| Summary: | We develop the mathematical theory of epistemic updates with the tools of
duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK),
introduced by Baltag-Moss- Solecki, without the common knowledge operator. We
dually characterize the product update construction of EAK as a certain
construction transforming the complex algebras associated with the given model
into the complex algebra associated with the updated model. This dual
characterization naturally generalizes to much wider classes of algebras, which
include, but are not limited to, arbitrary BAOs and arbitrary modal expansions
of Heyting algebras (HAOs). As an application of this dual characterization, we
axiomatize the intuitionistic analogue of the logic of epistemic knowledge and
actions, which we refer to as IEAK, prove soundness and completeness of IEAK
w.r.t. both algebraic and relational models, and illustrate how IEAK encodes
the reasoning of agents in a concrete epistemic scenario.
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