Election Algorithm for Random k Satisfiability in the Hopfield Neural Network

Abstract

Election Algorithm (EA) is a novel variant of the socio-political metaheuristic algorithm, inspired by the presidential election model conducted globally. In this research, we will investigate the effect of Bipolar EA in enhancing the learning processes of a Hopfield Neural Network (HNN) to generate global solutions for Random k Satisfiability (RANkSAT) logical representation. Specifically, this paper utilizes a bipolar EA incorporated with the HNN in optimizing RANkSAT representation. The main goal of the learning processes in our study is to ensure the cost function of RANkSAT converges to zero, indicating the logic function is satisfied. The effective learning phase will affect the final states of RANkSAT and determine whether the final energy is a global minimum or local minimum. The comparison will be made by adopting the same network and logical rule with the conventional learning algorithm, namely, exhaustive search (ES) and genetic algorithm (GA), respectively. Performance evaluation analysis is conducted on our proposed hybrid model and the existing models based on the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Sum of Squared Error (SSE), and Mean Absolute Error (MAPE). The result demonstrates the capability of EA in terms of accuracy and effectiveness as the learning algorithm in HNN for RANkSAT with a different number of neurons compared to ES and GA.

Modified election algorithm in hopfield neural network for optimal randomksatisfiability representation

“…where U jk is the synaptic connection matrix that established the strong connection between j and k neurons, L j defines the unit condition k and t j described the threshold function of neurons j. Several studies [16,28,[33][34][35][36][37][38] defined t j = 0 to verify that the HNN always leads to a decrease in energy monotonically. Each time neuron was connected with U jk , the value of the synaptic connection will be preserved as a stored pattern in an interconnected vector where U ð1Þ ¼ ½U ð1Þ jk n  n and U ð2Þ ¼ ½U [16,20] that the constraint of synaptic weight matrix U (1) and does not allow self-loop neuron connection U jk .…”

“…The value of H ℚ RÀkSAT signifies the value of the energy concerning the absolute final energy H min ℚ RÀkSAT obtained from random-kSAT. Hence the quality of the final neuron configuration can be properly examined based on the condition [15,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] defined in equation 7jH…”

“…For the case of k 2, random kSAT has the following properties: (a) A set of variables {x 1 , x 2 , x 3 , … , x n }, in a clause C i where i = 1, 2, 3, ..., n that consists of x or À x. (b) Randomly select 2 variables from the set of n variables with 0.5 probability of negating each variable in the clause [34,35]. (c) Each x i in C i is connected by a disjunction (∨) .…”

“…In general, the formulation can be generated in different combinations of neurons (atoms) as the number of neurons (atoms) fluctuated. Comparatively, a high number of neurons (atoms) per number of the clause would increase the probability of many neurons beings satisfied [16][17][18]34]. The costs function of the negated random 2SAT for bipolar follows.…”

“…The aim of selecting the traditional searching framework is to determine the degree of efficacy of HNN-R2SATES model in carrying out random 2SAT logic programming. Apart from that, there are feasible satisfiable assignments given for any random 2SA logical representation [34][35][36]. The satisfied clause for the ES heuristic is extracted after a brutal "trial and error" procedure is carried out.…”