Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis
Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah 21589, Saudi Arabia.
Received on December 08, 2016
Accepted on February 08, 2017
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.
Keywords- Multi-value qualitative comparative analysis, Conventional Karnaugh map, Multi-valued Karnaugh map, Variable-entered Karnaugh map, Prime implicants, Minimal sum, Complete sum.
Rushdi, A. M. A. (2018). Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis. International Journal of Mathematical, Engineering and Management Sciences, 3(1), 28-46. https://dx.doi.org/10.33889/IJMEMS.2018.3.1-004.
Conflict of Interest
Ali, M. H., Hassan-Alshiroofi, F. J., & Rotithor, H. G. (1996). A framework for design of multivalued logic functions and its application using CMOS ternary switches. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 43(4), 279-289.
Bahraini, M., & Epstein, G. (1988). Three-valued Karnaugh maps. In International Symposium on Multiple-Valued Logic (ISMVL),18, 178-185.
Brown, F. M. (1990). Boolean reasoning: The logic of Boolean equations, Kluwer Academic Publishers, Boston, USA.
Crama, Y., & Hammer, P. L. (2011). Boolean functions: Theory, algorithms, and applications, Cambridge, United Kingdom, Cambridge University Press.
Cronqvist, L. (2006). Tosmana: Tool for small-N analysis. Marburg: University of Marburg. Available online at: http://www.tosmana.net.
Duşa, A., & Thiem, A. (2015). Enhancing the minimization of Boolean and multivalue output functions with eQMC. The Journal of Mathematical Sociology, 39 (2), 92-108.
Fletcher, W. I. (1980), An engineering approach to digital design. Prentice Hall, Englewood-Cliffs, NJ, USA.
Ghiye, V., Bonde, S., & Dhande, A. (2014). Investigation of ternary function minimization. In 2014 IEEE Fourth International Conference on Communication Systems and Network Technologies (CSNT), pp. 1054-1058.
Hartmann, C., & Kemmerzell, J. (2010). Understanding variations in party bans in Africa. Democratization, 17(4), 642-665.
Hill, F. J., & Peterson, G. R. (1993). Computer aided logical design with emphasis on VLSI, 4th ed., Wiley, New York, USA.
Jordan, E., Gross, M. E., Javernick-Will, A. N., & Garvin, M. J. (2011). Use and misuse of qualitative comparative analysis. Construction Management and Economics, 29 (11), 1159–1173.
Kan, A. K. S., Adegbite, E., El Omari, S., & Abdellatif, M. (2016). On the use of qualitative comparative analysis in management. Journal of Business Research, 69 (4), 1458-1463.
Lee, S. C. (1978). Modern switching theory and digital design, Prentice-Hall, Englewood Cliffs, New Jersey, NJ, USA.
Marx, A., & Duşa, A. (2011). Crisp-set qualitative comparative analysis (csQCA), Contradictions and consistency benchmarks for model specification. Methodological Innovations, 6(2), 103–148.
Marx, A., Rihoux, B., & Ragin, C. C. (2014). The origins, development, and application of qualitative comparative analysis: The first 25 years. European Political Science Review, 6(1), 115–142.
Muroga, S. (1979). Logic design and switching theory, John Wiley & Sons, New York, NY, USA.
Ragin, C. C. (1987). The comparative method. moving beyond qualitative and quantitative strategies, University of California Press, Berkeley, CA, USA.
Ragin, C. C., Mayer, S. E., & Drass, K. A. (1984). Assessing discrimination: A Boolean approach. American Sociological Review, 49(2), 221–234.
Rathore, T. S. (2014). Minimal realizations of logic functions using truth table method with distributed simplification. IETE Journal of Education, 55 (1), 26-32.
Rathore, T. S., & Jain, A. (2014). A systematic map method for realizing minimal logic functions of arbitrary number of variables. In 2014 IEEE International Conference on Circuits, Systems, Communication and Information Technology Applications (CSCITA), pp. 81-86.
