The Birth of Fuzzy
Written by BLD.
My first encounter with fuzzy logic was in the context of geoinformation systems. The membership of a geo-object within a particular class can be determined using fuzzy logic (that is what I learned). But what is fuzzy logic? Well, before fuzzy logic, there were “fuzzy sets,” so let’s take a look first at these.
In 1965, Lotfi Asker Zadeh, a researcher in the Department of Electrical Engineering and Electronics Research Laboratory and professor emeritus at the University of California, Berkeley, published the article “Fuzzy Sets.” To get right to the definition, “[a] fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one” (Zadeh 1965). So, an unfuzzy set (or ordinary set ‒ the less cool, but more appropriate name) has no continuum and no grades of membership, but rather, it has the two possibilities of an object either being a member (1) or not (0) based on whether or not the object to be classified has a defining characteristic. Fuzzy sets, on the other hand, allow for an object to be a member of a class along a continuum of membership, without there being a crisp criterion defining this membership. Fuzzy sets are Zadeh’s contribution towards developing a new conceptual framework for mathematically modelling how humans recognize patterns and communicate and process information, when it comes to ambiguous classes.
A “fuzzy sets” example: we have a fuzzy set of numbers: 0, 1, 5, 10, 100, 500. The question is, which of these numbers is much greater than 1? Clearly, the word “much” makes this question a little less straight-forward to answer; namely, it makes it hard for us to define which number belongs to the class and which not. With an ordinary set, we’d have to define a strict criteria in order to conclude which numbers are definitively much greater than 1, and which numbers are not (assigning them a 1 or 0, respectively). With a fuzzy set however, we reach the following conclusion:
fA(0) = 0
fA(1) = 0
fA(5) = 0.01
fA(10) = 0.2
fA(100) = 0.95
fA(500) = 1
Where fA(x) is the membership function which characterizes a fuzzy set A (Zadeh 1965). With such a classification, we can say that some objects are members and some are not. But we can also say that some members have a larger degree of membership than others.
Fuzzy logic is an extension of the original fuzzy sets theorizing. As Zadeh explains:
“Humans have a remarkable capability to reason and make decisions in an environment of uncertainty, imprecision, incompleteness of information, and partiality of knowledge, truth and class membership. The principal objective of fuzzy logic is formalization/mechanization of this capability.”
Fast-forward to the 21st century: as of 2013 there were over 89,000 published articles on the topic of the theory or application of fuzzy logic. If you search fuzzy logic in a database of peer-reviewed publications, you’ll find papers that use fuzzy logic in diverse domains: environmental management, risk assessment, neural networks, social attitudes, climate change, political debates, et cetera. Mathematical fuzzy logic, based on an idea that was sparked in a researcher in 1965, has radically changed the way in which many fields address classification. Its application in science and technology is ubiquitous, affecting countless aspects of how our society functions today.
Links are posted below where you can read the original 1965 article as well as the fuzzy logic article, but be warned: the mathematical functions are abundant.
References
Singh, Harpreet, Madan M. Gupta, Thomas Meitzler, Zeng-Guang Hou, Kum Kum Garg, Ashu M. G. Solo, and Lotfi A. Zadeh. 2013. “Real-Life Applications of Fuzzy Logic.” Advances in Fuzzy Systems. https://doi.org/10.1155/2013/581879
Zadeh, Lotfi A. 1965. “Fuzzy Sets.” Information and Control. https://doi.org/10.1016/s0019-9958(65)90241-x
Zadeh, Lotfi A. 2008. “Fuzzy logic.” Scholarpedia, 3(3):1766. https://doi:10.4249/scholarpedia.1766
Gulp! I’m reading this after a rather sleepless night… will try again later when my logic is less fuzzy…
But nicely explained.
PS please feel free to delete my comments if you both of you want to engage at a higher level
PS Franck did one (?) of his master theses on fuzzy classification.