A Theory-based Typological Notion of Institutions

By adopting theories as primitive components of a logic and recognizing that formulae are just presentation details we arrive at the concept of topological institution. In a topological institution, we have, for each signature , a frame of theories, a set of interpretation structures and a satisfaction relation. More precisely, we have, for each signature, a topological system. We show how to extract a topological institution from a given institution and establish an adjunction. Illustrations are given within the context of equational logic. We study the compositionality of theories. Formulae are recovered when we establish a general technique for presenting topological institutions. Topological institutions with nitely observable theories are shown to be useful in temporal monitoring applications where we would like to be able to characterize the properties of the system that can be monitored. Namely, an invariant property (G') cannot be monitored because it cannot be positively established in nite time. On the contrary, a reactivity property (F') can be positively established in nite time.