Fuzzy - Probability Bridge I: Kolmogorov's Fields of Fuzzy Sets

Pavel Provinský

Abstract


Systems can be described in many ways. A very successful way is a probabilistic description. Another successful way is a description by fuzzy sets. Both of these approaches have some advantages and disadvantages. Our desire is to obtain a description of systems which combine the advantages of the both approaches. The first step towards achieving this goal and the topic of this article is a characterization of such systems of fuzzy sets which satisfy Kolmogorov's axioms.

Keywords


fuzzy sets, probability, Kolmogorov's axioms, Kolmogorov's fields of fuzzy sets

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References


BELLMAN R, KALABA R, ZADEH L. Abstraction and pattern classication.

Journal of Mathematical Analysis and Applications. 1966, vol. 13(issue 1), pp. 1{7,

doi: 10.1016/0022-247X(66)90071-0. ISSN 0022247x.

BELLMAN R. E., ZADEH L. A. Decision-Making in a Fuzzy Environment. Man-

agement Science. 1970, vol. 17(issue 4), pp. B{141{B{164, doi: 10.1287/mnsc.17.

B141. ISSN 0025-1909.

BOUCHON-MEUNIER B., DUBOIS D., GODO L., PRADE H. Fuzzy Sets and

Possibility Theory in Approximate and Plausible Reasoning. In: Fuzzy sets in ap-

proximate reasoning and information systems. Boston, MA: Springer, 1999, pp. 15{

ISBN 9781461552437.

BURKILL J. C. The Lebesgue integral. London: Cambridge University Press, 1963.

DUBOIS D., PRADE H. Fuzzy sets and probability: misunderstandings, bridges

and gaps. In: [Proceedings 1993] Second IEEE International Conference on Fuzzy

Systems, San Francisco, CA: IEEE, 1993, pp. 1059{1068, doi: 10.1109/FUZZY.

327367. ISBN 0-7803-0614-7.

DUBOIS D., PRADE H. Possibility Theory. Wiley Encyclopedia of Electrical and

Electronics Engineering. 2001-08-21, doi: 10.1002/047134608X.W3502.

DUBOIS D., PRADE H. M. Fundamentals of fuzzy sets. Boston: Kluwer Academic,

ISBN 07-923-7732-X.

DUBOIS D., PRADE H. M. Fuzzy sets and systems: theory and applications. New

York: Academic Press, 1980. ISBN 01-222-2750-6.

Fuzzy logic. Wikipedia: the free encyclopedia, San Francisco (CA): Wikimedia

Foundation, 2015 [wieved 2015-06-09]. Available also from: http://en.wikipedia.

org/wiki/Fuzzy_logic.

GAINES B. R. Fuzzy and probability uncertainty logics. Information and Con-

trol. 1978, vol. 38(issue 2), pp. 154{169, doi: 10.1016/S0019-9958(78)90165-1.

ISSN 00199958.

GILES R. Lukasiewicz logic and fuzzy set theory. International Journal of Man-

Machine Studies. 1976, vol. 8(issue 3), pp. 313{327, doi: 10.1016/S0020-7373(76.

-X. ISSN 00207373.

HERSH H. M., CARAMAZZA A. A fuzzy set approach to modiers and vague-

ness in natural language. Journal of Experimental Psychology: General. 1976, vol.

(issue 3), pp. 254{276, doi: 10.1037/0096-3445.105.3.254. ISSN 0096-3445.

KOLMOGOROV A. N. Grundbegrie der Wahrscheinlichkeitsrechnung. Berlin: J.

Springer, 1933.

KOPA

CEK J. Matematicka analyza nejen pro fyziky (III). 3., upr. vyd. Praha:

Matfyzpress, 2007. ISBN 978-80-7378-020-3.

KOSKO B. Fuzziness vs. Probability. International Journal of General Systems.

, vol. 17(2-3), pp. 211{240, doi: 10.1080/03081079008935108. ISSN 0308-1079.

MAISTROV L. E. Probability theory: a historical sketch. New York: Academic

Press, 1974. ISBN 01-246-5750-8.

MENGER K. Statistical Metrics. Proceedings of the National Academy of Sciences

of the United States of America. 1942, 28(12), pp. 535{537, doi: 10.1073/pnas.28.

535.

MONTES I., HERNANDEZ J., MARTINETTI D., MONTES S. Characterization

of continuous t-norms compatible with Zadeh's probability of fuzzy events. Fuzzy

Sets and Systems. 2013, vol. 228(October, 2013), pp. 29{43, doi: 10.1016/j.fss.

11.020. ISSN 01650114.

OKUDA T., TANAKA H., ASAI K. A formulation of fuzzy decision problems with

fuzzy information using probability measures of fuzzy events. Information and Con-

trol. 1978, vol. 38(issue 2), pp. 135{147, doi: 10.1016/S0019-9958(78)90151-1.

ISSN 00199958.

RESCHER N. Many-valued logic. New York: McGraw-Hill, 1969. ISBN 0070518939.

RENYI A. On a new axiomatic theory of probability. Acta Mathematica Academiae

Scientiarum Hungaricae. 1955, vol. 6(3-4), pp. 285{335, doi: 10.1007/BF02024393.

ISSN 0001-5954.

STIGLER S. M. The history of statistics: the measurement of uncertainty before

Cambridge, Mass.: Belknap Press of Harvard University Press, 1986. ISBN 06-

-0340-1.

STEPAN J. Teorie pravdepodobnosti: Matematicke zaklady. Praha: Academia, 1987.

VESELY

J. Zaklady matematicke analyzy: Prvn dl. Praha: Matfyzpress, 2004.

ISBN 978-80-7378-063-02.

ZADEH L. A. Fuzzy probabilities. Information Processing. 1984, vol. 20(issue 3),

pp. 363{372, doi: 10.1016/0306-4573(84)90067-0. ISSN 03064573.

ZADEH L. A. Fuzzy sets. Information and Control. 1965, vol. 8(issue 3), pp. 338{

, doi: 10.1016/S0019-9958(65)90241-X. ISSN 00199958.

ZADEH L. A. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and

Systems. 1978, vol. 1(issue 1), pp. 3{28, doi: 10.1016/0165- 0114(78)90029- 5.

ISSN 01650114.

ZADEH L. A. Probability Measures of Fuzzy Events. Journal of Mathematical Anal-

ysis and Applications. 1968, vol. 23(issue 2), pp. 421{427, doi: 10 . 1016 / 0022 .

X(68)90078-4. ISSN 0022247x.




DOI: http://dx.doi.org/10.14311/NNW.1901.%25x

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