Modularisation is important for managing the complex structures that arise in large theorem proving problems, and in large software and/or hardware development projects. This paper studies some properties of logical systems that support the definition, combination, parameterisation and reuse of modules. Our results show some new connections among: (1) the preservation of various kinds of conservative extension under pushouts; (2) various distributive laws for information hiding over sums; and (3) (Craig style) interpolation properties. In addition, we study differences between syntactic and semantic formulations of conservative extension properties, and of distributive laws. A model theoretic property that we call exactness plays an important role in some results. This paper explores the interplay between syntax and semantics, and thus lies in the tradition of abstract model theory. We represent logical systems as institutions. An important technical foundation is a new ...

Institutions formalize the intuitive notion of logical system, including syntax, semantics, and the relation of satisfaction between them. Our exposition emphasizes the natural way that institutions can support deduction on sentences, and inclusions of signatures, theories, etc.; it also introduces terminology to clearly distinguish several levels of generality of the institution concept. A surprising number of different notions of morphism have been suggested for forming categories with institutions as objects, and an amazing variety of names have been proposed for them. One goal of this paper is to suggest a terminology that is uniform and informative to replace the current chaotic nomenclature; another goal is to investigate the properties and interrelations of these notions in a systematic way. Following brief expositions of indexed categories, diagram categories, twisted relations, and Kan extensions, we demonstrate and then exploit the duality between institution morphisms in the original sense of Goguen and Burstall, and the "plain maps" of Meseguer, obtaining simple uniform proofs of completeness and cocompleteness for both resulting categories. Because of this duality, we prefer the name "comorphism" over "plain map;" moreover, we argue that morphisms are more natural than comorphisms in many cases. We also consider "theoroidal" morphisms and comorphisms, which generalize signatures to theories, based on a theoroidal institution construction, finding that the "maps" of Meseguer are theoroidal comorphisms, while theoroidal morphisms are a new concept. We introduce "forward" and "semi-natural" morphisms, and develop some of their properties. Appendices discuss institutions for partial algebra, a variant of order sorted algebra, two versions of hidden algebra, and...

GB92] to formally capture the notion of logical system. Interpreting institutions as functors, and morphisms and representations of institutions as natural transformations, we give elegant proofs for the completeness of the categories of institutions with morphisms and representations, respectively, show that the duality between morphisms and representations of institutions comes from an adjointness between categories of functors, and prove the cocompleteness of the categories of institutions over small signatures with morphisms and representations, respectively. Category: F.3, F.4

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.