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Reviewer: Paul Cull

Automata and language theory is one of the oldest parts of computer science. One might expect this field to have developed into a small, coherent set of techniques that can be used to derive all the results. Instead, this field contains a wide variety of techniques, with each result requiring its own technique. A look through the standard textbook, Hopcroft and Ullman [1], confirms the hodgepodge of techniques that are needed to present even the best known results. Eilenberg [2], in his series of books, takes the point of view that automata and language theory form a subfield of algebra. While his presentation is quite convincing that finite state machine theory can be neatly handled by algebraic techniques, he seems to need more ad hoc techniques when dealing with other types of automata and languages. Ginsburg and Greibach [3] have created abstract families of languages and have common proofs for some results about various classes of languages, but their abstraction scheme omits many quite useful results about languages. Salomaa has long been active in language theory [4,5,6] and he has championed the application of formal power series to languages. In the book under review, Kuich and Salomaa present us with the latest results on formal power series and their application to both automata and languages. The central construction is the formal power series. In the leading special case, &Sgr; is a finite alphabet and the symbols of &Sgr; are considered to be noncommuting variables; so, for each string over &Sgr;, there is a corresponding product of noncommuting variables. Any language over &Sgr; can be represented as an infinite sum of these products of noncommuting variables, in which a product has a coefficient of 1 when the corresponding string is in the language and a coefficient of 0 when the corresponding string is not in the language. For example, the language consisting of all strings of a's would be represented by the series &egr; + a + aa + aaa + aaaa + . . . . We would, of course, write this as a* because we are interpreting the coefficients of the series as belonging to the Boolean semiring. The idea of the semiring—in fact, the closed semiring—is familiar from the transitive closure algorithm. Since the Boolean semiring is closed, the * of each element exits. The authors want to cast a wider net, so they assume that the coefficients of their power series come from a semiring that is not necessarily closed. Why go to all this trouble when the Boolean semiring suffices to represent all languages__?__ The authors are trying to give a uniform description to various classes of languages. We are familiar with the fact that all regular languages can be represented in the closed semiring of regular expressions. Thus we might be able to express more general classes by formal series whose coefficients are regular expressions. If one is interested in ambiguity, series over IN rather than IB can be used, so that the coefficient of a string is the number of distinct derivations of a string using a particular grammar. By studying a restricted class of mappings between series over IN, it might be possible to determine if the language is inherently ambiguous. If one were interested in probabilistic automata or grammars, one could use series over [0,1] in which the coefficient of a string would represent the probability that the string is accepted or generated. Concretely this book is divided into three chapters containing a total of 16 sections. Chapter I, Linear Algebra, develops the mathematics of semirings and various objects over semirings. These objects include series, matrices, linear systems, and morphisms. A key observation in this chapter is that series and matrices over semirings naturally form their own semirings. To present all this material in 100 pages, the authors have to use a rather dense mathematical style. They suggest that the reader take a look at Chapter II to see the motivation for the mathematics of Chapter I. Chapter II, Automata, consisting of about 200 pages, is the core of this book. Starting from the idea of a finite automaton that is represented by a finite transition matrix whose entries are sets of symbols such that any symbol in the set in position ( i, j) will cause a transition from state i to state j, the authors generalize by allowing an infinite number of states, by replacing the semiring of the power set of the set of strings over an alphabet by an arbitrary semiring, and by allowing entries in the transition matrix to be power series. They then show that, with suitable replacements, the usual characterization theorems for finite automata are still valid. In particular, the sets accepted by the generalized automata can be represented by regular expressions with the base elements for the expressions drawn from the appropriate semiring. Unfortunately, they show that, unlike the finite case, various properties of these expressions are undecidable when the (semi)ring is the integers. They then characterize rational transductions that generalize generalized sequential machine mappings and are the mappings computed by their generalized automata (transducers). Since the generalized automata are too powerful (they can accept noncomputable languages), the authors introduce generalized pushdown automata that have a finite specification and will be used in Chapter III to discuss context-free languages. Next, the authors introduce AFP, abstract families of power series, which generalized the notion of abstract families of languages. They then consider the AFP properties of their pushdown automata and one-counter automata. The last chapter, Algebraic Systems, consisting of some 60 pages, deals with power series representing context-free languages. The chapter is called Algebraic Systems because the series are solutions to algebraic (polynomial) systems of equations, just as regular expressions are solutions of linear systems of equations. Using elimination and factoring of polynomials, the authors show that these systems can be effectively reduced to a single polynomial. This leads to a number of decidability results for power series and thence to decidability results about context-free grammars. In particular, if G is an unambiguous context-free grammar and R is a regular language, then it is decidable whether or not L(G) = R. The authors stress that every known proof of this result uses power series. Various people, for example Ehrig [7], have noticed that finite automata can be generalized by allowing the automata to work on more general structures than strings. These constructions allow the unification of various kinds of automata, but the constructions involve the use of categories, and the introduction of such abstract structures seems a heavy price to pay for some unification. The usual question raised is: Does the introduction give you any new results or do we simply see old results in a setting that, while unifying, tends to obscure the intuitive reasons for the results__?__ In a reasonable sense, the unification presented by Kuich and Salomaa is more practical because it generalizes by going only from the Boolean semiring to more general semirings without having to introduce the fuller abstraction of categories. Further, the authors present new results that they can prove with their generalized descriptions. These new results are difficult to prove in the usual language framework. This book displays the high standards for typesetting that one expects from Springer. Some of my colleagues talk confidently about sitting down at their terminals and producing camera-ready copy, but those of us who make heavy use of mathematical symbolism appreciate that there is, as yet, no (readily available) computer system that can produce such a beautiful book. There are exercises for each section suggesting that this book could be used as a text. The prerequisites for such a course would be knowledge of automata theory equivalent to the book by Hopcroft and Ullman [1] and that elusive quality—mathematical maturity. I would consider this book to be more of a research monograph than a text. The important question is whether the formal power series technique is the right (best, or a good) way to present automata theory. More importantly, are there more results that can be proved using these techniques that cannot be derived by the more picturesque traditional methods__?__ I can't answer either of these questions. I liked the approach, but I needed six months to digest it, so I would be skeptical of the usefulness of this approach in a course. The authors convinced me that they had one new result that was difficult, if not impossible, to prove using other approaches. But ten new results, or one master theorem from which ten standard results could each be derived in one line, would be more convincing.