We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as term rewriting.

model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum

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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.

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.