Selected group-theoretic aspects of confirmation measure symmetries

The paper considers symmetry properties of rule interestingness measures (in particular: measures of confirmation). Many authors have studied various symmetries, however the discussion on which sets of symmetries should be taken into account, let alone why the particular symmetries are desirable or not, has still not produced a generally recognized consensus. Furthermore, the results published so far neglect the fact that symmetries can be the subject of group theory-based considerations. This paper aims at solving those problems by introducing group-theoretic interpretations of symmetries, indicating that all symmetries can be treated as permutations, and compositions of symmetries as compositions of permutations. Such an interpretation allows us to apply the well-known group-theoretic results to symmetries. In particular, using this approach, we reveal the phenomenon of incompleteness occurring in sets of symmetries considered by different authors, and propose an effective way of controlling it. Moreover, we demonstrate that assessing the symmetries as either desirable or undesirable, as introduced by these authors, brings in inconsistencies, which become evident under the permutation-based interpretation. Finally, we present group-theoretic guidelines to the design of such symmetry sets that are free of the incompleteness and inconsistency phenomena but remain meaningful in the context of rule evaluation.