Hidden Algebra for Software Engineering

This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We rst review classical algebraic speciication for traditional \Platonic" abstract data types like integers, vectors, matrices, and lists. Software engineering also needs changeable \abstract machines," recently called \objects," that can communicate concurrently with other objects through visible \attributes" and state-changing \methods." Hidden algebra is a new development in algebraic semantics designed to handle such systems. Equational theories are used in both cases, but the notion of satisfaction for hidden algebra is behavioral, in the sense that equations need only appear to be true under all possible experiments; this extra exibility is needed to accommodate the clever implementations that software engineers often use to conserve space and/or time. The most important results in hidden algebra are powerful hidden coinduction principles for proving behavioral properties. This paper also includes some comparison with the closely related area called coalgebra, and some bits of history.