Min and Max are the Only Continuous $\&$- and $\vee$-Operations for Finite Logics
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Vladik Kreinovich
1
(Department of Computer Science, University of Texas at El Paso, El Paso, USA.)
کلید واژه: Max, Finite logic, Continuous logical operation, “And”-operation, “Or”-operation, min,
چکیده مقاله :
Experts usually express their degrees of belief in their statements by the words of a natural language (like “maybe”, “perhaps”, etc.). If an expert system contains the degrees of beliefs t(A) and t(B) that correspond to the statements A and B, and a user asks this expert system whether “A & B” is true, then it is necessary to come up with a reasonable estimate for the degree of belief of A & B. The operation that processes t(A) and t(B) into such an estimate t(A & B) is called an &-operation. Many different &-operations have been proposed. Which of them to choose? This can be (in principle) done by interviewing experts and eliciting a &-operation from them, but such a process is very time-consuming and therefore, not always possible. So, usually, to choose a &-operation, we extend the finite set of actually possible degrees of belief to an infinite set (e.g., to an interval [0, 1]), define an operation there, and then restrict this operation to the finite set. In this paper, we consider only this original finite set. We show that a reasonable assumption that an &-operation is continuous (i.e., that gradual change in t(A) and t(B) must lead to a gradual change in t(A & B)), uniquely determines min as an &-operation. Likewise, max is the only continuous ∨-operation. These results are in good accordance with the experimental analysis of “and” and “or” in human beliefs.
Experts usually express their degrees of belief in their statements by the words of a natural language (like “maybe”, “perhaps”, etc.). If an expert system contains the degrees of beliefs t(A) and t(B) that correspond to the statements A and B, and a user asks this expert system whether “A & B” is true, then it is necessary to come up with a reasonable estimate for the degree of belief of A & B. The operation that processes t(A) and t(B) into such an estimate t(A & B) is called an &-operation. Many different &-operations have been proposed. Which of them to choose? This can be (in principle) done by interviewing experts and eliciting a &-operation from them, but such a process is very time-consuming and therefore, not always possible. So, usually, to choose a &-operation, we extend the finite set of actually possible degrees of belief to an infinite set (e.g., to an interval [0, 1]), define an operation there, and then restrict this operation to the finite set. In this paper, we consider only this original finite set. We show that a reasonable assumption that an &-operation is continuous (i.e., that gradual change in t(A) and t(B) must lead to a gradual change in t(A & B)), uniquely determines min as an &-operation. Likewise, max is the only continuous ∨-operation. These results are in good accordance with the experimental analysis of “and” and “or” in human beliefs.
[1] J. Agusti' et al., Structured local fuzzy logics in MILORD, In: L. Zadeh, J. Kacrpzyk (eds.), Fuzzy Logic for the Management of Uncertainty, Wiley, New York, (1992).
[2] R. Belohlavek, J. W. Dauben and G. J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective, Oxford University Press, New York, (2017).
[3] B. G. Buchanan and E. H. Shortli e, Rule-Based Expert Systems, Addison-Wesley, Reading, Massachusetts, Menlo Park, California, (1984).
[4] N. Bourbaki, General Topology, Addison-Wesley, Reading, Massachusetts, (1967).
[5] H. M. Hersch and A. Camarazza, A fuzzy-set approach to modi ers and vagueness in natural languages, Journal of Experimental Psychology: General, 105 (1976), 254-276.
[6] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, New Jersey, (1995).
[7] V. Kreinovich et al., What non-linearity to choose? Mathematical foundations of fuzzy control, Proceedings of the 1992 International Fuzzy Systems and Intelligent Control Conference, Louisville, Kentucky, (1992) 349-412.
[8] J. M. Mendel, Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions, Springer, Cham, Switzerland, (2017).
[9] H. T. Nguyen, C. L. Walker and E. A. Walker, A First Course in Fuzzy Logic, Chapman and Hall/CRC, Boca Raton, Florida, (2019).
[10] V. Novak, I. Per lieva and J. Mockor, Mathematical Principles of Fuzzy Logic, Kluwer, Boston, Dordrecht, (1999).
[11] G. C. Oden, Integration of fuzzy logical information, Journal of Experimental Psychology: Human Perception Performance, 3(4) (1977), 565-575.
[12] J. Puyol-Gruart, L. Godo and C. Sierra, A specialisation calculus to improve expert systems communication, Research Report IIIA 92/8, Institut d'Investigacio en Intelligenicia Arti cial, Spain, (1992).
[13] G. Shafer and J. Pearl (eds.), Readings in Uncertain Reasoning, M. Kaufmann, San Mateo, California, (1990).
[14] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[15] H. J. Zimmerman, Results of empirical studies in fuzzy set theory, In: G. J. Klir (ed.), Applied General Systems Research, Plenum, New York, (1978), 303-312.
