Ultraproducts and possible worlds semantics in institutions

Abstract

We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called 'institutions'. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ultraproducts of [R. Diaconescu, Institution-independent ultraproducts, Fundamenta Informaticae55 (3-4) (2003) 321-348] to possible worlds semantics and prove a fundamental preservation result for abstract modal satisfaction. As a consequence we develop a generic compactness result for possible worlds semantics.

Hybridization of Institutions

“…Hence, an answer to this problem resorts to the decomposition of the hybridization process into two steps: a modalization followed by a hybridization. The former, may be defined as in [16] just making a straightforward generalization to sets of modal symbols Λ. The latter is then applied to the resulting institution.…”

“…Unfortunately this latter step does not work in general, however this scheme may work if we extended our theory by considering also a 'rigid' part for the signatures and models as in [16].…”

“…In particular we are thinking to extend the method of ultraproducts of [16] from modalized to hybridized institutions, to investigate a general method of diagrams and the existence of initial semantics for hybridized institutions. The latter has a special specification theoretic significance: it would give foundational support for classical algebraic specification style with hybrid(ized) logics.…”

“…Combination of logical system (or institutions), in which typically different roles are played by the different logics to be composed, is, in itself, a relevant research topic. The approach discussed in this paper is in line with the process of modalization of an institution, proposed in [16], in which a modal logic is combined with an arbitrary institution in a systematic way. We take a further step by replacing modal by hybrid logic and allowing multi-modalities.…”

“…In this context, the first contribution of this paper is an attempt to 'universalize' the hybridization idea. Following the lines of [16], where a method to modalize arbitrary institutions is presented, the paper introduces a method to hybridize logics at the same institution-independent level. The method extends arbitrary institutions with Kripke semantics (for multi-modalities with arbitrary arities) and hybrid features.…”