 6 displayed the range of optimal value of errors for each performance metric in the energy analysis.

C. SIMILARITY INDEX SETTING
According to [31], linear initial neuron states of DHNN may result in biasedness of the testing phase due to DHNN nature of directly memorizes the final neuron states without generating a new state. Thus, possible positive (PP) and possible negative (PN) can be reduced by producing all the neuron states non systematically by (29).
whereby S i was defined earlier in (8). Also, note that in this experiment, the false positive and the false negative can be traced in the local minimum ratio. Therefore, analyzing the similarity index (SI) (29) will be initiated to generate random clauses and literals for each δ k RAN 3SAT combination. Note that (30) and (31) are the total variation (TV) of DHNN-RAN3SAT. As mentioned earlier, the similarity analysis will be conducted by comparing the retrieved final neuron states with benchmark neuron states listed in Table 7.
The algorithm of DHNN-RAN3SAT for all orders of k is implemented in Dev C++ Version 5.11 and executed on VOLUME 9, 2021 an Intel Core i5 8 th Gen machine, 4GB RAM. The same medium specification was utilized to avoid any biasedness. Fig. 2 explains each configuration of DHNN-RAN3SAT. Here, the blue blocks represent the process in the training phase. However, the red blocks represent the testing phase.

V. RESULTS AND DISCUSSION
The purpose of this paper is to analyze the behavior of different combinations δ k RAN 3SAT by evaluating several performance evaluation metrics. The simulations were conducted with a specific number of neurons α. We specifically experimented up until α = 120 for training error, testing error, and energy analysis. Meanwhile, for SI, we stopped α = 110 since the simulation achieved a 100% local minima ratio.

A. TRAINING ERROR
In this section, four performance metrics are presented to analyze the change in fitness of neuron states. ES facilitates the training phase to check clause satisfaction. Throughout this section, we can observe the synaptic weight management by the proposed model for all logical combinations δ k RAN 3SAT . According to Figs. 3-6, for all logical combinations δ k RAN 3SAT , it is noticeable that the trend δ 2,3 RAN 3SAT is more consistent than the others. It needs to be mentioned from Fig. 3 and Fig. 4 that the value of errors is linearly increasing as α increase. Note that SSE train the formulation of SSE was expressed in log y to capture the sensitivity of accumulated errors in the training phase. Other than that, as α increases, the value of MAPE is close to 100% in the range of 70 ≤ α ≤ 120. 100% MAPE told us the difference between maximum fitness and achieved neuron fitness is relatively large. Moreover, having a lower percentage of MAPE indicates a high degree of accuracy, for which it can be said that the smaller the value MAPE, the better the forecast [36].   Consequently, the errors δ 1,3 RAN 3SAT and δ 1,2,3 RAN 3SAT are relatively close due to the nature of a similar logical structure Z (1) i . Nonetheless, δ 2,3 RAN 3SAT achieved more minor errors due to the existence of Z i . Here, we can conclude that as α it increases, the probability of getting Q δ k RAN 3SAT = 0 decreases which indicates the suboptimal training phase of DHNN-RAN3SAT. This is due to the nature of ES, which employs 'trial and error' that cannot keep the larger size of constraints, or it will accumulate more error indefinitely [37]. Apart from the technicality of ES, all combinations can achieve low errors at α = 10 which leads to an optimal training phase that emphasized good synaptic weight management by the proposed model. Meanwhile, the work by [38] emphasized that a restricted number of learning will not assure the training phase to achieve higher fitness of neuron states. Thus, it will be beneficial to study the behavior of a novel SAT. Therefore, in our simulation, we restricted the number of learning all possible combinations of δ k RAN 3SAT in DHNN. Also, when the number of neurons increases at the same time, training errors increase. This has happened as the number of neurons increases, the probability of getting satisfied interpretation will become lower, and at that time, it is difficult for neuron string to achieve zero cost function. The proposed model can be further improved by implementing a higher number of learning as an effort to achieve maximum fitness of neuron states by using ES. The consistent error attained δ 2,3 RAN 3SAT indicates that this structure is best in achieving the maximum fitness of neurons in the training phase.     management related to global or local minima solution. After DHNN-RAN3SAT completed checking clause satisfaction (minimization of the cost function), the synaptic weight will be generated through the WA method [11]. The zero value of cost function will retrieve optimal synaptic weight (optimal testing phase), resulting in a global minima solution. Referring to the figures above, DHNN-RAN3SAT mostly achieved zero errors 10 ≤ α ≤ 30. During this interval, DHNN-RAN3SAT is in the optimal testing phase and performs well by producing global minima solutions. From Figs 7-10, the testing error is zero at α = 10 in the basis of synaptic weight management, which brief that α = 10 with correct synaptic weight, the model is 100% right. It is well known that the critical challenge in training a neural network is how to train them. Introducing the small number of neurons training/testing will mean that the model will under-fit the train and test sets. So that why we have to add more neurons. The testing errors for 10 neurons are even smaller than the corresponding training errors, which is quite normal since training error focuses on the neurons' fitness while testing error focuses on energy analysis. On the other hand, DHNN-RAN3SAT is in the suboptimal testing phase for 90 ≤ α ≤ 120 that achieved the highest errors. We can always validate the optimal and suboptimal testing phase with the number of global and local minima solutions produced by the proposed model. In short, all logic combinations δ k RAN 3SAT show a similar energy profile which is decreasing monotonically towards equilibrium states [4]. This indicates at α = 100 the number of global minima solution produced by the proposed model is similar. Linking to previous information, a study by [39] highlighted the advantages of adding a bias or noise in an ANN development that provides a small impact in retrieving more global minima solution that helps in designing new associative memory of impulsive DHNN.
Furthermore, the choice of the searching algorithm used in investigating the quality of the solution of synaptic weight management is crucial. Our study utilized ES as a searching algorithm. The ''trial and error'' nature of ES could affect the minimization of the cost function [15]. If ES failed to retrieve optimal synaptic weight, the testing phase would be affected, thus resulting in local minima solution. Alternatively, the number of learning can be increased to aid the proposed model in the resulting optimal testing phase. This is because more number of iterations can lead to a global minimal solution. One may ponder why the local minima solution is considered ''bad'' in our approach? The local minima solution is insignificant in our simulation because the higher number of local solutions will disrupt the similarity measurement of the final states of the neurons. Therefore, our mentioned experimental setup helps the network to avoid several local minima solutions.
Another interesting point is different combinations δ k RAN 3SAT generate different results. δ 1,3 RAN 3SAT Has reached the highest errors compared with others. Meanwhile, δ 2,3 RAN 3SAT is the most stable and produces less error in terms of synaptic weight management. This is due to the logical structure of various orders of the SAT. The inclusion of third-order logic is the best combination δ k RAN 3SAT because the probability of getting a satisfying interpretation for third-order logic is higher than first-order logic. Besides, the logical structure of first-order logic can perturb the process of retrieving correct synaptic weight, thus leading to high testing errors.

