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Classification of possibilistic uncertainty and information functions. (English) Zbl 0637.94027
The U-uncertainty introduced by Higashi and Klir is proved to be of the form $V(p_ 1,...,p_ n)=z(p_ 1)\log n+(z(p_ 2)-z(p_ 1))\log (n-1)+(z(p_ 3)-z(p_ 2))\log (n-2)+...$ where $$p_ 1\leq p_ 2\leq..$$. is an ordered possibility distribution and $$z: [0,1]\to [0,1]$$, $$z(0)=0$$, $$z(1)=1$$ and z monotonic. It is proved that any other form of U-uncertainty is a particular form of the above and some properties are derived. A related paper is by G. J. Klir and M. Mariano [ibid. 24, 197-219 (1987; Zbl 0632.94039)]. Here, the authors prove that the uniqueness is related to the assumption of linearity of V, i.e. $$V(a+b,1)=V(a,1)+V(b,1).$$
Reviewer: H.Teodorescu

##### MSC:
 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory) 94A17 Measures of information, entropy
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##### References:
 [1] Dubois, D.; Prade, H., Properties of measures of information in evidence and possibility theories, Fuzzy sets and systems, 24, 161-182, (1987), (this issue) · Zbl 0633.94009 [2] Guiasu, S., Information theory with applications, (1977), McGraw-Hill New York · Zbl 0379.94027 [3] Higashi, M.; Klir, G., Measures of uncertainty and information based on possibility distributions, Internat. J. gen. systems, 9, 43-58, (1982) · Zbl 0497.94008 [4] Klir, G., Where do we stand on measures of uncertainty, ambiguity, fuzziness and the like?, Fuzzy sets and systems, 24, 141-160, (1987), (this issue) · Zbl 0633.94026 [5] Klir, G.; Mariano, M., On the uniqueness of possibilistic measure of uncertainty, Fuzzy sets and systems, 24, 197-219, (1987), (this issue) · Zbl 0632.94039 [6] Ramer, A., Uniqueness of information measure in the theory of evidence, Fuzzy sets and systems, 24, 183-196, (1987), (this issue) · Zbl 0638.94027 [7] A. Ramer, Interrelation of axioms for uncertainty measures, Fuzzy Sets and Systems (forthcoming). [8] Rényi, G., The theory of probability, (1967), North-Holland Amsterdam [9] Shannon, C.E.; Shannon, C.E., A mathematical theory of communication, Bell systems technical J., Bell systems technical J., 27, 623-656, (1948) · Zbl 1154.94303 [10] Zadeh, L., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
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