**Fuzzy logic is one of the important research areas at the Department of Computer Science at the UP Faculty of Science. This scientific field, which has a huge number of applications, is the subject of an extensive volume, Fuzzy Logic and Mathematics: A Historical Perspective, which was recently published by prestigious Oxford University Press.**

**One of its three authors is the Head of the UP Department of Computer Science, Radim Bělohlávek, who gave us an interview.**

**Prof Bělohlávek, what exactly is fuzzy logic? Could you please explain this concept?**

If we speak of logic, we speak of a discipline which has been evolving since Aristotle, if not earlier. It was developed by philosophers, since the 19th century by mathematicians, and since the mid-20th century – with the development of computers, e.g. artificial intelligence – to a great extent by computer scientists. That however was almost entirely about “classical” logic. Logic was considered the pillar of exact science – even the queen of all sciences, mathematics, is based on classical logic. Fuzzy logic refutes the basic principle of classic logic, the principle of bivalence. Bivalence, or Boolean algebra, states that every statement is true or false. For example every cardinal number is or is not even, every person is or is not tall – always just completely true or false, nothing else is allowed.

**So how does fuzzy logic get over this black and white view of logic?**

It states that expressions which relate to practical life are for the most part only partly true – only to a certain degree of truth. Let’s imagine it’s summer and it’s 26 degrees Celsius out, and I say, “It’s very hot outside.” With classical logic, this statement if true will have the value 1, or if false will have the value 0. With fuzzy logic, it might be something in between, so the statement may have the value 0.8. And if the temperature outside were higher, say 28 degrees, than it would be more true, with a value of 0.9. So while classical logic is only working with two values of truth – true and false – fuzzy logic allows more values of truth. This is what fuzzy logic is all about.

**What is the relationship between classical and fuzzy logic? What advantages does fuzzy logic have?**

Simply put, fuzzy logic is a generalisation of classical logic. Formally, mathematically, fuzzy logic came from classical logic by replacing some classical axioms with others. The world of fuzzy logic which that opens up is much richer, but also more complicated. As opposed to the world of classical logic, there are new, often even conceptually different problems. In terms of the contribution of fuzzy logic, what is most important is the informal aspect. Fuzzy logic makes it possible to work in situations where classical logic shuts down or can solve things only in a tangential or roundabout way. A simple example is the famous paradox of the heap: One grain of sand does not form a heap; when we add another grain to a number of grains which do not form a heap, it still will not make a heap. Thus according to classical logic, no amount of grains can form a heap. Fuzzy logic is able to easily solve this brain teaser, which has been confounding people since antiquity. The paradox of the heap is a famous puzzle, but it’s merely a puzzle, a game. Fuzzy logic however is capable of solving important, practical problems. It normally covers human terms like “high temperature” – which are vague, indistinct, “fuzzy” and not at all black and white nor mathematically exact as classical logic would like to have it. These practical situations led to the development of fuzzy logic.

**How and when did fuzzy logic originate?**

Its creation is considered to date back to 1965, when the American mathematician and electrical engineer Lotfi Zadeh published the article “Fuzzy Sets”. It’s very well written, simple – but in terms of content, revolutionary. I give it to every student of fuzzy logic to read. According to a recent evaluation by the magazine *Nature*, it’s among the 100 most cited papers of all time, across all disciplines, and in the area of mathematics and computer science it’s number three. Problems which he came across when working with information led Zadeh to the proposition of fuzzy logic. He considered classical logic inadequate, too delimiting. According to him, people are guided by a different logic, in which there is a place for inaccuracy. That’s why he came up with “fuzzy” logic.

Fuzzy logic represents quite a radical change, a new paradigm. There was a fairly strong reaction against it from the start by some mathematicians, engineers, even philosophers and psychologists. By the way, Prof Zadeh passed away a few days ago. He was 96, and in great condition up to the very end. It’s interesting that several proposals very similar to Zadeh’s fuzzy logic appeared already in the first half of the 20th century. Aristotle’s *Organon* even mentions fuzzy concepts and the limitations of classical logic. Several philosophers since the time of Aristotle have dealt with it, but in an exceptional way, and always on the periphery. Vague terms such as “old age”, which are at the centre of fuzzy logic, were even considered as something inappropriate, something necessary to rid ourselves of. Even Frege, one of the fathers of modern logic, at the end of the 19th century coined the uncompromising opinion that there is no place for vague expressions in logic – and by extension, in exact sciences. Despite the fact that in ordinary human communication, and that includes scientific, the vast majority of the terms are vague – unacceptable in classical logic. It was only until Zadeh came along, and he was able to show that even vague terms can be dealt with mathematically, and usefully in practise.

**Where does the importance of fuzzy logic lie? How is it used in practise?**

The first non-academic applications of fuzzy logic started in the early 1980s in the area of industrial control of smelting furnaces. The turning point however was the “fuzzy boom” in Japan. It was there in the 1980s that they began to press for various systems based on fuzzy logic. For example in the city of Sendai, train engineers in the metro were completely replaced by an automated system based on fuzzy logic. The result was not only about a ten percent savings in energy, but also more precise stopping of trains and smoother acceleration and braking. The system used “fuzzy controllers”, which could be mathematically described and operated by a control algorithm, one normally used by a person – and in that sense they simulated human intelligence. Fuzzy controllers were then put into consumer electronics in Japan. Thanks to brilliant Japanese engineers and the Japanese market, which loves innovation, fuzzy logic drivers cropped up in washing machines, vacuum cleaners, cameras, dishwashers, and a number of other products. According to the Japanese government, at that time the volume of appliances on the market with fuzzy logic made up one percent of the global computer technology market, which is an enormous share.

