# Exploring Variability of Visual Accessibility Options in Operating Systems
## 5 Interpretation
### Logical Relations between Features  
We retained three types of logical relations: co-occurrences, binary implications, and mutual exclusion (called _mutex_). For each logical relation, we give respectively a definition, the extraction method, and one example. Table 6 presents the number of features, as well as the exact and the upper bound of binary implications, co-occurrences, and mutex number. The upper bounds are computed using the formulas in [36]. We notice that the Brightness mechanism has the smallest values; the Contrast mechanism has the average values; the Text style mechanism, the Zoom mechanism and the central lattice have the largest values. We also notice that the number of binary implications and co-occurrences seems to depend from the total number of features, which is not always the case with mutex relationships. All logical relations are available online([[https://gite.lirmm.fr/huchard/visualaccessibilityoptionsinos](https://gite.lirmm.fr/huchard/visualaccessibilityoptionsinos)]([https://gite.lirmm.fr/huchard/visualaccessibilityoptionsinos](https://gite.lirmm.fr/huchard/visualaccessibilityoptionsinos)), accessed on 1 September 2021). Table 7 presents the number of objects, attributes and concepts. * **A Co-occurrence** is a pair of features that always appear together in configurations. A simple method can be used to extract the co-occurrences from a conceptual structure. The method considers the attribute introducer concept set \(AC_{K}\). Each concept of \(AC_{K}\) is analyzed: if it contains more than one feature, the introduced features are co-occurrent. An example of co-occurrence relation from \(Concept\_Zoom\_2\) in Figure 7a is \(FullScreen\leftrightarrow MagnificationGestures\). The worst case is when all features appear together, thus any feature pair is a co-occurence. There are at most \(\dfrac{\mid A\mid(\mid A\mid-1)}{2}\) co-occurrences (where \(A\) is the feature number) [36]. * **Binary implications** correspond to the case where when a first feature is present, a second one is mandatorily present too. A method to extract binary implications is described in [36]. The method uses the set of all pairs \((C_{i},C_{j})\in AC_{K}\times AC_{K}\) such that \(C_{i}{\leq}_{s}C_{j}\). For each feature \(a_{i}\) introduced in \(C_{i}\) and each feature \(a_{j}\) introduced in \(C_{j}\), \(a_{i}\to a_{j}\) is a valid implication. An example of binary implication relation is \(ReduceTransparency\to NormalContrast\) because \(Concept\_Contrast\_3{\leq}_{s}Concept\_Contrast\_5\) (see Figure 7b). In a concept lattice, there are at most \(\mid A\mid^{2}\) binary implications (where \(A\) is the feature number) [36]. * **Mutex** are pairs of mutually exclusive features \(\{a_{1},a_{2}\}\) which satisfy \(a_{1}\to\
eg a_{2}\) (or equivalently \(a_{2}\to\
eg a_{1}\)). In a concept lattice, when the extent of the bottom concept is not empty, this means that there exists at least one configuration which owns all the features, and that there is no mutual exclusion situation. To extract mutex in a conceptual structure, each feature introducer concept pair \((C_{i},C_{j})\) with \(i\
eq j\) is analyzed to check if their extents intersect. If they do not intersect, a feature \(a_{i}\) introduced in \(C_{i}\) and a feature \(a_{j}\) introduced in \(C_{j}\) are mutually exclusive and correspond to a mutex. An example of mutex relation is \(HighContrastText\to\
eg LowContrast\), because the intersection of the extents of the feature introducers is empty. This can be seen in Figure 7b, where \(Concept\_Contrast\_4\) introduces \(HighContrastText\), and \(Concept\_Contrast\_2\) introduces \(LowContrast\). The upper bound of the mutex number is \(\dfrac{\mid A\mid(\mid A\mid-1)}{2}\)[36]. Examples of co-occurrences, implications and mutex involving relational attributes are given and interpreted in the next section. The interpretation step is summarized as follows (Box 5).