Classification by cause-and-effect relationships

By definition, property $A$ in object $O$ is the cause for the occurrence of consequence $B$ which is available for observation in information space $I$ if characteristics of $A$ can generate an object in space $I$ containing consequence $B$. In this case, $B$ determinedly appears with the appearance of $A$. Therefore, one can consider the classification problem as calculating the consequences of the characteristics of the object where the consequences act as characteristics of the class. In this case, the characteristics of the classification object can be considered as the cause that deterministically (classification as mapping) generates consequences (characteristics of the class). Each of the properties $A_i$, $i=1,\ldots , k$, is the cause of the deterministic appearance of a nonempty set of its consequences. If the number of classes is large as well as the sets of consequences of each, then the classification problem can be complex to compute due to the fact that repetitions of consequences in the sets of consequences are possible. Therefore, it is advisable to look for simplified schemes for classifying objects according to the causes for the consequences in them. For this, an apparatus of systems of various representatives can be used. In the context of the problem of classifying causes due to consequences, it is impossible to directly use F. Hall's theorem on systems of various representatives, since elements of cause-and-effect chains cannot be broken. The paper shows that the transformation of each of the same chains of cause-and-effect relationships into one common new element in the sets of consequences forms the possibility of applying the conditions of F. Hall's theorem.