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Reviewer: Jonathan Samuel Golan

Semirings (essentially, "rings without subtraction") are algebraic structures that have found many important applications in computer science, both on the theoretical and applied levels. Especially useful are complete lattice-ordered semirings, which also have the structure of a complete lattice for which the semiring addition and the lattice max operation coincide. One of the major applications of such semirings is shown here, in the area of constraint satisfaction problems (CSPs). The extension of CSPs to the more general notion of soft constraint satisfaction problems (SCSPs), by using semirings, is the subject of this monograph, which constitutes an extended and revised version of Bistarelli's Ph.D. thesis, written under the direction of two acknowledged masters in this area: Ugo Montanari, of the University of Pisa, and Francesca Rossi, of the University of Padova. The definitions of constraint satisfaction problems and soft constraint satisfaction problems are highly technical, and beyond the scope of a short review. Very roughly, however, we note that the distinction between them is that SCSPs allow us to cope with incomplete knowledge of real problems, to solve over-constrained problems, or to represent cost-optimization problems. This, therefore, provides a framework that includes, not only (crisp) CSPs, but also fuzzy, probabilistic, and weighted CSPs. After a brief introduction to constraint satisfaction problems and constraint logical programming (CLP), the book covers the following major topics: SCSPs and their solutions, SCSP abstraction, higher-order semiring-based constraints, soft CLP and its application to generalized shortest path problems, soft concurrent constraint programming, interchangeability in SCSPs, and SCSPs for modeling attacks on security protocols. It ends with short conclusions, and suggested directions for future research. There is no index, which is a definite drawback. Another unfortunate point is the author's lack of familiarity with the (very extensive) algebraic literature on semirings, which would have considerably simplified some of his definitions and proofs. It would be very useful to have available a deeper mathematical treatment of some of the topics covered here, though this (I definitely wish to emphasize) does not detract from the importance of this volume. The author begins by quoting Eugene Freuder: "Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it." Soft constraint programming takes us closer still, though at the cost of demanding considerably more mathematical sophistication. It is a cost that computer scientists will have to learn to pay.