## 2 Method to Extract the Rule-Based Knowledge
### Notations and Definitions
Moreover, this study also takes some properties of PRs into consideration toward NAARs as the main focus. An FCI is a frequent itemset which has no superset with the same support values. Formally, an FCI is defined as follows:  
**Definition 3**. : _(Frequent Closed Itemset)_  
Given a frequent itemset \(X\), \(X\) is an FCI \(\Leftrightarrow\
exists Z\
i Z\
i X\wedge sup(X)=sup(Z)\). One previous study states that a generator item set should be selected according to the minimum description length [24]. An item set is called a generator if there is no subsequence with the same support values. The formal definition of a generator is presented below. **Definition 4**. : _(Generator Frequent Closed Itemset)_  
Given an FCI, where \(X\) is a GFCI \(\Leftrightarrow\
exists Z\
i X\subseteq Z\wedge sup(X)=sup(Z)\). According to Definition 4, a set of GFCIs may still contain redundant generators, e.g., two GFCIs that share the same information; thus, the longest one can be pruned to enhance the mining process. Assume we have a set of GFCIs, such that \(G=\{(a:a1,\,\mathit{sbp}:\,\mathit{sbp2}),\,(bmi:bmi2,\mathit{fg}:\,\mathit{fg1}),\, (a:a1,\,bmi:bmi2,\mathit{sbp}:\,\mathit{sbp2}),\,(a:a1,\,\mathit{sbp}:\,\mathit{sbp2 },\mathit{fg}:\,\mathit{fg1}),\,(bmi:bmi2,\mathit{fg}:\,\mathit{fg1},\,\mathit{sbp }:\,\mathit{sbp1}).\) After that, we have a set of minimal generators \(mG=\{(a:a1,\,\mathit{sbp}:\,\mathit{sbp2}),\,(bmi:bmi2,\mathit{fg}:\,\mathit{fg1})\}\), which, by definition, can be described as follows. **Definition 5**. : _(Minimal Generator Frequent Closed Itemset)_  
Let \(G=\left\{X_{g1}\,\ X_{g2},\ldots\right\}\) be a set of GFCIs. \(X_{g_{i}}\) can be said to be an mGFCI if \(\
exists X_{g_{i}}\
i X_{g_{i}}\subseteq X_{g_{j}}\), where \(i=j\) and \(X_{g1}\,\ X_{g2}\in G\). Regarding Definition 5, we can clearly state that a set of mGFCIs (\(mG\)) is not an empty set. Additionally, there is a property of \(mG\) as follows. **Property 1**. : _If \(mG\) is a set of mGFCIs, then: (i) \(mG=\varnothing\); and/or (ii) \(mG=G\) when no proper generator \(X_{g}\in G\) can be found._  
By adopting Definition 2, an AR from an mGFCI set can be generated, which is further called a non-redundant association rule (NRAR). In conventional ARM, ARs can be obtained by randomly putting all possible FCIs on the antecedent or consequent sides. However, because the number of mGFCIs is small, it may difficult to construct a rule only from mGFCIs. Conforming to [17], a NRAR can be formed by two combinations between FCIs and mGFCIs. To construct a NRAR, mGFCIs are considered a premise, while items in FCI--excluding that of its corresponding mGFCI--are put as the conclusion. The definition of a NRAR is described below. **Definition 6**. : _(A Non-Redundant Association Rule)_  
Let \(mG\) be a set of mGFCIs and \(F\) is a set of FCIs. A NRAR can be written as \(\mathcal{R}:\,X\rightarrow(Y\backslash X)\) where \(X\in mG,\ Y\in F\) and \(X\
eq Y\). Regarding Property 1 and Definition 6, we ensure that \(Y\backslash X\) is a finite set, which means that we do not need to worry about losing any information, e.g., \((X\rightarrow\varnothing)\). A construction rule, \(\mathcal{R}:\,X\to Y\) with \(X,\,Y\in F\) and \(X=Y\), can still be applied and considered to be a NRAR, even when \(mG=F\). On the other hand, since we are trying to obtain concise rule bases, then PRs are also discussed as one of main focuses of this study. **Definition 7**. : _(Production Rules)_  
A rule-based mode of knowledge representation is termed a PR. The rule is basically in the form of if _<premises>_ then _<conclusions>_ or similarly if _<evidence (E)>_ then _<hypotheses (H)>_. PRs can be applied as rule bases after the collection of rules in PRs is verified through a logical testing procedure. For simplicity, we need to ensure that there is no single rule that can be implied with other rules, called non-subsumption rules (or non-implication rules).