C. ENERGY ANALYSIS
In this section, the energy profile and types of solution (global or local) produced by DHNN-RAN3SAT will be further discussed. The number of global and local minima solutions attained by the proposed model is illustrated in Figs. 11-12. In terms of energy profile, the difference in energy was analyzed by observing the value of RMSE and SSE between the minimum energy and final energy, which can be seen in Figs. 13-14. As α (the number of neurons) increases for all δ k RAN 3SAT logical structures, the number of global minimum solutions decreases. Note that the logical structure contains more literals as more clauses exist; thus, more iterations are needed to produce feasible solutions. Through Fig. 11, we can observe that δ 2,3 RAN 3SAT achieved more consistent global minima solutions due to the nonexistence structure of k = 1 clauses which has the lowest probability of getting satisfied interpretations compared to δ 2SAT and δ 3SAT .   This signifies that δ k RAN 3SAT of δ 1,3 RAN 3SAT and δ 1,2,3 RAN 3SAT is prone to more neuron oscillations. Another fruitful finding is α = 10 for both Figs. 11-12; we can observe that DHNN-RAN3SAT was able to retrieve the highest number of global minimum solutions with no difference in energy. This may occur due to optimal synaptic weight management that leads to the optimal testing phase in retrieving the consistent final neuron states. One may ponder, what does the energy profile signify? The work by Veerasamy et al. [40] incorporates the numerical method Runge-Kutta 4 to form a modified Hopfield neural network for a power flow analysis of = 0, which leads to higher energy. The minimum energy of various δ k RAN 3SAT logic is dependent on the existence of different SAT clauses.
From the findings, it can be seen that δ 1,3 RAN 3SAT attained the lowest difference in energy which may explain the logic being able to achieve the minimum energy compared to other δ k RAN 3SAT logics. In this study, bipolar neuron representation was utilized instead of the binary that consists of {1, 0}. The existence of the value zero will eliminate a certain coefficient that will lead to zero energy. However, in this study, the Lyapunov energy function is significant to indicate the process of energy minimization by the proposed model. In this study, our focus is υ ≤ ϕ (restricted training). In line with the previous study by [17], many DHNN models were reported in the nonrestricted training environment achieved a lower energy profile. Thus, the conclusion is a higher number of training iteration can lead to global minima energy. Modifications can be done to improve the quality of the solutions of DHNN-RAN3SAT. One of the efforts that can be executed in the testing phase is implementing Boltzmann Machine to improve the effectiveness of the updating rule in DHNN, which can reduce neuron oscillation and generate more global minima solutions.

