**3. Satisfiability Representation**

SAT is a representation of determining the interpretation that satisfies the given Boolean formula. According to [32], SAT is proven to be an NP-complete problem and is included to cover wide range of optimization problem. Extensive research on SAT leads to the creation of variant SAT which is 2SAT. In this paper, the choice of *k* = 2 is due to the two-dimensional decision-making system. Generally, 2SAT consist of the following properties [19]:


By taking property (a) into account until (c), one can define the explicit definition of *Q*2*SAT* as follows:

$$Q\_{2SAT} = \underset{i=1}{\stackrel{y}{\wedge}} \mathbb{C}\_i \tag{1}$$

where *Ci* is a list of clause with two variables each

$$\mathbb{C}\_{i} = \bigvee\_{i=1}^{x} (m\_{i}, n\_{i}) \tag{2}$$

By considering the Equations (1) and (2), a simple example of *Q*2*SAT* can be written as

$$Q\_{\rm 2SAT} = (A \lor \neg B) \land (\neg M \lor D) \land (\neg E \lor \neg F) \tag{3}$$

where the clauses in Equation (3) are *C*<sup>1</sup> = (*A* ∨ ¬*B*), *C*<sup>2</sup> = (¬*M* ∨ *D*), and *C*<sup>3</sup> = (¬*E* ∨ ¬*F*). Note that each clauses mentioned above must be satisfied with specific interpretations [10]. For example, if the interpretation reads (*M*, *D*) = (1, −1), *Q*2*SAT* will evaluate false or −1. Since *Q*2*SAT* contains an information storage mechanism and is easy to classify, we implemented *Q*2*SAT* into ANN as a logical system.
