## 4 Concept Stability Calculation Based on Maximal Non-Generator
### Linking Stability Calculation to Maximal Non-Generator  
In this part, we first provide a formula (Corollary 2) for calculating stability value leveraging the non-generators. Then, we make use of the special properties (Property 1) of maximal non-generators to obtain a novel perspective (Theorem 1) for calculating the stability of a concept.  
**Corollary 2**: _Given a formal concept \((A,B)\), the stability of \((A,B)\) is the proportion of generator in the power set of \(A\). On the contrary, it also can be computed by the proportion of the total number of non-generator \(|NonGen|\). Therefore, Equation (1) of the stability is simplified as follow:_  
\[\sigma\left(A,B\right)=1-\frac{|NonGen|}{2^{|A|}} \tag{3}\]  
_where \(|NonGen|\) is the total number of non-generator of \((A,B)\)._  
**Proof.** According to Corollary 1, \(|Gen|\) is the total number of \((A,B)\). Then we have the following equality: \(|NonGen|+|Gen|=2^{|A|}\). Naturally, \(\sigma\left(A,B\right)=\frac{|Gen|}{2^{|A|}}=\frac{2^{|A|}-|NonGen|}{2^{|A|}}=1- \frac{|NonGen|}{2^{|A|}}\).  
Definition 3 (Maximal Non-generator): Given a formal context \(K=(G,M,I)\) and a concept \((A,B)\) of \(K\). If there is a subset \(P\) of \(A\) which makes \(f(P)\
eq B\), then \(P\) will be a maximal non-generator if there no exists \(P_{1}\subseteq G\) and \(P\subset P_{1}\) such that \(f(P_{1})\
eq B\).  
**Property 1 (Property of Definition 3 [18]):** Let \((A,B)\) be a formal concept. If \(P\) is a maximal non-generator of the extent \(A\), then \(P\) fulfills that \(\forall P_{1}\subseteq P\), \(P_{1}\) is also a non-generator of \(A\).  
**Proof.** It could be found in [18].  
Based on the special Property 1 of maximal non-generator, we have known all subsets of maximal non-generator are non-generators. On this basic, we can naturally calculate the number of subsets of maximal non-generators to obtain the \(|NonGen|\) in Equation (3). Therefore, a new theorem for calculating the stability of a concept is proposed. The theorem is as follows:  
Theorem 1: Given a formal context \(K=(G,M,I)\) and a concept \((A,B)\) of \(K\). If all maximal non-generators of \((A,B)\) are \(\{P_{1},P_{2},P_{3}\ldots P_{n}\}\), then  
\[\sigma\left(A,B\right)=1-\left(\sum_{i=1}^{n}2^{|P_{i}|}-\sum_{1\leq i<j\leq n }2^{|P_{i}\cap P_{j}|}\ +\cdots+\left(-1\right)^{n-1}2^{|P_{1}\bigcap P_{2}\cap\cdots\cap P_{n}|} \right)\left(2^{|A|}\right)^{-1} \tag{4}\]  
**Proof.** Since \(\{P_{1},P_{2},P_{3}\ldots P_{n}\}\) are the maximal non-generators of \((A,B)\), based on Property 1, all subsets of \(P_{i}\) are non-generators. Thus, the total number of non-generator is \(|NonGen|=|\bigcup_{i=1}^{n}\{S|S\subseteq P_{i}\}|\). Based on Corollary 2 and the Inclusion-Exclusion principle, the total number of non-generators could got via \(|\bigcup_{i=1}^{n}\{S\}|=\sum_{i=1}^{n}2^{|P_{i}|}-\sum_{1\leq i<j\leq n}2^{ |P_{i}\cap P_{j}|}+\cdots+\left(-1\right)^{n-1}2^{|P_{1}\bigcap P_{2}\cap \cdots\cap P_{n}|}\). Naturally, Equation (4) can be directly derived.  
Example 3: Continue the Example 2, we have known the generators of \((\{1,2,3\},\{a,b\})\) are \(\{1,3\},\{2,3\},\{1,2,3\}\) and its stability is equal to 3/8. Obviously, all the subsets of \(\{1,2,3\}\) except these three generator are non-generators. Thus, the non-generator of \((\{1,2,3\},\{a,b\})\) are \(\{1,2\},\{1\},\{2\},\{3\},\emptyset\). According to the Definition 3, the maximal non-generators are \(\{1,2\},\{3\}\).  
Suppose we first have know the maximal non-generators \(\{1,2\},\{3\}\), according the Property 1, the number of subsets of \(\{1,2\}\) and \(\{3\}\) is equal to the number of non-generators of the concept. According the Theorem 1, we can get the number of non-generator is 5 and the stability of \((\{1,2,3\},\{a,b\})\) is equal to \(1-\frac{5}{8}=\frac{3}{8}\).