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Research Article Open access CC BY 4.0

Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis

Ali Muhammad Ali Rushdi, Raid Salih Badawi

Asian Journal of Research in Computer Science · pp. 57–84 · Published 7 May 2021

10.9734/ajrcos/2021/v8i230199

Abstract

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According  to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and  postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

Qualitative Comparative Analysis karnaugh map prime implicant irredundant disjunctive form minimal sum complete sum partially-defined boolean function set of supporting variables

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Cited by 3

Derivation of Minimal Cutsets from Minimal Pathsets for a Multi-State System and Utilization of Both Sets in Checking Reliability Expressions

A. Rushdi, Motaz H. Amashah · Journal of Engineering Research and Reports · 2021

Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

A. Rushdi, Raid Mohammad Salih Badawi · Journal of Engineering Research and Reports · 2021

Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

A. Rushdi, Raid Mohammad Salih Badawi · Asian Journal of Education and Social Studies · 2021

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