# Exploring Variability of Visual Accessibility Options in Operating Systems
## 6 Case Study
### Results
#### 6.3.4 Relationships between Operating Systems (RQ4)  
For RQ4, we analyze the extents of the object introducer concepts. The concept lattice excerpts highlight a few examples. We analyze into details the Brightness lattice. Then we show how relationships are combined in the central lattice.  
The Brightness lattice (shown Figure 10) is small, but it contains the main possible situations. We can observe for example:  
* System option set equivalence:
* _iOS5_Brightness_ and _MacOS1010_Brightness_ are in the same simplified extent (_Concept_Brightness_2_). This means that _iOS5_ and _MacOS1010_ provide exactly the same Brightness features.
* System option set specialization:
* _Concept_Brightness_2\(\leq_{s}\)_Concept_Brightness_3_ highlights the fact that _iOS5_ and _MacOS10.10_ have more Brightness features than _Android5.1_, _Android6.0_, _Windows7.0_ and _Windows8.1_.
* Incomparable option sets with shared intersection:
* As _Concept_Brightness_3_ is incomparable for \(\leq_{s}\) with _Concept_Brightness_1_, this indicates that _Android51_Brightness_, _Android60_Brightness_, _Windows70_Brightness_, _Windows81_Brightness_, _iOS5_Brightness_ and _MacOS1010_Brightness_ only share the top concept features with _Ubuntu12.04 and _Ubuntu14.10_.  
The central concept lattice (excerpt in Figure 13) allows us to analyze the different systems taking into account all mechanisms together. Systems are respectively introduced in separate concepts, e.g., _Ubuntu12.04_ is introduced in _Concept_Systems_4_. This means that there are no identical systems.  
For example, from the concept specialization, we obtain the following information:  
* _Concept_Systems_7\(\leq_{s}\)_Concept_Systems_15_ meaning that _Android5.1_ provides more features than _Android6.0_, which means that the option set has been reduced during system upgrade;
* _Concept_Systems_2\(\leq_{s}\)_Concept_Systems_8_ meaning that _Windows8.1_ provides more options than _Windows7.0_;
* _Concept_Systems_11_ is the least upper bound (and the direct cover) of _Concept_Systems_4_ and _Concept_Systems_5_, highlighting some proximity of _Ubuntu12.04 and _MacOS10.10_;
* _Concept_Systems_13_ is the least upper bound (and the direct cover) of _Concept_Systems_5 and _Concept_Systems_6_, revealing _MacOS10.10 and _iOS5_ closeness.  
The results of this analysis illustrate the system diversity and the needs to propose guidance to OS manufacturers. From the analysis of the various systems, and by navigating the concept lattices, they can overview common options (that they should implement) and  
Figure 12: Excerpt of several concept lattices (redrawn from the lattices output of the tool for the sake of explainability) to illustrate mutex in the central lattice.  
more rare ones (that they can decide to add or not). It also informs end users about the existing configurations in the different systems, that can guide them in the choice of a system, or guide the migration.  
The concept lattices are the search space for solving tasks, such as comparing OS, classifying user settings, either independently in each mechanism, or considering all mechanisms together. Tables 7 and 8 show the size of this search space for the fictitious example and our case study, respectively. The number of concepts in a concept lattice varies between 1 (i.e., when all OS have an identical description) and \(2^{min(\#objects,\#attributes)}\) (i.e., when the concept lattice is isomorphic to the powerset of the OS set). This number of concepts thus indicates a variability level between the OS, and also the size of the search space. The figures presented in Tables 7 and 8 show that this search space is of reasonable size (between 5 and 34 concepts for the case study) and practical for navigation purposes. For example, for contrast and color, the concept lattice groups attributes into a structure with 21 concepts which is far more easy to comprehend and exploit, compared to a table with 88 columns.