Extra Theory Morphisms for Institutions: Logical Semantics for Multi-Paradigm Languages

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) á 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.