Map Visualization of Boolean Manipulations in Qualitative Comparative Analysis

We use a regular and modular version of the eight-variable Karnaugh map to demonstrate and visualize some technical details of the Boolean minimization procedures usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, which looks better than the published solution in two different aspects, namely: (a) that it is based on a smaller minimal set of supporting variables, and (b) that it provides a more compact Boolean formula. However, we deliberately refrain from claiming that our solution is a better one, but instead we stress that it is simply un-comparable with the published solution. We emphasize that the comparison between any two rival solutions should be context-specific and not be tool-specific. This paper is part of an ongoing activity striving to streamline the use of Boolean minimization techniques in QCA applications. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as it is in the somewhat different QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming or other equivalent approaches), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum (Blake Canonical Form). Moreover, each of them is a unate function as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.