## 2 Method to Extract the Rule-Based Knowledge
### NAARs as PRs  
This study introduces NAARs that satisfy logical testing in KBs; thus, they can be considered PRs. This paper was motivated by project work in the area of medical diagnoses, called MYCIN, which began in 1972 [20]. It is a rule-based ES for diagnosing infectious blood diseases. MYCIN inferred conclusions from a given condition through certain factor values, which are a calculus of uncertainty for measuring the credibility of a rule, since reasoning in uncertainty is the most important part of MYCIN. Uncertain knowledge representation is also introduced in this PRS by identifying its certainty factor value [25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Type of Circularity**} & **Condition** & **Definition** \\ \hline
1. & DC Type I & \(\mathcal{R}:X\to Y\in\mathbb{R}\) & There is a pair of rules \(\mathcal{R}:X\to Y\in\mathbb{R}\) and \\  & \(\mathcal{R}^{*}:X^{*}\to Y^{*}\in\mathbb{R}\) & \(\mathcal{R}^{*}:X^{*}\to Y^{*}\in\mathbb{R}\
i Y  
As implied, the generation of an AR differs from PRs as there are two quality measurements (i.e., support and confidence) in ARM. However, we found that quality measurements in ARM and PRs share similar statistical-approach concepts, in which Bayesian probability, i.e., conditional probability, is employed. The certainty factor (\(CF\)) is denoted by  
\[CF(\mathcal{R})=\frac{MB(Y,X)-MD(Y,X)}{1-\min\{MB(Y,X),MD(Y,X)\}} \tag{3}\]  
where \(MB\) is a measurement of belief and \(MD\) is a measurement of disbelief which are denoted below. \[MB(Y,X)=\left\{ \begin{array}{lcl}1&,&P(Y)=1\\ \frac{\max\{P(Y|X),P(Y)\}-P(Y)}{\max\{0,\ 1\}-P(Y)}&,&\text{otherwise}\\ \end{array} \right. \tag{4}\]  
\[MD(Y,X)=\left\{ \begin{array}{lcl}1&,&P(Y)=0\\ \frac{\min\{P(Y|X),P(Y)\}-P(Y)}{\min\{0,\ 1\}-P(Y)}&,&\text{otherwise}\\ \end{array} \right. \tag{5}\]  
where, \(P(Y)\) is a probability of \(Y\) and \(P(Y|X)\) is a conditional probability of \(Y\) given \(X\). To relate quality measurements in NRARs with \(CF\) values in PRs, the gain value of a rule \(\mathcal{R}\) represents the difference between confidence and support values of an AR, which is denoted as \(Gain_{AR}(\mathcal{R})\). Later, \(Gain_{AR}(\mathcal{R})\) can be utilized to relate to \(CF(\mathcal{R})\). Based on [26], we show the derivation of \(Gain_{AR}(\mathcal{R})\). **Lemma 1**. : _(Gain Value of an AR)_  
Suppose that \(\mathcal{R}:\ X\to Y\) is an AR. The gain value of \(\mathcal{R}\) is given by  
\[Gain_{AR}(\mathcal{R})=conf(\mathcal{R})-sup(Y). \tag{6}\]  
**Proof. **  
\[Gain_{AR}(\mathcal{R})=Gain(X\to Y)\] \[\triangleq H(Y|X)-H(Y)\] \[\approxeq P(Y|X)-P(Y)\] \[=conf(\mathcal{R})-sup(Y)\]  
As in ARM [23], a set of ARs is extracted by satisfying confidence and/or support threshold values, where \(sup\) and \(conf\in\ [\ 0,\ 1]\). Consequently, the range value of \(Gain_{AR}(\mathcal{R})\) can be determined as a bounded value of \([-1,\ 1]\). **Corollary 1**. : _Given an AR \(\mathcal{R}\), the bounded value of \(Gain_{AR}(\mathcal{R})\) is in range \([-1,\ 1]\)._  
**Proof. ** Since \(conf(\mathcal{R})\in[0,\ 1]\) and \(sup(Y)\in[0,\ 1]\), \(-1\leq Gain_{AR}\ (\mathcal{R})\leq 0\) when \(supp(Y)=1\), and \(0\leq Gain_{AR}\ (\mathcal{R})\leq 1\) when \(conf(\mathcal{R})=1\). From Equation (6), certainty factors of ARs can be viewed as the \(Gain_{AR}\) value over the \(sup\) value of the consequent part. We denote certainty factor values of ARs as \(CF_{AR}\). The derivation of \(CF_{AR}\) is described as follows. **Theorem 1**. : _(Certainty Factors of ARs)_Let \(\mathcal{R}:\ X\to Y\) be an AR. If \(Gain_{AR}(\mathcal{R})=conf(\mathcal{R})-supp(Y)\), then we have \(CF_{AR}(\mathcal{R})\in[-1,\ 1]\), such that  
\[CF_{AR}(\mathcal{R})=\left\{ \begin{array}{ll}\frac{Gain_{AR}(\mathcal{R})}{1- sup(Y)},&Gain_{AR}(R)\geq 0\\ \frac{Gain_{AR}(\mathcal{R})}{sup(Y)},&Gain_{AR}(R)<0\end{array} \right.\]  
And the range interval of \(CF_{AR}\) is \([-1,\ 1]\). Proof.