Why complex-valued fuzzy? Why complex values in general? A computational explanation

Kosheleva, Olga; Kreinovich, Vladik; Ngamsantivong, Thavatchai

doi:10.1109/ifsa-nafips.2013.6608577

Cited by 12 publications

(7 citation statements)

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“…Complex-valued fuzzy sets have been indeed successfully used in practical applications; see, e.g., [1,4,6,13,24] -and our analysis explains why. Let us show, on several examples, that they indeed help to reconstruct the membership values d i .…”

Section: How Good Are Complex-valued Degrees In Practice?mentioning

confidence: 96%

50 Years of fuzzy: from discrete to continuous to – Where?

Kreinovich

Nguyen

Kosheleva

et al. 2015

IFS

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Abstract. While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.Keywords: fuzzy techniques, discrete, continuous, interval-valued fuzzy, complex-valued fuzzy, computing with words, dynamical fuzzy logic, chemical kinetics, non-additive measures, symmetry Fuzzy Techniques as an Easier-to-Compute Continuous Approximation for Difficult-to-Compute Discrete Objects and ProcessesDiscrete objects and processes are ubiquitous. Many real-life objects are processes are discrete. On the macro level, there is an abrupt transition in space between physical bodies, there is an abrupt transition in time when, e.g., a glass breaks or a person changes his/her opinion. On the micro level, matter consists of discrete atoms and molecules, with abrupt transitions between different states of an atom.Continuous problems are easier to compute. While discrete objects and processes are ubiquitous in nature, from the computational viewpoint, it is often much easier to handle continuous problems. This may sound * Corresponding author. E-mail: vladik@utep.edu.counter-intuitive, since intuitively, if we restrict our search or optimization to only integer values, the problem would become easier -but it is not. For example, in the continuous case, it is relatively easy to find a solution x 1 , . . . , x n to a system of linearmany known feasible algorithms for that), the problem becomes NP-hard (computationally intractable) if we only allow discrete values of x i ; see, e.g., [9,28].Similarly, in the continuous case, it is relatively easy to find the values x 1 , . . . , x n that minimize a given quadratic function f (x 1 , . . . , x n ): it is sufficient to solve the corresponding system of linear equations ∂f ∂x i = 0. However, optimization of quadratic functions for discrete inputs, e.g., for x i ∈ {0, 1}, is NPhard [9,28]. Continuous approximations of discrete objects and processes are ubiquitous in physics. Because dealing with discrete objects and processes is often computationally complicated, physicists often approximate discrete objects with continuous ones. For example, it is not feasible to describe the changes in atmosphere by tracing all 10 23 molecules, but approximate equations that describe the atmosphere as a continuous field leads to many useful weather predictions. Similar, a solid body -in effect, a collection of atoms -is well described by a continuous density field, and an atomic nucleus -a collection of protons and neutrons -is well described by a continuous (liquid) model; see, e.g., [8].Such approximations are also useful in analyzing social phenomena. For example, in analyzing how epidemics spread, ...

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Section: How Good Are Complex-valued Degrees In Practice?mentioning

confidence: 96%

50 Years of fuzzy: from discrete to continuous to – Where?

Kreinovich

Nguyen

Kosheleva

et al. 2015

IFS

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“…Prirodna interpretacija semantike ovakvih sustava je vrednovanje s kompleksnim brojevima. Premda je ovo relativno novo područje s računalnim motivacijama (Kosheleva, Kreinovich i Ngamsantivong, 2013), naša motivacija i pristup su specifični po tome što se na ove sustave gleda kroz primarno filozofijske probleme. Međutim, bez obzira na različitu početnu motivaciju, dolazimo do istih zaključaka.…”

Section: Jezična Suprapozicija I Dimenzionalnost Jezikaunclassified

Neizrazito usvajanje jezika

Skansi

2016

Metod. ogl.

