## Abstract Semantics for Module Composition

Grigore Rosu
CS2000-0653
May 8, 2000

Programmers and software engineers agree in unanimity that a useful characteristic of the programming languages they use for implementations (C++, Java, etc.) is their support for both {\em public} and {\em private} features (types, functions). The public features are often called {\em interfaces}. The private part is not visible outside the module (class, package) that declares it, but it can be used internally to define the visible part. Such distinction helps software engineers abstract their work and ignore details which are a main source of confusion and errors. We claim that the distinction between private and public features might also be a desirable characteristic of formal specifications, not only from the practical point of view but also because of at least two important theoretical reasons. One is the possibility to finitely specify theories which do not admit finite standard presentations. For example, Bergstra and Tucker showed that any recursive algebra is a restriction of an initial algebra presented by a finite number of equations over a larger signature. Another reason is that it allows the user to specify behavioral properties of systems, in the sense that every behavioral specification is equivalent to hiding some operators (i.e., making them private) in a usual specification. We introduce the notion of {\em module specification} as a generalization of the standard specification, having both public (or visible) and private features, and then explore their properties at an abstract level, categorical. To formalize the notion of logical system'' we use {\em institutions} enriched with {\em inclusions}, an abstraction of the natural notion of inclusion of signatures from particular logics. The {\em visible theorems} (or the visible consequences) of a module are those theorems which contain only visible symbols, and a model of that module is a model of its visible consequences. Five basic operations on module specifications are explored: renaming, hiding, enriching, aggregation and parameterization. An internal property of modules, {\em conservativeness}, seems to have a decisive role in giving semantics for module composition.

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