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Reviewer: Simon John Thompson

Induction and recursion lie at the heart of computer science: grammars are defined inductively, as are proofs in logic, typing in type systems, and functions and procedures in programming languages. Every computer science graduate should be familiar with these and other examples, and be able to reason inductively and make recursive definitions without a second thought. It can be harder to work with other objects: what is the right way to characterize the behavior of a potentially infinite stream of data, or an interactive process that can evolve indefinitely__?__ The answer is to use coinduction to define and reason about them. In particular, bisimulation is used to characterize equality between infinite objects of this sort. While these notions are well known to researchers in process calculi and domain theory, they are less familiar to computer scientists in general. Sangiorgi's book looks set to change that, with an introduction to the subject that is lucid, comprehensive, and approachable. Abstract concepts are illustrated with a variety of well-chosen examples, including the lambda calculus and data types (such as infinite lists), as well as a set of process calculi, which are used throughout the book. A general introduction promises to "liberate us from the constraints of stratification" and sets the work in the context of 20th century mathematics as governed by Russell's discovery and repair of paradoxes in foundational theories through stratification: larger objects have to be built up from smaller constituents. Coinduction offers a way out of this, by making it as equally possible to reason well about "non well-founded" infinite structures, such as a nonterminating process, as about the more traditional structures that produce finite outputs from finite inputs in a finite amount of time. Chapter 1 establishes the definition of bisimulation as the coarsest relation between processes that distinguishes only those with observably different behavior, and shows clearly that this is of a different flavor from traditionally defined equalities based on the structure (or syntax) of objects. Rather, bisimulation is based on more "semantic" considerations. For the general reader, the heart of the book is chapter 2, where Sangiorgi presents a set of examples of induction and coinduction, including terminating and nonterminating behavior in the lambda calculus, finite lists and infinite streams, and finite and infinite process traces. Building on these, he presents the many-faceted duality between induction and coinduction, and in particular their characterizations as least and greatest fixed points in models of various kinds. Moreover, he shows how typical reasoning principles-induction over structure, coinduction by unwinding-are themselves dual. It is a bravura exposition, both compelling and convincing, and should be on the reading list of anyone aiming to master the foundations of theoretical computer science. The remaining chapters consolidate the message of chapter 2, concentrating more on bisimulation than on the wider topic of coinduction, but nevertheless these chapters offer a wealth of valuable graduate-level material in process algebra and theory in general. The text contains numerous examples, and exercises are scattered throughout the book. Many have worked solutions in the appendix, supporting self-study. Inevitably, there are topics that are not covered, and some papers, such as Andy Gordon's seminal 1994 dissertation on functional programming and input/output, are missing from the bibliography; however, Sangiorgi and Rutten have edited a companion volume on advanced topics in bisimulation and coinduction, which gives a broader context to the introduction under review here. In conclusion, this is an outstanding introduction to an important topic, and I hope and expect that in years to come it will become a standard text for advanced students and researchers in theoretical computer science, programming languages, concurrency, logic, and beyond. Online Computing Reviews Service