1.1.6. Abstract Model Theory

From our various studies in Abstract Model Theory, the technical report [29] is particularly relevant for the present paper. That time, we proved in detail and in a systematic way that four specification formalisms are indeed institutions. Our main finding was that the proof of the satisfaction condition always boiled down to the existence of what we called corresponding assignments and corresponding evaluations, respectively. This finding has been integrated later by Pawlowski in his concept of context institutions [30]. One of the main motivations for context institutions was to incorporate open formulas in the abstract description of specification formalisms and the term context has been coined as an abstract pendant for a "set of variables".

What we call feature expressions in our framework are nothing but a generalization of open formulas. We differentiate, however, conceptually between variable declarations and contexts. In some specification formalisms, both concepts may denote the same thing. In other formalisms, any variable declaration will be also a context but not vice versa. In addition, there can be formalisms where variable declarations and contexts are kept apart, as in Description Logic for example. We use the term context as an abstract pendant for things like a set of generators in Group Theory, an underlying graph of a sketch in Category Theory, an underlying graph of a software model in Software Engineering, a set of literals (atomic values) in Logic Programming and a set of individual names in Description Logic, for example.
