TY - GEN

T1 - Introducing reactive Kripke semantics and arc accessibility

AU - Gabbay, Dov M.

PY - 2008

Y1 - 2008

N2 - 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 of a reactive Kripke model contains not only pairs 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 , meaning that the arc (a,b) is accessible to t, or even connections of the form . 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 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.

AB - 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 of a reactive Kripke model contains not only pairs 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 , meaning that the arc (a,b) is accessible to t, or even connections of the form . 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 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.

UR - http://www.scopus.com/inward/record.url?scp=48949083679&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-78127-1_17

DO - 10.1007/978-3-540-78127-1_17

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AN - SCOPUS:48949083679

SN - 3540781269

SN - 9783540781264

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 292

EP - 341

BT - Pillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday

A2 - Avron, Arnon

A2 - Dershowitz, Nachum

A2 - Rabinovich, Alexander

ER -