Introducing reactive Kripke semantics and arc accessibility

Abstract

Ordinary Kripke models are not reactive. When we evaluate (test/ measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the process. The additional device we add to Kripke semantics to make it reactive is to allow the accessibility relation to access itself. Thus the accessibility relation \({\cal R}\) of a reactive Kripke model contains not only pairs \((a, b) \in {\cal R}\) of possible worlds (b is accessible to a, i.e., there is an accessibility arc from a to b) but also pairs of the form \((t, (a, b))\in {\cal R}\), meaning that the arc (a,b) is accessible to t, or even connections of the form \(((a,b), (c,d))\in {\cal R}\). This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality \(\square\)) by a class of reactive frames. There are logics which cannot be characterised by ordinary frames but which can be characterised by reactive frames. We also discuss the manifestation of the ‘reactive’ idea in the context of automata theory, where we allow the automaton to react and change it’s own definition as it responds to input, and in graph theory, where the graph can change under us as we manipulate it.