Epistemic Actions as Resources

We provide an 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. due to secrecy and deceit, as well as nested knowledge. 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 $A,B,\dots\in\mathcal A$, are modeled as elements of a quantale. The quantale $(Q,\bigvee,\bullet)$ acts on an underlying Q-right module $(M,\bigvee)$ of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair $f^M_A:M\rightarrow M$ and $f^Q_A:Q\rightarrow Q$ of (lax) morphisms for each agent $A\in\mathcal A$, which preserve the module and quantale structure respectively. By adjunction, they give rise to epistemic modalities, capturing the agents' knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities—cf. weakest precondition. This model subsumes the crucial fragment of Baltag, Moss and Solecki's dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions.