Epistemic Updates on Algebras

TL;DR

This paper develops a duality-theoretic framework for epistemic updates in algebraic models, generalizing existing constructions and applying it to axiomatize and analyze an intuitionistic epistemic logic.

Contribution

It introduces a dual characterization of the product update in EAK, extending it to broader algebraic classes and establishing an axiomatic system for IEAK.

Findings

Dual characterization of epistemic updates in algebraic models.

Axiomatization and completeness proof for IEAK.

Application to modeling agent reasoning in epistemic scenarios.

Abstract

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.