学术活动发布与订制系统

We introduce a methodology, based on duality theory, which makes it possible to study epistemic updates from an algebraic perspective. We focus on the case study of public announcements, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given model shifts to its submodel relativized to the announced proposition. We give the dual characterization of the corresponding submodel-injection map, as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide 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). In this way, we access the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL; we provide relational semantics for IPAL, we prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and we show that the well-known Muddy Children Puzzle can be encoded in IPAL. (Based on joint work with M. Ma and M. Sadrzadeh)