Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis

Abstract

A recent debate in the literature of Qualitative Comparative Analysis (QCA) concerns the potentials and pitfalls of the multi-value variant (mvQCA) in comparison with the more established crisp-set QCA (csQCA) and fuzzy-set QCA (fsQCA) variants. So far, the mvQCA methodology has been implemented either algebraically or via specific software tools such as TOSMANA. The main goal of this paper is to enhance the mvQCA methodology through the utilization of several varieties of Karnaugh maps including (a) the Conventional Karnaugh Map (CKM), (b) the Multi-Valued Karnaugh Map (MVKM), and (c) the Variable-Entered Karnaugh Map (VEKM). The paper offers a tutorial exposition of each of these maps in terms of two recently-published problems concerning the legal provision (introduction) and implementation of party bans in sub-Saharan Africa. Results obtained via various map techniques agree exactly among themselves, and are generally more compact than those obtained earlier via elementary algebraic manipulations, or even via software tools. We show, by way of example, that coding multi-valued variables by binary ones has a harmful primary effect of increasing the input domain. This effect is partially counterbalanced by a (contrarily to common belief) beneficial secondary effect of introducing genuine don’t-care configurations. We also address the issue of unresolved contradictory configurations, and propose two strategies to cope with them. The maps used tackle seven binary variables (or their equivalent), a number beyond the typical map limit of six variables. They are used to produce not only the minimal sum of a Boolean function but the complete sum as well. Though this paper is basically intended as a contribution to mvQCA methodology, it is also of significant utility in any field that demands the use of the Karnaugh map. It serves as a unification/exposition of three fundamental variants of the map, and has a definite pedagogical advantage for the wide spectrum of map users.

Switching-Algebraic Analysis of Multi-State System Reliability

“…However, one of the 'good' alternatives for handling problems of multi-valued logic is to reduce them to problems of binary logic. There is already an unsettled debate (extending to areas beyond the scope of reliability), on whether problems of multi-valued logic should be better solved in the multi-valued domain, or should be alternatively replaced by equivalent problems in the binary domain [45]. We reiterate herein our belief that one might prefer one of these two alternatives to the other only as a matter of personal discretion, taste, and background.…”

“…In passing, we observe that the Karnaugh map proved to be a handy and powerful tool for our current application. Other related uses of the Karnaugh map (beyond its conventional use in digital design) are also available (see, e. g., [28,29,31,32,[45][46][47][48][49] [28,29,31,34,45]), and the Multi-Valued Karnaugh Map (MVKM) (see, e. g. , [45]).…”

Checking Correctness of a Symbolic Reliability Expression for a Capacitated Network

“…A mapping {0, 1} → where is the field of real numbers, i.e. ( ) is an assignment of a real number for each of the possible 2 values of [4,21,12,20,[22][23][24][25][26][27][28].…”

“…Multiaffine functions include: (a) Certain algebraic functions such as system reliability/ unreliability and system availability/unavailability [29]. (b) Pseudo-Boolean (switching) functions [23][24][25][26][27][28] such as source-to-terminal capacity or the squared capacity as a function of link successes.…”

Reliability Analysis of a Commodity-Supply Multi-State System Using the Map Method

“…There is a long history of utilization of the Karnaugh map as a probability map (or reliability map) in the binary case [51][52][53][54][55][56][57][58][59]. There are also some notable applications of the Karnaugh map as a multi-value map [60][61]. Our work herein combines the probability and multi-value notions by adapting the map to multi-valued reliability calculations.…”