Introducing reactive Kripke semantics and arc accessibility

Dov M. Gabbay

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

32 Scopus citations


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.

Original languageEnglish
Title of host publicationPillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday
EditorsArnon Avron, Nachum Dershowitz, Alexander Rabinovich
Number of pages50
StatePublished - 2008
Externally publishedYes

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4800 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


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