*3.3. Modalised (Stratified) Institutions*

The modalisation of institutions, already discussed as an item in the list of examples of stratified institutions, constitutes an example of reversing the decomposition concept in which <sup>S</sup><sup>0</sup> is rather concrete—its models being Kripke frames—while <sup>B</sup> is kept abstract, and it goes back essentially to [61].

In this context, the work [66] generalises the famous encoding of modal logic into first-order logic [67] in the sense that any abstract encoding B → FOL becomes lifted to an encoding S<sup>∗</sup> → FOL (the precise notion of encoding being what is known as *theoroidal comorphism*). This highly general encoding constitutes the foundations for the formal specification and verification language *H* [68], which is a language that is institutionindependent in the sense that in principle, the base institution B can be any institution that can be plugged into the system.

Although the modalisation of institutions has been defined in the way presented above, in fact, it can be extended to a construction that takes an arbitrary stratified rather than an ordinary institution as input. So, it becomes a method for building new stratified institutions on top of proper stratified institutions. A brief description of this method is as follows:


In order to capture precisely various relevant examples, this construction can be refined in various ways by considering constrained models (axiomatically in the manner described in [63] or more concretely as in [61]), or by considering nominals structures or polyadic modalities. In the case of the latter two extensions, of course, the new category of signatures is a product between *Sign*S and some category of signatures for relations.
