Springer Publishing Company, Incorporated, 2009.

I.2.4 KR formalisms and methods - modal logic, relation systems, representations

I.2.3 Nonmonotonic reasoning and belief revision

F.4.0 Mathematical logic - general

This book covers a broad range of ideas related to nonmonotonic reasoning. This review summarizes some of its nice ideas--or, as the authors say, its “logical tools.”

Reactive diagrams are used to model complex arguments in Section 1.3, essentially smooth preferential structures in chapter 6, and Touretzky's inheritance systems [1] in chapter 9. It would be fair to call this whole book an exposition on reactive diagrams in various application areas. An earlier paper by the authors [2] illustrates this idea.

Another idea presented is how sets can be logically classified as big or small. This is important in nonmonotonic consequence, where one talks of minimal models. Unfortunately, this is clubbed into the densely mathematical chapter 3, which could use a lot more motivation and explanation. One of the authors' earlier papers [3] discusses other logical approaches to the same problem, which is lacking here.

The book's core—chapters 4 to 6—examines this representation of nonmonotonic consequence using preference relations. Readers should be aware of Kraus, Lehmann, and Magidor [4], who were the first to deal with these representations, in order to follow this book. Some chapters include the authors' own overview of an area, discursive essays on how available techniques address specific problems, and then how their own techniques do so. The authors often reference Schlechta's earlier book [5], from which many ideas are derived or are improved upon. (That being said, I haven't read it [5].)

The other chapters deal primarily with issues of representation and completeness. The authors show how some technical problems in theory revision (chapter 8) and deontic logic (chapter 7) can be solved using these methods. When all of this is taken into account, it can be said that there is substantial material for researchers.

There is little of interest for novices in the field. The little there is, is marred by the authors' writing style—for example, they begin a chapter with a large table that typically runs across many pages, expresses many different kinds of syntactic and semantic conditions, and concludes with dense proofs of all the correspondences. To a certain extent, these large tables are exactly what the authors want to convey; they can also be found in the authors' earlier papers. Even though agents are mentioned in the title and appear in some examples, the book is primarily concerned with semantic structures.

The final impression I was left with is that of a somewhat esoteric collection of ideas applied to diverse areas within the authors' ken. Any future edition will require a great deal of editing and rewriting, in order to meet the authors' goal of creating a repository of the correspondence theory of nonmonotonic consequence with binary relations.

[1] Touretzky, D.S.
*The mathematics of inheritance systems*.
Morgan Kaufmann Publishers, Los Altos, CA, 1986.

[2] Gabbay, D.M.; Schlechta, K.
Reactive preferential structures and nonmonotonic consequence.
*Review of Symbolic Logic* **2**,2 (2009), 414–450.

[3] Gabbay, D.M.; Schlechta, K.
Size and logic.
*Review of Symbolic Logic* **2**,2 (2009), 396–413.

[4] Kraus, S.; Lehmann, D.; Magidor, M.
Nonmonotonic reasoning, preferential models and cumulative logics.
*Artificial Intelligence* **44**, (1990), 167–207.

[5] Schlechta, K.
*Coherent systems*. Elsevier, Boston, MA, 2004.