Quotient spaces in mathematics are wonderful, powerful, and useful. I personally remember (after many years) how the notion of a quotient illuminated group theory and made it come alive. Since then, I’ve encountered quotient structures in several different places and thought about them in more contexts as they provide for a way to step back from messy details to look at a larger picture. But I’ve never found a way to use quotients in software as much more than a thought experiment, though certainly this has often been useful in framing solutions.

The authors of this book offer a theoretical framework for using quotient spaces as an organizing principle for many kinds of tasks often encountered in computing, from problem representation to reasoning to search.

There are seven chapters:

(1) “Problem Representations” is an introduction to representing problem domains as hierarchies of variously sized granularities. This is the fundamental framework underlying the whole work.

(2) “Hierarchy and Multi-Granular Computing” is about extracting information on different granular levels and fuzzy quotient spaces.

(3) “Information Synthesis in Multi-Granular Computing” concerns the synthesis of structures in different domains.

(4) “Reasoning in Multi-Granular Worlds” addresses the relation between uncertainty and granularity, and fuzzy reasoning based on quotient space structures.

(5) “Automatic Spatial Planning” is about methods of motion planning and dimension reduction.

(6) “Statistical Heuristic Search” discusses heuristic search methods and a specific statistical search algorithm with comparisons to A*.

(7) “The Expansion of Quotient Space Theory” provides a discussion of quotient space theory and systems analysis, fractal geometry, and other domains.

There are two addenda, one on the basics of point-set topology, which is used extensively in the book, and one on statistical inference.

The book is aimed primarily at graduate students (and academicians) with strong mathematical maturity and an interest in mathematical modeling in the fields around artificial intelligence (AI). It is unlikely to be of much use or interest to most practitioners.

This is certainly interesting material and there are some things to like in here, but more than a few sections are complicated by typographical errors, inconsistent or even strange layouts of the mathematics, and other minor but complicating problems. The book as a whole could profit from better editing (this seems to be a common enough problem, but responsible publishers should take it seriously). These and other problems make things harder to follow than they should be. There are good examples, but not always where the examples are needed, and a good textual summary could be added to many sections to good effect.

There is also a bit of the “these techniques can solve everything” effect, where the authors apply it to domains where the fit may not be that good (for instance, the section on fractal geometry).