Skip to content
Research Article Open access CC BY 4.0

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

Ali Muhammad Ali Rushdi, Raid Salih Badawi

Journal of Engineering Research and Reports · pp. 117–137 · Published 24 May 2021

10.9734/jerr/2021/v20i717348

Abstract

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a 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 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 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 QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

Partially-defined boolean function set of supporting variables Karnaugh map prime implicant irredundant disjunctive form minimal sum complete sum

References (29)

  1. 1 Learning Decision Lists [DOI]
  2. 2 Boolean Functions: Theory, Algorithms, and Applications
  3. 3 Often Trusted but Never (Properly) Tested: Evaluating Qualitative Comparative Analysis [DOI]
  4. 4 Model Ambiguities in Configurational Comparative Research [DOI]
  5. 5 Computational limitations on learning from examples [DOI]
  6. 6 Boolean Feature Discovery in Empirical Learning [DOI]
  7. 7 Cause-effect relationships and partially defined Boolean functions [DOI]
  8. 8 Boolean Reasoning: The Logic of Boolean Equations
  9. 9 Logical analysis of data—An overview: From combinatorial optimization to medical applications [DOI]
  10. 10 Maximum patterns in datasets [DOI]
  11. 11 Karnaugh-Map Utility in Medical Studies: The Case of Fetal Malnutrition [DOI]
  12. 12 Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis [DOI]
  13. 13 Reliability Analysis of a Commodity-Supply Multi-State System Using the Map Method [DOI]
  14. 14 On the learnability of Boolean formulae [DOI]
  15. 15 Analogy-preserving functions: A way to extend Boolean samples [DOI]
  16. 16 Derivation of the Complete Sum of a Switching Function with the Aid of the Variable-entered Karnaugh Map [DOI]
  17. 17 Logical Design of n-bit Comparators: Pedagogical Insight from Eight-Variable Karnaugh Maps [DOI]
  18. 18 On learning Boolean functions [DOI]
  19. 19 Map differentiation of switching functions [DOI]
  20. 20 A purely map procedure for two-level multiple-output logic minimization [DOI]
  21. 21 Mathematics and Examples of the Modern Syllogistic Method of Propositional Logic [DOI]
  22. 22 Computer Engineers Look at Qualitative Comparative Analysis [DOI]
  23. 23 Effective computer methods for the calculation of Rademacher-Walsh spectrum for completely and incompletely specified Boolean functions [DOI]
  24. 24 A pseudo Karnaugh mapping approach for datasets imbalance [DOI]
  25. 25 Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis [DOI]
  26. 26 Useless prime implicants of incompletely specified multiple-output switching functions [DOI]
  27. 27 A Comparison of Two Methods for Realizing Minimal Function of Several Logic Variables [DOI]
  28. 28 Utilization of Eight-Variable Karnaugh Maps in the Digital Design of n-bit Comparators [DOI]
  29. 29 Horn Extensions of a Partially Defined Boolean Function [DOI]

Cited by 0

No indexed citations yet.

Related research

Article metrics

Real usage data collected on this platform.

42

Page views

0

PDF downloads

8

Outbound clicks

0

Citations

Views over time

Views by country

Approximate, from request IP at view time — not citizenship or institution. Countries with fewer than 5 views are grouped as "Other".

Traffic sources

Referring site, by host.

Views and downloads exclude known bots/crawlers. Citations combines this platform's own DOI-resolved index with each external source's own reported total — see Cited by above for individually listed citing works. Last refreshed 0 seconds ago.