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UvodUsvajanje jezika jedno je od temeljnih filozofijskih ali i metodič-kih pitanja koje je prvi, u današnjem obliku, postavio Ludwig Wittgenstein u svojim Filozofijskim istraživanjima (Wittgenstein, 1998, par. 2). Wittgenstein je stipulirao tzv. jezične igre kao okružje usvajanja jezika. U prikazu komunikacije zidara pokazao je kako ne postoji jedan konkretan način upotrebe tzv. singularnih termina, odnosno jezičnih izraza koji se doživljavaju kao jednoznačni. Oni mogu biti referencijalni i direktno referirati na predmet ili mogu biti deskriptivni pa opisom hvatati referenciju ili mogu biti imperativni zadaci koji ne referiraju uopće na predmet nego na neki problem, pa se zaključuje na predmet za njegovo rješavanje. Ono što je Wittgenstein izvukao, kao jedinu preciznu definiciju, da je značenje neke riječi njena upotreba u jeziku, u kontekstu određene jezične igre gdje sudionici znaju pravila igre (odnosno upotrebe riječi) ili se na njih utreniravaju (ako su novi igrači igre) (Wittgenstein, 1998, par. 43).

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“…Empirically, it is sometimes helpful to use numerical degrees which are not from the interval [0, 1] -e.g., complexvalued degrees; see, e.g., [1], [2], [3], [5], [6].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Approximate nature of traditional fuzzy methodology naturally leads to complex-valued fuzzy degrees

Kosheleva

Kreinovich

2014

2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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Abstract-In the traditional fuzzy logic, the experts' degrees of confidence in their statements is described by numbers from the interval [0,1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical degrees -e.g., complex-valued degrees. While these complex-valued degrees are helpful in solving practical problems, their intuitive meaning is not clear. In this paper, we provide a possible explanation for the success of complex-valued degrees which makes their use more intuitively understandable -namely, we show that these degrees naturally appear due to the approximate nature of the traditional fuzzy methodology. I. FORMULATION OF THE PROBLEMFact: complex-valued fuzzy degrees are sometimes useful in practice. One of the main motivations for fuzzy logic was to take into account that experts are often not 100% sure about their statements, especially about statements which use imprecise ("fuzzy") words from a natural language, such as "John is young".Traditionally, in fuzzy logic, to describe the expert's degree of certainty in a statement, we use numbers from the interval [0, 1]; see, e.g., [4], [7], [8]. These numerical degrees have a clear intuitive sense. For example, we can ask the expert to mark, on a scale from 0 to 10, to what extend he or she believes that 29 years old is young. If the expert marks 8 on a scale of 0 to 10, we take 8/10 = 0.8 as the desired degree. In general, if an expert marks m on a scale from 0 to n, we take m/n as the desired degree.Empirically, it is sometimes helpful to use numerical degrees which are not from the interval [0, 1] -e.g., complexvalued degrees; see, e.g., Challenge: why complex-valued fuzzy degrees? From the pragmatic viewpoint, this is good, since the use of complex numbers helps in solving practical problems. However, from the intuitive viewpoint, the meaning of such degrees is not clear.One of the advantages of fuzzy logic (as opposed to, e.g., neural networks) is the intuitively clear nature of its operations and results. It is therefore desirable to come up with an intuitive explanation for complex-valued fuzzy degrees.Olga Kosheleva and Vladik Kreinovich are with the University of Texas at El Paso, 500 W. University, El Paso, TX 79968, USA (email: {olgak, vladik}@mae.cuhk.edu.hk).This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant N62909-12-1-7039 from the Office of Naval Research.What we do in this paper. In this paper, we deal with the above challenge by proposing a possible explanation for the use of complex-valued degrees.The structure of this paper. To provide this explanation, we first recall the motivation behind "and" and "or" operations (t-norms and t-conorms) and remind readers that these operations provide only an approximate description of the expert's c...

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Why complex-valued fuzzy? Why complex values in general? A computational explanation

Cited by 12 publications

References 11 publications

50 Years of fuzzy: from discrete to continuous to – Where?

50 Years of fuzzy: from discrete to continuous to – Where?

Neizrazito usvajanje jezika

Approximate nature of traditional fuzzy methodology naturally leads to complex-valued fuzzy degrees

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