Map Visualization of Cause-Effect Relations in Qualitative Comparative Analysis | Chapter 1 | Novel Perspectives of Engineering Research Vol. 3

 This chapter investigates cause-and-effect interactions modeled by partially-defined Boolean functions (PDBF) using a recent regular and modular version of the eight-variable Karnaugh map (known also as incompletely-specified switching functions). First, we show how to use a simple Karnaugh-map test to determine whether a certain variable must be included in a list of supporting variables for a function, or whether it should be excluded from such a set. This exclusion is achieved through certain don't-care assignments that ensure the equivalence of the Boolean quotient with respect to the variable and 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 either of these two half maps replaces the original full map as a representation of the function. This type of variable exclusion could be repeated with respect to other variables until a minimal set of supporting variables is found. The work deals with obtaining all of a PDBF's minimal sets of supporting variables without explicitly using integer-programming techniques. Standard map methods for extracting prime implicants allow the development of all irredundant disjunctive forms for each of the minimal sets obtained (IDFs). We avoid the effort of drawing prime-implicant loops on the initial eight-variable map by following this approach of first finding a minimal set of supporting variables, and we postpone this task until the map is (typically drastically) reduced in size. The approach described here has significant implications for the recently founded field of Qualitative Comparative Analysis (QCA). The mainstream QCA community is unlikely to appreciate these repercussions, as they plainly show that the too-often impressive outcomes reported by QCA adherents must be reviewed and investigated. More observations, in our opinion, are needed to limit down the possibilities and reduce the number of possible IDFs.

Author(S) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

Raid Salih Badawi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

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