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Reviewer: Richard John Botting

The current theory of how to define what a language means is presented. This text has 11 chapters and covers the main areas: the &lgr;-calculus, call-by-value versus call-by-name, recursive definitions, simple types, theories of finite approximations, full abstraction and adequacy, structured types, <__?__Pub Fmt nolinebreak>domain<__?__Pub Fmt /nolinebreak> theory, universal types, type inference, generic definitions, and polymorphism (subtype and parametric). The book develops from uncomplicated models but ends with highly abstract theories (such as Grothendieck fibrations). The length of the book fits the subject. The author uses “natural operational semantics,” which is also called “structural operational semantics” and is derived from the use of the mathematics of denotational semantics to define the behavior of a program. Semantics are given as tables of derivation rules for equations, relations, or type judgments. Instead of repeating previous rules, the author mentions their source in an informal discussion (p.<__?__Pub Fmt interword-space>xvii). This approach is a good idea. It would be even better to make the cross-references formal and highly visible—a mathematical version of the #include, IMPORT, or USE/WITH clauses. The preface claims that the “book is designed as a text for upper-level and graduate-level students from all areas of computer science.” A familiarity with discrete mathematics is needed, and experience of category theory would help. The book has many exercises. The later chapters go well beyond the needs of upper-level students. The last four chapters, on domain theory and polymorphism, describe recent work and make the book a useful addition to a research library. The paradoxes that crop up are only mentioned in the end-of-chapter notes. These chapters are suited to a graduate seminar, not to undergraduate lectures. Excellent bibliographic notes are provided in each chapter, and the author gives 274 references to papers, books, and theses at the end of the book. The book has two indexes, an index of notations and an index of subjects. These looked adequate at first glance, but I could not find several notations and subjects that I needed as I read the text. Lists of figures and of tables are given, but many diagrams of categories in theorems and definitions are not listed as figures. The typography is good, but the book lacks editing. In the first part of the book, terms are sometimes used without definition or before they are defined. The phrases “semantic function” and “meaning function” appear on page 15 but are not in the index and, as far as I can see, are never defined. Similarly, “equivalence classes” are mentioned on page 32 but are not defined, although “equivalence relations” are defined on page 36. I noted a half dozen typos: the scrambled subscripts on the last line of page 79 got in the way of under<__?__Pub Fmt hyphen-point>standing. Gunter gives no definition of the meaning of syntactic definitions. Syntactic precision is sacrificed on the altar of parsimony. So, informal rules clarify the parsing of the languages defined. The Programming language for Computable Functions (PCF) defined on page 100 includes expressions such as “succ(true)” and “zero__?__(false).” No semantics are provided for these until 128 pages later, where, by default, they are defined to evaluate to “tyerr” [sic]. The book is concerned almost entirely with the semantics of applicative languages like LISP and ML. Most programmers will find that their language is never mentioned. Ada, however, is mentioned briefly. Algol 60 is the only non–LISP-like example. On page 253, the author states that <__?__Pub Fmt nolinebreak>“ALGOL<__?__Pub Fmt /nolinebreak> 60 has the typed &lgr;-calculus as a sublanguage." It does not. Hennessy's text [1] gives an introduction to natural semantics aimed at undergraduates. Meyer's book [2] will appeal more to practitioners. These two texts use more familiar languages. They cover less than half of the material in <__?__Pub Caret>Gunter's book, since they do not tackle the categorical and domain theories of data types. Gunter's text is a good graduate text and reference for researchers in languages.