Paper:

# How to Describe Conditions Like 2-out-of-5 in Fuzzy Logic: A Neural Approach

## Olga Kosheleva^{*}, Vladik Kreinovich^{**}, and Hoang Phuong Nguyen^{***,†}

^{*}Department of Teacher Education, University of Texas at El Paso

500 West University Avenue, El Paso, Texas 79968, USA

^{**}Department of Computer Science, University of Texas at El Paso

500 West University Avenue, El Paso, Texas 79968, USA

^{***}Division Informatics, Math-Informatics Faculty, Thang Long University

Nghiem Xuan Yem Road, Hoang Mai District, Hanoi, Vietnam

^{†}Corresponding author

In many medical applications, we diagnose a disease and/or apply a certain remedy if, e.g., two out of five conditions are satisfied. In the fuzzy case, i.e., when we only have certain degrees of confidence that each of *n* statement is satisfied, how do we estimate the degree of confidence that *k* out of *n* conditions are satisfied? In principle, we can get this estimate if we use the usual methodology of applying fuzzy techniques: we represent the desired statement in terms of “and” and “or,” and use fuzzy analogues of these logical operations. The problem with this approach is that for large *n*, it requires too many computations. In this paper, we derive the fastest-to-compute alternative formula. In this derivation, we use the ideas from neural networks.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.24, No.5, pp. 593-598, 2020.

- [1] R. Bělohlávek, J. W. Dauben, and G. J. Klir, “Fuzzy Logic and Mathematics: A Historical Perspective,” Oxford University Press, 2017.
- [2] G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic: Theory and Applications,” Prentice Hall, 1995.
- [3] J. M. Mendel, “Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions,” 2nd Edition, Springer, 2017.
- [4] H. T. Nguyen, C. L. Walker, and E. A. Walker, “A First Course in Fuzzy Logic,” 4th Edition, CRC Press, 2019.
- [5] V. Novák, I. Perfilieva, and J. Močkoř, “Mathematical Principles of Fuzzy Logic,” Kluwer Academic Publishers, 1999.
- [6] L. A. Zadeh, “Fuzzy sets,” Information and Control, Vol.8, Issue 3, pp. 338-353, 1965.
- [7] K.-P. Adlassnig, “Fuzzy methods in medical research and patient care,” Abstracts of the Int. Conf. on Artificial Intelligence and Computational Intelligence (AICI 2020), pp. 8-9, 2020.
- [8] W. Koller, A. Rappelsberger, B. Willinger, G. Kleinoscheg, and K.-P. Adlassnig, “Artificial Intelligence in Infection Control – Healthcare Institutions Need Intelligent Information and Communication Technologies for Surveillance and Benchmarking,” V. Kreinovich and N. H. Phuong (Eds.), “Soft Computing for Biomedical Applications and Related Topics,” Springer (in press).
- [9] J. Zeckl, M. Wastian, D. Brunmeir, A. Rappelsberger, S. B. Arseniev, and K.-P. Adlassnig, “From Machine Learning to Knowledge-Based Decision Support – A Predictive-Model-Mmarkup-Language-to-Arden-Syntax Transformer for Decision Trees,” V. Kreinovich and N. H. Phuong (Eds.), “Soft Computing for Biomedical Applications and Related Topics,” Springer (in press).
- [10] V. Kreinovich, “From traditional neural networks to deep learning: towards mathematical foundations of empirical successes,” Proc. of the World Conf. on Soft Computing, 2018.
- [11] V. Kreinovich and A. Bernat, “Parallel Algorithms for Interval Computations: An Introduction,” Interval Computations, No.3, pp. 6-62, 1994.
- [12] V. Kreinovich and O. Kosheleva, “Deep Learning (Partly) Demystified,” Proc. of the 2020 4th Int. Conf. on Intelligent Systems, Metaheuristics & Swarm Intelligence (ISMSI’20), pp. 30-35, 2020.