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...

We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphism map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of logical multiparadigm languages, especially structuring specifications (module systems) a la OBJ-Clear. They include model reducts, free constructions (liberality), co-limits, model amalgamation (exactness), and inclusion systems. We outline a general logical semantics for languages whose semantics satisfy certain "logical" principles by extending the institutional semantics developed within the Clear-OBJ tradition. Finally, in the Appendix, we briefly illustrate it with the concrete example of CafeOBJ. Keywords Algebraic specification, Institutions, Theory morphism. AMS Classifications 68Q65, 18C10, 03G30, 08A70 2 1 Introduction Computing Motivation This work belongs to the research are...

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 composition of such modules 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 the visible and hidden parts of modules, while the second uses only the visible properties of the components; the semantics agree when the visible consequences of hidden information are enough to determine the result of the composition. A number of "laws of software composition" are proved relating the composition operations. Inclusive institutions simplify many proofs.

This paper surveys the logical and mathematical foundations of CafeOBJ, which is a successor of the famous algebraic specification language OBJ but adding several new primitive paradigms such as behavioural concurrent specification and rewriting logic. We first give a concise overview of CafeOBJ. Then we focus on the actual logical foundations of the language at two different levels: basic specification and structured specification, including also the definition of the CafeOBJ institution. We survey some novel or more classical theoretical concepts supporting the logical foundations of CafeOBJ together with pointing to the main results but without giving proofs and without discussing all mathematical details. Novel theoretical concepts include the coherent hidden algebra formalism and its combination with rewriting logic, and Grothendieck (or fibred) institutions. However for proofs and for some of the mathematical details not discussed here we give pointers to relevant publications. ...

This paper presents axiomatizability results for Inclusive Equational Logic, a categorical generalization of equational logic that further generalizes local equational logic (Cazanescu 1993). The word "inclusive" is motivated by our pervasive use of inclusion systems, invented by Diaconescu, Goguen and Stefaneas (Diaconescu et al. 1993), and further developed in (Hilberdink 1996; Cazanescu and Rosu 1997); these are categories with extra structure to capture the notion of "inclusion". Inclusion systems can be very useful in computing science, e.g., in specification theory (Diaconescu et al. 1993)

Acharacterization result for behaviorally de nable classes of hidden algebras shows that a class of hidden algebras is behaviorally de nable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a nal object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the di erent approaches in the literature. 1

A general critical pair theory is given for rewriting many sorted terms with overloaded operations modulo equations. A main notion is sunification, which yields a set of scritical pairs, such that a set of rules is locally confluent iff they all converge. We prove a sufficient condition for overlaps to work instead of sunification, show that complete sunifier sets always exist, and are finite in important special cases. We also sketch a generalization based on category theory, for rewriting in free objects, e.g., algebras with additional structure, such as many sorts, ordered sorts, equationally defined subsorts, or continuity.

semantics is given for modularization in [13], based on strong inclusions; no factorization (see Definition 9) is involved, which means that, perhaps, strong inclusions are good enough technical tools to handle complex modularization concepts. Definition 9 hI; Ei is a weak inclusion system of A, or A has a weak inclusion system hI; Ei, iff I is a subcategory of inclusions of A, E is a subcategory of A having the same objects as A, and every morphism f in A has a unique factorization f = e; i with e 2 E and i 2 I. hI; Ei is called an inclusion system if E contains only epics, and it is called a regular inclusion system if E contains only coequalizers. 2 Example 10 All structures in Example 8 have weak inclusion systems: Set with I the set of inclusions and E the set of surjective functions. It is regular as each surjective function is a retract, so a coequalizer. Top has two interesting weak inclusion systems (see [1]). One is hI 1 ; E 1 i, where I 1 is the set of continuous inclu...

New properties and implications of inclusion systems are investigated in the present paper. Many properties of lattices, factorization systems and special practical cases can be abstracted and adapted to our framework, making the various versions of inclusion systems useful tools for computer scientists and mathematicians.

This paper introduces the novel concept of inclusive institution as a foundational framework for studying logic-independent module compositionality, defines specification modules as specifications allowing both public and private signatures, and shows that an internal property of modules, called conservatism, is crucial for compositional semantics.