Fuzzy Discrete Structures

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex­ cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets.

The subject of the book could be described as fuzzy discrete structures. The book deals with fuzzy logic, fuzzy switching functions, fuzzy decision trees, fuzzy networks, fuzzy petri nets, fuzzy path problems, fuzzy automata, and fuzzy languages. The notion of a fuzzy decision tree and a branch-bound-backtrack algorithm is presented. In the Appendix, this algorithm is described in C++. Network flow problems with various capacities are examined. Fuzzy shortest path techniques are presented whereby there is an actual path which corresponds to the fuzzy shortest path. Recent results on fuzzy automata and fuzzy languages are also presented.