Reactive Kripke Semantics and Arc Accessibility

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. This is reminiscent of game theoretic semantics where the two sides react to each other. However, reactive Kripke models do not go as far as that. The only additional device we add to Kripke semantics to make it reactive is to allow the accessibility relation to access itself. Thus the accessibility relation R of a reactive Kripke model contains not only pairs (a, b) ∈ 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)) ∈ R, the arc (a, b) is accessible to t. This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality ) 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 use such models to fibre logics which disagree on their common language. 1 Motivation and Background Traditional modal logic uses possible world semantics with accessibility relation R. When we evaluate a formula such as B = 2 p∧ 3q in a Kripke model m = (S ,R, a, h) (S is the set of possible worlds, a ∈ S ,R ⊆ S 2 and h is the assignment) the model m does not change in the course of evaluation of B. We say the model m is not reactive. It stays the same during the process of evaluation. To make this point absolutely clear, consider the situation in Figure 1 below To evaluate a 3q, we have to check b 2q. We can also check another formula at b, say, b 2 p. In either case the world accessible to b are c and d. We do not say that since b 2q started its evaluation at world a as a 3q and continued to b 2q, then the accessible worlds to b are now different. In other words the model does not react to our starting the evaluation of a 3q by changing