D. SIMILARITY INDEX
The SI can be expressed as a metric to check the similarity/dissimilarity of various data items. The work by [21] VOLUME 9, 2021 suggested that the performance of the SAT with DHNN can be assessed by using SI. Here we introduce three standard indexing parameters, which are the Jaccard Index, Kulczynski Measure, Ochiai Coefficient, and also consider the useful parameter known as the total variation of neurons (TV ). In Fig. 15 is zero at α = 110. Even though δ k RAN 3SAT is satisfactory, ES will interrupt the proposed model to achieve the optimal training phase (learn inconsistent interpretation). Note that, from Fig. 12, local solutions attained by the proposed model increase as α an increase. Thus, the simulation stops α = 110. Linking to the previous findings δ k RAN 3SAT is believed to undergo higher training complexity compared to 2SAT [6], 3SAT [18], and HornSAT [12]. As discussed, this is due to the fact that the stated studies only utilized systematic, logical rules where each clause is feasibly easier to achieve Q δ RAN 3SAT = 0. It is noticeable that α = 50 the trend δ 1,3 RAN 3SAT is zero and at α = 60 the trend of δ  [42] and [43] indicated that a nonsystematic structure of satisfiability capable of generating diversified solutions. Higher-order of k and random nonredundant literals δ k RAN 3SAT helps in promoting the high value of TV . Above all, TV relates to the existence of other neuron states that leads to global minima solution in other solution spaces. Fig. 16 JAC is presented. Significantly the total trend in between 0.46 ≤ JAC ≤ 0.70. Here the highest-lowest   oscillation is noticed in the combination of δ 1, 2,3 RAN 3SAT whereas δ 2,3 RAN 3SAT the trend is more constant where the indexing range in between 0.55 ≤ JAC ≤ 0.57. Besides, from Fig. 17, we depict OHI where the indexing range is 0.43 ≤ OHI ≤ 0.90. In the SI, the highest-ranking metrics are shown OHI . In this trend, a more consistent fluctuation is also shown δ 2,3 RAN 3SAT . Finally, in Fig. 18, we have drawn KZI δ k RAN 3SAT different structures k. The range for KZI is 0.64 ≤ KZI ≤ 0.84 which is slightly higher than JAC. It needs to mention that the highest-lowest oscillation occurs in OHI and KZI is for δ 1,2,3 RAN 3SAT . As the purpose of SI metrics is to investigate neuron variation produced by DHNN-RAN3SAT, a more dynamic SI metrics formulation should be utilized. Promising formulation of [44] and [45] can analyze true negatives of benchmark neuron states and explore a more diversified solution. At this stage of understanding, we presume (−1, −1) the solution will create solutions that are linearly independent of each other. It needs to be mentioned that the SI part TV shows the highest number of fluctuations. From Fig. 15 to Fig. 18, we see that fluctuation δ 2,3 RAN 3SAT is more constant than δ 1,3 RAN 3SAT and δ 1,2,3 RAN 3SAT . Meanwhile the logical rule that achieves the highest global minimal solutions is very effective for logic mining which can be imposed on numerous fields. The final state of the neuron can be converted into maximum induced logic. The benefits from the results of this research can provide benefits to human resources [7], financial sectors [6], to medical sciences [1].
From the above discussion, we provided a clear concept of cost minimization, synaptic weight, energy profile, and neuron variation that is the overall behavior of δ k RAN 3SAT DHNN. It can be concluded that all combinations of δ k RAN 3SAT were successfully implemented in DHNN by achieving higher global minima solutions and diversified final neuron states. Through our findings gives better insights into a nonsystematic logical rule than the work of Sathasivam et al. [21] in terms of energy analysis and similarity of final neuron states. From our results, δ 2,3 RAN 3SAT is the best logical combination of δ k RAN 3SAT producing lower errors, more consistency in achieving global minima solutions, and more neuron variations. However, aligned with the no free lunch theorem [46], there is no best combination or model we can conclude since each δ k RAN 3SAT provides individually promising findings. On the other hand, this study limits less than 120 neurons, unlike [21] that takes up to 300 neurons. This is because of the learning search of ES. Therefore, the limitations of this work are that we can improve the proposed model by adding robust metaheuristics such as a Genetic algorithm or an Election algorithm. Despite the flexibility of DHNN-RAN3SAT, more modifications can be made to improve the quality of solutions produced. We can increase the number of learning that also affected the number of iterations needed in our simulations. With more iterations, the proposed model can generate more neuron variations, fewer errors, and a global minimum solution. Table 8 illustrates a comparison with existing methods such as 2 Satisfiability (δ 2SAT ) [22], 3 Satisfiability (δ 3SAT ) [23], and Random 2 Satisfiability (δ RAN 2SAT ) [21] with Random 3 Satisfiability δ 2,3 RAN 3SAT . The reason for choosing δ 2,3

