Totally optimal decision rules

Optimality of decision rules (patterns) can be measured in many ways. One of these is referred to as length. Length signifies the number of terms in a decision rule and is optimally minimized. Another, coverage represents the width of a rule's applicability and generality. As such, it is desirable to maximize coverage. A totally optimal decision rule is a decision rule that has the minimum possible length and the maximum possible coverage. This paper presents a method for determining the presence of totally optimal decision rules for “complete” decision tables (representations of total functions in which different variables can have domains of differing values). Depending on the cardinalities of the domains, we can either guarantee for each tuple of values of the function that totally optimal rules exist for each row of the table (as in the case of total Boolean functions where the cardinalities are equal to 2) or, for each row, we can find a tuple of values of the function for which totally optimal rules do not exist for this row.