Rihoux, B., & de Meur, G. (2009). Crisp-set qualitative comparative analysis (csQCA). In B. Rihoux, & Charles C. Ragin (Eds.), Configurational Comparative Methods: Qualitative Comparative Analysis (QCA) and Related Techniques. (pp. 33-69). Sage Publications, Thousand Oaks, CA, USA.
Rohlfing, I. (2012). Analyzing multilevel data with QCA: A straightforward procedure. International Journal of Social Research Methodology, 15(6), 497-506.
Roth, Jr., C. H., & Kinney, L. L. (2014). Fundamentals of logic design, 7th Ed. Cengage Learning, Stamford, CT, USA.
Rushdi, A. M., & Al-Yahya, H. A. (2000). A Boolean minimization procedure using the variable-entered Karnaugh map and the generalized consensus concept. International Journal of Electronics, 87(7), 769-794.
Rushdi, A. M., & Al-Yahya, H. A. (2001a). Further improved variable entered Karnaugh map procedures for obtaining the irredundant forms of an incompletely-specified switching function, Journal of King Abdulaziz University: Engineering Sciences, 13 (1), 111-152.
Rushdi, A. M., & Al-Yahya, H. A. (2001b). Derivation of the complete sum of a switching function with the aid of the variable-entered Karnaugh map, Journal of King Saud University: Engineering Sciences, 13 (2), 239-26.
Rushdi, A. M., & Amashah, M. H. (2011). Using variable–entered Karnaugh maps to produce compact parametric general solutions of Boolean equations, International Journal of Computer Mathematics, 88 (15), 3136-3149.
Rushdi, A. M. (1983). Symbolic reliability analysis with the aid of variable-entered Karnaugh maps, IEEE Transactions on Reliability, R-32 (2), 134-139.
Rushdi, A. M. (1985). Map derivation of the minimal sum of a switching function from that of its complement, Microelectronics and Reliability, 25 (6), 1055-1065
Rushdi, A. M. (1987). Improved variable-entered Karnaugh map procedures. Computers & Electrical Engineering, 13(1), 41-52.
Rushdi, A. M. (1997). Karnaugh map, Encyclopedia of mathematics, Supplement Volume I, M. Hazewinkel (Editor), Boston, Kluwer Academic Publishers, pp. 327-328. Available online at http://eom.springer.de/K/k110040.html.
Rushdi, A. M., Zarouan, M., Alshehri, T. M., & Rushdi, M. A. (2015a). The incremental version of the modern syllogistic method. Journal of King Abdulaziz University: Engineering Sciences, 26 (1), 25-51.
Rushdi, A. M., Zarouan, M., Alshehri, T. M., & Rushdi, M. A. (2015b). A modern syllogistic method in intuitionistic fuzzy logic with realistic tautology, The Scientific World Journal, 2015, Article ID 327390, 12 pages.
Thiem, A. (2013). Clearly crisp, and not fuzzy: A reassessment of the (putative) pitfalls of multi-value QCA. Field Methods, 25 (2),197–207.
Thiem, A., & Duşa, A. (2013a). Boolean minimization in social science research: A review of current software for qualitative comparative analysis (QCA). Social Science Computer Review. 31 (4), 505-521.
Thiem, A., & Duşa, A. (2013b). QCA: A package for qualitative comparative analysis. The R Journal, 5 (1), 87-97.
Vingron, S. P. (2004). Reduced karnaugh maps. chapter 20 in switching theory, pp. 207-218. Springer-Verlag, Berlin-Heidelberg, Germany.
Vingron, S. P. (2012). Karnaugh maps. Chapter 5 in logic circuit design: Selected methods, pp. 51-66. Springer-Verlag, Berlin-Heidelberg, Germany.
Vink, M. P., & Van Vliet, O. (2009). Not quite crisp, not yet fuzzy? Assessing the potentials and pitfalls of multi-value QCA. Field Methods, 21 (3), 265-289.
Vink, M. P., & Van Vliet, O. (2013). Potentials and pitfalls of multi-value QCA: Response to Thiem. Field Methods, 25 (2), 208-213.