Circular coinductive rewriting is a new method for proving behavioral properties, that combines behavioral rewriting with circular coinduction. This method is implemented in our new BOBJ behavioral specification and computation system, which is used in examples throughout this paper. These examples demonstrate the surprising power of circular coinductive rewriting. The paper also sketches the underlying hidden algebraic theory and briefly describes BOBJ and some of its algorithms.

We argue for an algorithmic approach to behavioral proofs, review the hidden algebra approach, develop circular coinductive rewriting for conditional goals, extend it with case analysis, and give some examples.

This paper describes the Tatami project at UCSD, which is developing a system to support distributed cooperative software development over the web, and in particular, the validation of concurrent distributed software. The main components of our current prototype are a proof assistant, a generator for documentation websites, a database, an equational proof engine, and a communication protocol to support distributed cooperative work. We believe behavioral specification and verification are important for software development, and for this purpose we use first order hidden logic with equational atoms. The paper also briefly describes some novel user interface design methods that have been developed and applied in the project

Circular coinduction is a technique for behavioral reasoning that extends cobasis coinduction to specifications with circularities. Because behavioral satisfaction is not recursively enumerable, no algorithm can work for every behavioral statement. However, algorithms using circular coinduction can prove every practical behavioral result that we know. This paper proves the correctness of circular coinduction and some consequences.

: The benefits of the object, logic (or relational), functional, and constraint paradigms

recent advances in web technology, interface design, and specification. Our effort to improve the usability of such systems has led us into algebraic semiotics, while our effort to develop better formal methods for distributed concurrent systems has led us into hidden algebra and fuzzy logic. This paper discusses the Tatami system design, especially its software architecture, and its user interface principles. New work in the latter area includes an extension of algebraic semiotics to dynamic multimedia interfaces, and integrating Gibsonian affordances with algebraic semiotics. 1

Following condensed introductions to classical and behavioral algebraic specification, this paper discusses the verification of behavioral properties using BOBJ, especially its implementation of conditional circular coinductive rewriting with case analysis. This formal method is then applied to proving correctness of the alternating bit protocol, in one of its less trivial versions. We have tried to minimize mathematics in the exposition, in part by giving concrete illustrations using the BOBJ system.

The behavioral equivalence of hidden terms in an equational specification logic is not itself specifiable in general (Buss and Ro¸su 2000). But much recent work has been done on its partial specification, in particular using coinduction. In this paper we consider the more general notion of conditional behavioral equivalence introduced by Reichel in 1984. We investigate the behavioral proof theory of a general class of equational specification logics, the hidden equational logics. Among other things we characterize the behaviorally valid conditional equations of a hidden equational logic as those conditional equations which, in a natural sense, do not increase the deductive power of the logic when they are added as new rules of inference. For a special kind of hidden equational logic (the equivalential logics) we obtain methods for proving behavioral validity that work well in practice. Those hidden equational logics whose behavioral is specifiable by a (non-hidden) equational logic are characterized in terms of a special class of equivalential logics—equivalently as those hidden equational logics that have a cobasis (Ro¸su and Goguen 2001) of a special form.

Behavioral rewriting differs from standard rewriting in taking account of the weaker inference rules of behavioral logic, but it shares much with standard rewriting, including notions like termination and confluence. We describe an efficient implementation of behavioral rewriting that uses standard rewriting. Circular coinductive rewriting combines behavioral rewriting with circular coinduction, giving a surprisingly powerful proof method for behavioral properties; it is implemented in the BOBJ system, which is used in our examples. These include several lazy functional stream program equivalences and a behavioral refinement. 1 Introduction Behavioral specification is the area where models (implementations) only behaviorally satisfy specifications; it also supports infinitary data structures, behavioral refinement, and coinductive proof methods. Behavioral specifications distinguish visible from hidden sorts, with equality being strict on visible sorts and behavioral on hidden sorts,...

We show that for any behavioral Sigma-specification B there is an ordinary algebraic specification ~ B over a larger signature, such that a model behaviorally satisfies B if and only if it satisfies ~ B, where is the information hiding operator exporting only the Sigma-theorems of ~ B. The idea is to add machinery for contexts and experiments (sorts, operations and equations), use it, and then hide it. We develop a procedure, called unhiding, that takes a finite B and produces a finite ~ B. The practical aspect of this procedure is that one can use any standard equational or inductive theorem prover to derive behavioral theorems, even if neither equational reasoning nor induction is sound for behavioral satisfaction.