Structured Theories and Institutions

Category theory provides an excellent foundation for studying structured speciica-tions and their composition. For example, theories can be structured together in a diagram, and their composition can be obtained as a colimit. There is, however, a growing awareness, both in theory and in speciication practice, that structured theories should not be viewed just as the \scaaolding" used to build unstructured theories: they should become rst-class citizens in the speciication process. Given a logic formalized as an institution I, we therefore ask whether there is a good definition of the category of I-structured theories, and whether they can be naturally regarded as the ordinary theories of an appropriate institution S(I) generalizing the original institution I. We answer both question in the aarmative, and study good properties of the institution I inherited by S(I). We show that, under natural conditions, a number of important properties are indeed inherited, including cocompleteness of the category of theories, liberality, and extension of the basic framework by freeness constraints. The results presented here have been used as a foundation for the module algebra of the Maude language, and seem promising as a semantic basis for a generic module algebra that could be both speciied and executed within the logical framework of rewriting logic.