Modal Logic of Transition Systems in the Topos of Trees

Colin Riba 1, 2 Guilhem Jaber 1, 2
2 PLUME - Preuves et Langages
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We investigate the adjunction between the category of transition systems (with not nec. bounded morphisms) and the topos of trees S from the perspective of the basic modal logic K on transition systems. More specifically, we show how the modal logic K over an object of S seen as a transition system can be lifted back as a subobject of S. This relies on the usual mutually transpose formulations of modal satisfaction as either a map of transition system or as a map of Boolean algebras with operators. Moreover, thanks to the usual geometric morphism from the Boolean topos Psh(|N * |) of presheaves over the discrete category of natural numbers to S , we can give a characterization of modal satisfaction within S as a map from formulae to total subobjects of S whose image in the Boolean topos Psh(|N * |) is an internal map of Boolean algebras.
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https://hal.archives-ouvertes.fr/hal-01620992
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Submitted on : Sunday, October 22, 2017 - 6:00:03 PM
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Colin Riba, Guilhem Jaber. Modal Logic of Transition Systems in the Topos of Trees. 2017. ⟨hal-01620992⟩

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