ar X iv : m at h / 06 08 16 6 v 1 [ m at h . L O ] 7 A ug 2 00 6 Epistemic actions as resources

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A,B, . . . ∈ A, are modeled as elements of a quantale, hence conceiving them as resources. Indeed, quantales are to locales what monoidal closed categories are to Cartesian closed categories, respectively providing semantics for Intuitionistic Logic, and for non-commutative Intuitionistic Linear Logic, including Lambek calculus. The quantale (Q, ∨ , •) acts on an underlyingQ-right module (M, ∨ ) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair f A : M → M and f A : Q → Q of (lax) morphisms for each agent A ∈ A. By adjunction, they give rise to epistemic modalities [12], capturing the agents’ knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities [19] — cf. weakest preconditions. This model subsumes the crucial fragment Baltag, Moss and Solecki’s [6] dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions.