This paper studies the composition of modules that can hide information, over a very general class of logical systems called inclusive institutions. Two semantics are given for compositions using five familiar operations, and a property called conservativity is shown necessary and sufficient for these semantics to agree. The first semantics extracts the visible properties of the result of composing both the visible and hidden parts of modules, while the second uses only the visible properties of the components. Several "laws of software composition" are given, which demonstrate the power of inclusive institutions to simplify proofs.

Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logic-independent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothendieck flattenings to inclusion systems. An important practical significance of the resulting Grothendieck inclusion systems is that they allow the development of module algebras for multi-logic heterogeneous specification frameworks. At another level, we show that several inclusion systems in use in some syntactic (signatures, deductive theories) or semantic contexts (models) appear as Grothendieck inclusion systems too. We also study several general properties of Grothendieck inclusion systems.

In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic approach provides a uniform framework for the study of structured specifications independently from any actual choice of specification building operators, and moreover it unifies the theory and the model oriented approaches. Within this framework we develop concepts and results about ‘abstract structured specifications ’ such as co-limits, model amalgamation, compactness, interpolation, sound and complete proof theory, and pushout-style parameterization with sharing, all of them in a top down manner dictated by the upper level of institution independence. 1.

We develop module algebra for structured specifications with model oriented denotations. Our work extends the existing theory with specification building operators for non-protecting importation modes and with new algebraic rules (most notably for initial semantics) and upgrades the pushout-style semantics of parameterized modules to capture the (possible) sharing between the body of the parameterized modules and the instances of the parameters. We specify a set of sufficient abstract conditions, smoothly satisfied in the actual situations, and prove the isomorphism between the parallel and the serial instantiation of multiple parameters. Our module algebra development is done at the level of abstract institutions, which means that our results are very general and directly applicable to a wide variety of specification and programming formalisms that are rigorously based upon some logical system. 1.