## Possibility theory. An approach to computerized processing of uncertainty. With the collaboration of Henri Farreny, Roger Martin- Clouaire and Claudette Testemale. Transl. from the French by E.F. Harding in assoc. with the 1st edition.(English)Zbl 0703.68004

New York etc.: Plenum Press. xvi, 263 p. \$ 39.50 (1988).
See also the review of the French original (1985; Zbl 0674.68059). The book presents an approach to uncertainty based on fuzzy set theory. The basic concept is the measure of possibility, defined as follows. Let $$\Omega$$ be the universe, also called sure event. The emtpy set $$\emptyset$$ is the impossible event. An expert assigns a real number g(A)$$\in [0,1]$$ to any subset A of $$\Omega$$. The number g(A) reflects the confidence in the occurrence of event A. First, $$g(\emptyset)=0$$ and $$g(\Omega)=1$$. Additionally, it is assumed that g is monotonic with respect to inclusion, i.e., for all A,B$$\subseteq \Omega$$ $A\subseteq B\quad implies\quad g(A)\leq g(B).$ Moreover, g should satisfy $\lim_{n\to \infty}g(A_ n)=g(\lim_{n\to \infty}A_ n)$ for $$A_ 0\subseteq A_ 1\subseteq...\subseteq A_ n\subseteq..$$. or $$A_ 0\supseteq A_ 1\supseteq...\supseteq A_ n\supseteq...$$, where $$A_ 0,A_ 1,...,A_ n$$, are subsets of $$\Omega$$. Thus defined function g: 2$${}^{\Omega}\to [0,1]$$ is called a confidence measure. Hence, for all A,B$$\subseteq \Omega$$, $g(A\cup B)\geq \max (g(A),g(B)),$ and $g(A\cap B)\leq \min (g(A),g(B)).$ As a limiting case of confidence measures, a function $$\Pi$$ : 2$${}^{\Omega}\to [0,1]$$ such that for all A,B$$\subseteq \Omega\Pi (A\cup B)=\max (\Pi (A),\Pi (B))$$is called a possibility measure by L. Zadeh. For finite universe $$\Omega$$, a possibility measure $$\Pi$$ may be defined by its values on the single members of $$\Omega$$, i.e., for all $$A\subseteq \Omega$$, $\Pi (A)=\sup \{\Pi (\{\omega \})| \omega \in A\}.$ A function $$\Pi$$ : $$\Omega\to [0,1\}$$, defined by $$\Pi (\omega)=\Pi (\{\omega \})$$ is called a possibility distribution. Another limiting case of confidence measures is a function N: 2$${}^{\Omega}\to [0,1]$$ such that for all A,B$$\subseteq \Omega$$ $$N(A\cap B)=\min (N(A),N(B)).$$Obviously, relations with Dempster- Shafer theory are clear.
Function N is called a necessity measure. The confidence measure P such that for all A,B$$\subseteq \Omega$$ with $$A\cap B=\emptyset$$ $$P(A\cup B)=P(A)+P(B)$$is a probabilistic measure.
Then the book defines fuzzy sets with basic properties. The calculus of fuzzy quantities, represented by possibility distribution is presented. Many applications from operational research, artificial intelligence, and data base management are described.
The book is well written and is the best book on fuzzy sets I ever seen. Since a book review should contain some criticism, let me remark that the weakest point of the book are programs given at the end of some chapters. First of all, the authors do not need to convince the reader that what they are presenting is programmable - it is obvious. It would be much better to quote algorithms written in pseudo-Pascal, for example. Furthermore, if they decided to use programs, they should use the same programming language instead of two (Basic and Lisp). I skip criticism of the choice of both of these languages. Besides, some of these programs are not readable because of technical problems of printing.
Reviewer: J.Grzymala-Busse

### MSC:

 68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science 68T01 General topics in artificial intelligence 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory) 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc.

Zbl 0674.68059