Different types of arrow between logical frameworks

Abstract

We argue that there are problems with the distinction between signatures and sentences within the notion of institution. While the distinction is useful and necessary, maps between institutions ofter have to go beyond this distinction. Therefore, three new notions of map between institutions, conjunctive maps, weak maps and semi-maps, have been introduced. These allow to relax the distinction between signatures and sentences while still being sentence-structured, opposed to mere maps of specification frames.

When introducing these types of arrow we have followed some guidelines which allow to relate different categories of logical framework by pair of adjoint functors. Along two of those pairs, there is a useful borrowing of sentences from an institution, extendeding the work of Cerioli and Meseguer on borrowing logical structure along maps [7] to the relation between specification frames and institutions.

Thus, there is not one common type of logical framework, but there is the chance to introduce new types of logical framework (driven by a systematic study of examples) in a manner that different types of logical framework are well-related via nice general mathematical tools and theorems.

The technical results of the paper are summarized in lower half of Fig. 2, where the arrows ∃!,, U _Ext ° K ⁻¹ and K ° F _Ext only exist when everything is restricted to the case with amalgamation.