Map Visualization of Cause-Effect Relations in Qualitative Comparative Analysis

Abstract

This chapter utilizes a modern regular and modular version of the eight-variable Karnaugh map in a systematic and exhaustive investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely-specified switching functions). First, we present a simple Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without explicit resort to integer-programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is (usually dramatically) reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the mainstream QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized. In our opinion, more observations have to be made in order to narrow down the possibilities and decrease the number of candidate IDFs.