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Why Complex-Valued Fuzzy? Why Complex Values in General? A Computational Explanation

Olga Kosheleva and Vladik Kreinovich

University of Texas at El Paso, El Paso, TX 79968, USA

• lgak@utep.edu, vladik@utep.edu

Thavatchai Ngamsantivong

King Mongkut’s Univ. of Technology North Bangkok, Thailand tvc@kmutnb.ac.th

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1. Fuzzy Logic: Reminder

• In classical (2-valued) logic, every statement is either

true or false.

• In the computer, “true” is usually represented as 1,

and “false” as 0.

• The resulting 2-valued logic {0, 1} is well equipped to

represent: – situations when we are completely confident that a given statement is true, and – situations when we are completely confident that a given statement is false.

• 2-valued logic cannot adequately represent situations

when we only have some degree of confidence.

• To describe such situations, L. Zadeh invented fuzzy

logic.

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2. From the Mathematical Viewpoint, Complex- Valued Fuzzy Sets Are Natural

• In the original version of fuzzy logic, the set of possible

truth values is an interval [0, 1].

• [0, 1]-based logic assumes that we can describe an ex-

pert’s degree of confidence by an exact number.

• Real-life experts often cannot meaningfully distinguish

between nearby numbers.

• We thus need to generalize the set of all real numbers

from the interval [0, 1].

• In mathematics, one of the natural generalizations of

real numbers are complex numbers.

• Not surprisingly, complex-valued generalizations of fuzzy

sets have been proposed and successfully used.

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3. From the Intuitive Viewpoint, Complex-Valued Fuzzy Sets Remain a Puzzle

• Fuzzy sets are not just a mathematical theory.
• They are an intuitively clear way to describe how we

humans deal with uncertainty.

• The original idea is very natural: describe possible ex-

pert’s degrees of confidence ranging all the way: – from “absolutely false” (0) – to “absolutely true” (1).

• In contrast, the idea of using complex numbers is not

clear at all.

• Why complex-valued fuzzy numbers are useful is thus

still largely a mystery.

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4. Fuzzy Data Processing: Computational Chal- lenges

• The most widely used fuzzy-related techniques are tech-

niques of fuzzy control and fuzzy modeling.

• In these techniques, we start with rules, then:

– we use t-norm and t-conorm to combine member- ship functions corresponding to these rules, and – we apply defuzzification to the resulting member- ship function to get a control value u.

• For defuzzification, we can choose u for which µ(u) →

max or take u =

• u · µ(u) du
• µ(u) du .
• Combining membership functions can be done in par-

allel and thus, really fast.

• Integration and global optimization are NP-hard, so

they are the main challenges in fuzzy data processing.

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5. When Are Integration and Maximization Prob- lems Feasible

• NP-hard means that we cannot solve all particular

cases of the integration and maximization problem.

• A natural question is: for what classes of functions can

we feasibly solve these problems?

• These problems are known to be feasible for triangular

functions and not feasible for general ones.

• Where is the “threshold” separating feasible from non-

feasible cases?

• Such a threshold have discovered in a recent paper by

Kawamura et al.: – both integration and optimization problems are NP- hard for smooth (differentiable) functions; – these two problems become feasible (= polynomial- time) for analytical functions.

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6. Relation to Complex Numbers

• For functions of a real variable, analytical functions are

power series: f(x) = a0+a1·(x−x0)+a2·(x−x0)2+. . .

• All such functions (exp(x), sin(x), etc.) can be natu-

rally extended to complex numbers.

• In the complex domain, analytical functions can be

defined as functions f(z) differentiable w.r.t. z.

• So, a function f(x) is analytical if it can be extended

to a smooth functions of a complex variable.

• Thus, the above threshold result can be formulated as

follows: – for general smooth functions of a real variable, both integration and optimization problems are NP-hard; – for f-s extendable to smooth functions of a complex variable, integration and optimization are feasible.

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7. Resulting Computational Explanation of Why Complex-Valued Fuzzy

• We thus arrive at the computational explanation of

why complex-valued fuzzy sets are practically useful:

• The most computationally intense operations involved

in fuzzy techniques are: – integration and – optimization.

• These operations are computationally feasible only when:

– the corresponding functions – can be extended to smooth functions of complex variables.

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8. Other Reasons Why Complex Numbers Are Useful: They Are Easier to Use

• Many real-life phenomena – from planets orbiting the

Sun to waves – are periodic in time.

• A periodic signal x(t) can be expanded into Fourier

series: x(t) = x0 +

∞

• n=1

(sn · sin(n · ω · t) + cn · cos(n · ω · t)).

• This same linear combination can be equivalently rep-

resented in complex-valued form as x(t) = Re ∞

• j=0

zn · exp(i · n · ω · t)

• .
• Here, e.g., shifting t → t + t0 is easy for complex num-

bers, somewhat more complex for sin(x) and cos(x).

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9. Why Complex Numbers in General?

• Complex numbers are ubiquitous in engineering and in

physics (quantum physics, control); why?

• Physical theories are described in terms of differential

equations.

• To solve the corr. physical problems, we need to solve

(integrate) the corresponding differential equations.

• Alternatively, physical theories can be describing in

terms of optimization principles.

• Control also often means optimization.
• Integration and optimization become feasible only if

the functions are extendable to complex domain.

• Here is our answer to the “why?” question: the use of

complex numbers guarantees that we have feasibility.

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10. Why Complex Numbers: An Answer That We Gave So Far

• We explained the effectiveness of complex numbers by

citing a result about integration and optimization: – in the general case, the problem is NP-hard, so no feasible algorithm is possible; – on the other, when functions can be extended to the complex domain, feasible algorithms are possible.

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11. Complex Numbers Also Make Feasible Com- putations Faster

• A typical example is the use of Fast Fourier Transform

(FFT), an efficient algorithm that: – transforming a real function x(t) into – its complex-valued Fourier transform ˆ X(ω).

• The use of FFT leads to most efficient algorithms:

– for multiplying polynomials, – for multiplying large integers, and – for solving linear differential equations with con- stant coefficients.

• FFT is also very useful for arithmetic operations with

fuzzy numbers.

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12. Acknowledgment This work was supported in part:

• by the National Science Foundation grants HRD-0734825,

HRD-1242122, and DUE-0926721,

• by Grants 1 T36 GM078000-01 and 1R43TR000173-01

from the National Institutes of Health, and

• by a grant on F-transforms from the Office of Naval

Research. The authors are greatly thankful:

• to Scott Dick for his encouragement,
• to Martine Ziegler for valuable references and discus-

sions, and

• to the anonymous referees for important suggestions.

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13. Main References

• A. Kawamura, N. Th. M¨

uller, C. R¨

• snick, and M. Ziegler,

“Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime”, The Computing Research Repository (CoRR), November 2012,

• No. 1211, paper abs/1211.4974.
• O. Kosheleva, S. D. Cabrera, G. A. Gibson, and M. Koshelev,

“Fast Implementations of Fuzzy Arithmetic Operations Using Fast Fourier Transform (FFT)”, Fuzzy Sets and Systems, 1997, Vol. 91, No. 2, pp. 269–277.