**Did it find application in other sectors?**

Fuzzy logic began to be used routinely in many other areas, such as in the automobile industry. These examples for me are demonstrations of a more important fact, which is that fuzzy logic represents a new paradigm in the fundamentals of exact sciences. It makes possible the creation of mathematical models close to natural human reasoning. And there lies its importance. On the other hand, this understandably does not mean that everything founded on classical logic is bad, nor does it mean that fuzzy logic provides a solution to every problem.

**Can one find fuzzy logic in Czechia?**

In shops one commonly comes across washing machines or dishwashers which use fuzzy logic, and even have the phrase written on them. Whoever buys a Volkswagen Group car with Triptronic transmission, has inside of it a fuzzy logic control unit. There are lots of other examples. In addition, in our country there are several workplaces which are using it at the top international level.

**You deal with fuzzy logic together with your colleagues in the book Fuzzy Logic and Mathematics: A Historical Perspective**.

**What specifically does it discuss?**

The book describes the development of fuzzy logic from its inception to the present. We provide both the theoretical fundamentals and the applications of fuzzy logic. But we also are trying to document the development of the infrastructure of fuzzy logic and the processes which accompanied its development. The book is fairly comprehensive, it comes to about 550 pages. We wanted it to be quite specific and detailed. For example, it contains some 1000 footnotes. When we write about something, we describe it specifically; a mere reference to the relevant literature is not enough. We also attempted to put things directly into context. When describing the ideological roots of fuzzy logic, for example, we go back to Aristotle. Some parts, especially on the theoretical foundations, to a certain extent have a textbook feel to them.

**For whom is the book intended?**

I see two basic groups of readership. The first are those who work with fuzzy logic. The second group are not experts in fuzzy logic, typically people in the natural sciences or technical fields, who want to learn something about fuzzy logic. We wrote the book so that it can be read selectively.

**How did its publication come about?**

The initiator was my long-time colleague, Prof George Klir of the State University of New York, an important figure in fuzzy logic. He approached me in 2010 and asked whether I’d be interested in the project. Later we invited the leading maths historian Joseph Dauben of the City University of New York on board. We signed a contract with Oxford University Press in about 2011. Then it took us five years to write, about two of them practically non-stop. The difficulties came with areas in which we had only a cursory overview or of which we knew nothing. For example the prehistory of fuzzy logic, from Aristotle to the inception of fuzzy logic itself. Or an overview of areas in which we do not work. I was responsible primarily for the theoretical portions. Despite the fact that I think I am capable of reading texts from various areas of mathematics and computer science quite quickly, studying the original works was quite exhausting. There were times when I wondered whether I had not taken on too big a piece of the pie. But we had a brilliant editor at the publishing house. He was accommodating to us in everything and we always felt we had his support. We also did the layout for the book ourselves, and in that we had significant help from my colleague Ed Bartl. But I think we did a good job and the book was finally published in the spring of 2017. Professor Klir however did not live to see its publication. He died eight months before it came out.

**How long have you been dedicating yourself to this subdiscipline of logic? What captivated you about it?**

I became interested in fuzzy logic in my fourth year at university. At that time, in 1992, I coincidentally left for a year to Switzerland, and practically did not do anything else but submerge myself in literature about fuzzy logic. Then for several more years I was self-taught and tried to do some thinking on it myself. I was fascinated by the feeling that fuzzy logic is a breakthrough discovery, something really huge. What is more, it fit quite well with my interests in computer science, mathematics and even philosophy to some extent.

**Fuzzy logic is also one of the main themes of research at the Department of Computer Science. What are you dealing with there?**

The focus of our research is theory. Our typical results are theorems and their proofs, new algorithms, analysis of their computational complexity and so forth. Algorithms are for the most part inspired by various problems in analyses of data processing. Several new methods for analysing data which use fuzzy logic are ones we invented and are devoting quite a lot of time to at present. Our methods are still awaiting applications, but usually that is done by somebody else. Nevertheless we are looking at concrete applications ourselves, but to a lesser degree. For example, during my time in the United States I took part in the development of a system for modelling the sedimentation in the mouth of the Mississippi River. With Prof Milan Adamus of the Olomouc University Teaching Hospital, we developed a system for automatic dosage of anaesthesia during general anaesthesia, which is controlled by fuzzy logic. Prof Adamus, who was the initiator and key figure, then used it successfully during long brain operations.

**What are the other areas of academic activities in your department?**

One large area concerns relational data, which means the data saved in contemporary database systems. We are interested especially in its analysis. Another area are basic issues in computer science which deal with computational complexity and questions of algorithmic solvability in general, i.e. the question of which problems in principle can be solved by the help of computers at all. Some practical problems, even if they can be formulated simply, are so complex that no algorithm for their solution exists. Whether this is true or not is something we have to discover and prove mathematically. Another area is the design of compilers for several programming languages.

**What is the future of fuzzy logic, in your opinion?**

Its founder, Lotfi Zadeh, said that fuzzy logic will gradually enter into the majority of areas where mathematical models are used. However, he meant fuzzy logic in the sense of what we have at hand today, after fifty years of development. To a certain extent I would agree, but I would warn against the notion that fuzzy logic is some all-powerful panacea. Furthermore, I would venture to say that what we now call fuzzy logic is only the beginning. In a certain sense, those first, easier steps were the ones which led from classical logic to Zadeh. The less easier steps, but ones which are more essential from the perspective of fundamentals, I think can only be taken from here on.