# Linked Coupling Faults Detection by Multirun March Tests
## 3 Necessary and Sufficient Conditions for \(LCF\) Detection
The absence of a connection (roles) between cells \(a_{i}\) and \(a_{j}\) is denoted by the symbol (-). Thus, in the description of the _LCF_, all participating cells and all their respective roles are presented. In the formal description of _LCF_(\(a_{i},a_{j},a_{k},\ldots,a_{z}\)), the information is sequentially presented in the form of a set of symbols (-, \(A\), \(V\), and \(B\)) for all cells \(a_{i},a_{j},a_{k},\ldots,a_{z}\). When describing cells, their sequence \(a_{i},a_{j},a_{k},\ldots,a_{z}\) is also preserved (i.e., the first position of all the descriptions contains information about cell \(a_{i}\), then \(a_{j}\), and so on). This sequence is determined by the cell access time, corresponding to the sequence of their addresses for single march tests. As an example, we consider _LCF_6 presented in graphical form and the form of a new formal model (Figure 4). In the above example, _LCF_6 cells \(a_{k}\) and \(a_{l}\), included in this fault, form two simple _CF_s and act in relation to each other in both roles, indicated by the symbol \(B\) in the description of _LCF_6 (Figure 4). As explained, \(B\) indicates that the cell has two roles, and which one it performs is determined by the scenario, set by the sequence of accessing the cells within the framework of the march test. Depending on the direct \(\Uparrow\) addressing, when the addresses are related as follows \(i<j<k<\ldots<z\), or are reversed \(\Downarrow\), when \(i>j>k>\ldots>z\), the _LCF_6 description is presented as follows:  
\[ \begin{array}{l}\textit{LCF6(a_{i},a_{j},a_{k},a_{l})=\{\langle a_{i},A,A,- \rangle;\langle V,a_{j},-,-\rangle;\langle V,-,a_{k},B\rangle;\langle-,-,B,a_{ l}\rangle\}=\\ \Uparrow\Downarrow\{a_{i},A,A,-\rangle;\langle V,a_{j},-,-\rangle;\langle V,-,a _{k},A\rangle;\langle-,-,V,a_{l}\rangle\},\\ \Downarrow\{a_{i},A,A,-\rangle;\langle V,a_{j},-,-\rangle;\langle V,-,a_{k},A \rangle;\langle-,-,V,a_{l}\rangle\}.\end{array}  \tag{3}\]  
As noted, the ordered ratio of cell addresses involved in the _LCF_ failure corresponds to the time sequence of access to them within the framework of one-run march tests with a fixed address sequence. The relation of the values of the physical cell addresses in the _LCF_ and the relation of the access times to these cells can be different. An example of such a situation is the failure presented in Figure 2. Physically, this is the same fault. However, its manifestation depends on the address sequence, which is displayed in the description of this fault for two cases of address sequences:  
\[ \begin{array}{l}\textit{LCF1}=\textit{LCF(a_{i},a_{j},a_{k})=\{\langle a_{ i},-,A\rangle;\langle-,a_{j},A\rangle;\langle V,V,a_{k}\rangle\}}\\ \textit{LCF2}=\textit{LCF(a_{i},a_{k},a_{j})=\{\langle a_{i},A,-\rangle;\langle V,a_{k},V\rangle;\langle-,A,a_{j}\rangle\}}\end{array}  \tag{4}\]  
Similarly, any of the possible _LCF_s are described. Examples of such descriptions for the previously given _LCF_s are provided below:  
\[ \begin{array}{l}\textit{LCF3}=\textit{LCF(a_{i},a_{j},a_{k})=\{\langle a_{i},A,-\rangle;\langle V,a_{j},A\rangle;\langle-,V,a_{k}\rangle\};\\ \textit{LCF4}=\textit{LCF(a_{i},a_{j},a_{k})=\{\langle a_{i},A,A\rangle;\langle V,a_{j},-\rangle;\langle V,-,a_{k}\rangle\};\\ \textit{LCF5}=\textit{LCF(a_{i},a_{j},a_{k})=\{\langle a_{i},B,B\rangle;\langle B,a_{j},B\rangle;\langle B,B,a_{k}\rangle\}.}\end{array} \]  
For _LCF_6 (3), the _LCF_5 failure is represented as \(\Uparrow\{\langle a_{i},A,A\rangle;\langle V,a_{j},A\rangle;\langle V,V,a_{k}\rangle\}\), \(\Downarrow\{\langle a_{i},V,V\rangle;\langle A,a_{j},V\rangle;\langle A,A,a_ {k}\rangle\}\). The main advantage of the formal model for describing the configurations of linked faults is its compact notation containing complete information about all single _CF_s that  
Figure 4: Linked coupling fault _LCF_6 with description. form the \(LCF\).