Conditional random k satisfiability modeling for k =1,2 (CRAN2SAT) with non-monotonic Smish activation function in discrete Hopfield neural network

<jats:p xml:lang="fr">&lt;abstract&gt; &lt;p&gt;The current development of logic satisfiability in discrete Hopfield neural networks (DHNN)has been segregated into systematic logic and non-systematic logic. Most of the research tends to improve non-systematic logical rules to various extents, such as introducing the ratio of a negative literal and a flexible hybrid logical structure that combines systematic and non-systematic structures. However, the existing non-systematic logical rule exhibited a drawback concerning the impact of negative literal within the logical structure. Therefore, this paper presented a novel class of non-systematic logic called conditional random &lt;italic&gt;k&lt;/italic&gt; satisfiability for &lt;italic&gt;k&lt;/italic&gt; = 1, 2 while intentionally disregarding both positive literals in second-order clauses. The proposed logic was embedded into the discrete Hopfield neural network with the ultimate goal of minimizing the cost function. Moreover, a novel non-monotonic Smish activation function has been introduced with the aim of enhancing the quality of the final neuronal state. The performance of the proposed logic with new activation function was compared with other state of the art logical rules in conjunction with five different types of activation functions. Based on the findings, the proposed logic has obtained a lower learning error, with the highest total neuron variation &lt;italic&gt;TV&lt;/italic&gt; = 857 and lowest average of Jaccard index, &lt;italic&gt;JSI&lt;/italic&gt; = 0.5802. On top of that, the Smish activation function highlights its capability in the DHNN based on the result ratio of improvement &lt;italic&gt;Zm&lt;/italic&gt; and &lt;italic&gt;TV&lt;/italic&gt;. The ratio of improvement for Smish is consistently the highest throughout all the types of activation function, showing that Smish outperforms other types of activation functions in terms of &lt;italic&gt;Zm&lt;/italic&gt; and &lt;italic&gt;TV.&lt;/italic&gt; This new development of logical rule with the non-monotonic Smish activation function presents an alternative strategy to the logic mining technique. This finding will be of particular interest especially to the research areas of artificial neural network, logic satisfiability in DHNN and activation function.&lt;/p&gt; &lt;/abstract&gt;</jats:p>