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A Logical Framework for Graded Predicates Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Convergence, Continuity and Recurrence in Dynamic Epistemic Logic

Authors : Dominik Klein, Rasmus K. Rendsvig

Published in: Logic, Rationality, and Interaction

Publisher: Springer Berlin Heidelberg

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Abstract

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.

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Footnotes
1
\(\varvec{\mathcal {L}_{\varLambda }}\) is isomorphic to the domain of the Lindenbaum algebra of \(\varLambda \).
 
2
That all models in X are image-finite is a sufficient condition, cf. the Hennessy-Milner Theorem. See e.g. [14] or [27].
 
3
Space does not allow for a discussion of the remaining metrics of [1, 17], but see [31].
 
4
A logic \(\varLambda \) is logically compact if any arbitrary set A of formulas is \(\varLambda \)-consistent iff every finite subset of A is \(\varLambda \)-consistent.
 
5
An \(\varvec{\mathcal {L}}_{\varLambda }\) modal space \(\varvec{X}\) is saturated iff for each \(\varLambda \)-consistent set of formulas A, there is an \(\varvec{x}\in \varvec{X}\) such that \(x\vDash A\). Saturation relates to the notion of strong completeness, cf. e.g. [14, Proposition 4.12]. See [31] for its use in a more general context.
 
6
Multi-pointed action models are also referred to as epistemic programs in [2], and allow encodings akin to knowledge-based programs [22] of interpreted systems, cf. [42].
 
7
The precondition of \(\sigma \) specify the conditions under which \(\sigma \) is executable, while its postcondition may dictate the posterior values of a finite, possibly empty, set of atoms.
 
8
I.e. a conjuction of literals, where a literal is an atom or a negated atom.
 
9
Or \(\omega \)-limit point. The \(\omega \) is everywhere omitted as time here only moves forward.
 
10
We paraphrase van Benthem and Sadzik using the terminology introduced.
 
11
See [16] for an elegant and generalizing exposition.
 
12
The details differ depending on whether \(\varSigma {\scriptstyle \varGamma }\) must be static, but non-Boolean for Proposition 10, or Boolean, but non-static for Proposition 11. See resp. [15, 32].
 
13
For Proposition 11, tape cell content may be encoded using atomic propositions, changeable through postconditions, cf. [15]; for Proposition 10, cell content is written by adding and removing additional states, cf. [32].
 
14
The exact form is straightforward from the constructions used in [15, 32].
 
15
A similar argument shows that all \(x^{Z}\) with \(Z\subseteq \mathbb {N}\) co-infinite are recurrent points. Hence \(\omega _{f}(\varvec{x}'_{k})\) for any \(\varvec{x}'_{k}\in (\varvec{x}'_{n})_{n\in \mathbb {N}_{0}}\) contains uncountably many recurrent points.
 
16
Hence also the multi-agent belief revision policies lexicographic upgrade and elite change, also known as radical and conservative upgrade, introduced in [9], cf. [5].
 
17
In the omitted part of the quotation from the introduction.
 
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Metadata
Title
Convergence, Continuity and Recurrence in Dynamic Epistemic Logic
Authors
Dominik Klein
Rasmus K. Rendsvig
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Book
Logic, Rationality, and Interaction

Print ISBN: 978-3-662-55664-1

Electronic ISBN: 978-3-662-55665-8

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-662-55665-8_8

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