RAN 3SAT
is the most optimal logical rule in terms of all performance metric. We compared our proposed logical rule to different existing established rules. In this section, we consider the Number of Global Solutions (N Global ) to verify the effectiveness of the proposed logical rule with standard methods. The aim of comparison is to achieve for 100% N Global . Based on Table 8, we observe that for α = 30, δ 3SAT and δ 2,3 RAN 3SAT generated the highest N Global compared to the logical rule for δ 2SAT and δ RAN 2SAT . It means that these logical rules (δ 3SAT and δ 2,3 RAN 3SAT ) attained 100% global minima solutions. At α = 30, the N Global for δ 2SAT and δ RAN 2SAT is lower than δ 3SAT and δ 2,3 RAN 3SAT . This is due to the lower-order logical combinations that will decrease N Global . It is important to notice that δ 2,3 RAN 3SAT constructs a three-dimensional decision system improves retrieval phase at DHNN. By introducing the element of three-dimensional decision system, δ 2,3 RAN 3SAT is able to utilize the optimal synaptic weight during the training phase. Furthermore, δ 2,3 RAN 3SAT logic offers more structure variability, avoids repeated neuron states, and retrieve optimal final state of neuron compared to the existing methods. As a consequence, we can establish a good agreement that this specific number of neurons δ 2,3 RAN 3SAT is consistent with δ 3SAT and also these logical rules outperform δ 2SAT and δ RAN 2SAT . Meanwhile, if the number of neurons increases that is α > 100, the result will be different because increasing the number of neurons in the logical rule decreases the probability of P(E P 2,3 RAN 3SAT ) = 0. This weakness can be remedied by employing metaheuristics algorithm such as reported in [14] and [15].

VI. CONCLUSION
This paper has proposed a novel RAN3SAT with a different logical structure as a symbolic instruction in DHNN. The proposed DHNN-RAN3SAT reported different logical combinations, which gave new insights on the behavior of nonsystematic logical rules for higher-order dimensional decision systems in achieving the objective functions in each phase. Since RAN2SAT [21] has a lower probability of minimizing the cost function as well as comparatively higher Lyapunov energy function. So, by utilizing several structures of RANkSAT, we can get a higher probability of achieving VOLUME 9, 2021 the minimum cost function. Consequently, zero-cost function resulting in optimal synaptic weight management that also leads to lower energy. In terms of energy profile, it is clear that the combination for k = 2, 3 in RANkSAT has shown more consistent interpretation phenomena, which is one of the other important directions for the next research. Besides, the high dimensional values imparted a low variation value in the final neuron variation in any DHNN model. Here, having the almost exact amount of global minimum energy δ 2,3 RAN 3SAT generated more different neuron variations rather than others.
In the future expansion of this study, based on the obtained result, the manuscript will benefit the following papers [6], [7], [16] for further research extension. Meanwhile, the new metaheuristic approach such as the Grey-Wolf optimization [47] and the Billiard Game algorithm [48] will be significant to optimize the training phase DHNN-RAN3SAT. The application of DHNN-RAN3SAT can be investigated in real life by utilizing various data sets from health to the financial sector.