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)

CafeOBJ is an executable industrial strength multi-logic algebraic speci cation language which is a modern successor of OBJ and incorporates several new algebraic speci cation paradigms.

model theoretic framework of the theory of institutions. For this semantics we develop a generic interpolation result which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of important applications to computing science especially in the area of structured specifications. 1.

We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the so-caled ‘institution theory ’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system ’ and a mathematical concept of ‘homomorphism ’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here. 1.