Generalization of Final Algebra Semantics by Relativization

We consider the semantics of algebraic specifications viewed as evolving rather than static entities. This leads to a relativization of final algebra semantics with respect to the space of evolutionary possibilities open to a given specification. The evolutionary space itself, which we call a language, has significant semantic content. The unit of application for our semantics is such a language of specifications. We formalize relevant notions of languages and language models and derive conditions to establish the existence of a final object in the category of models for a language. The results can be used to broaden the applicability of final algebra semantics and to fine tune the balance between the desired strength of semantically valid assertions and the range of evolutionary possibilities to be considered.