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Logical Dynamics and Dynamical Systems

Rendsvig, Rasmus Kraemmer LU (2018)
Abstract
This thesis is on information dynamics modeled using *dynamic epistemic logic* (DEL). It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.

Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting the standard machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.

Paper II... (More)
This thesis is on information dynamics modeled using *dynamic epistemic logic* (DEL). It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.

Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting the standard machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.

Paper II extrapolates from Paper I's construction, showing how the augmentation and its natural peers may be construed as maps. It argues that under the restriction of dynamics produced by DEL dynamical systems still falls a collection rich enough to be of interest.

Paper III compares the approach of Paper II with *extensional protocols*, the main alternative augmentation to DEL. It concludes that both have benefits, depending on application. In favor of the DEL dynamical systems, it shows that extensional protocols designed to mimic simple, DEL dynamical systems require infinite representations.

Paper IV focuses on *topological dynamical systems*. It argues that the *Stone topology* is a natural topology for investigating logical dynamics as, in it, *logical convergence* coinsides with topological convergence. It investigates the recurrent behavior of the maps of Papers II and III, providing novel insigths on their long-term behavior, thus providing a proof of concept for the approach.

Paper V lays the background for Paper IV, starting from the construction of metrics generalizing the Hamming distance to infinite strings, inducing the Stone topology. It shows that the Stone topology is unique in making logical and topological convergens coinside, making it the natural topology for logical dynamics. It further includes a metric-based proof that the hitherto analyzed maps are continuous with respect to the Stone topology.

Paper VI presents two characterization theorems for the existence of *reduction laws*, a common tool in obtaining complete dynamic logics. In the compact case, continuity in the Stone topology characterizes existence, while a strengthening is required in the non-compact case. The results allow the recasting of many logical dynamics of contemporary interest as topological dynamical systems. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • professor van Benthem, Johan, Stanford University, USA
organization
publishing date
type
Thesis
publication status
published
keywords
Formal epistemology, Modal logic, Dynamic epistemic logic, Dynamical systems, General topology
pages
229 pages
publisher
Media-Tryck, Lund University, Sweden
defense location
C121, LUX, Helgonav├Ągen 3, Lund
defense date
2018-03-16 13:15
ISBN
9789188473714
9789188473707
language
English
LU publication?
yes
id
08f57b4b-428c-4af9-b31b-2762e8fd7c00
date added to LUP
2018-02-19 10:41:20
date last changed
2018-02-20 08:46:08
@phdthesis{08f57b4b-428c-4af9-b31b-2762e8fd7c00,
  abstract     = {This thesis is on information dynamics modeled using *dynamic epistemic logic* (DEL). It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.<br/><br/>Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting the standard machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.<br/><br/>Paper II extrapolates from Paper I's construction, showing how the augmentation and its natural peers may be construed as maps. It argues that under the restriction of dynamics produced by DEL dynamical systems still falls a collection rich enough to be of interest. <br/><br/>Paper III compares the approach of Paper II with *extensional protocols*, the main alternative augmentation to DEL. It concludes that both have benefits, depending on application. In favor of the DEL dynamical systems, it shows that extensional protocols designed to mimic simple, DEL dynamical systems require infinite representations. <br/><br/>Paper IV focuses on *topological dynamical systems*. It argues that the *Stone topology* is a natural topology for investigating logical dynamics as, in it, *logical convergence* coinsides with topological convergence. It investigates the recurrent behavior of the maps of Papers II and III, providing novel insigths on their long-term behavior, thus providing a proof of concept for the approach.<br/><br/>Paper V lays the background for Paper IV, starting from the construction of metrics generalizing the Hamming distance to infinite strings, inducing the Stone topology. It shows that the Stone topology is unique in making logical and topological convergens coinside, making it the natural topology for logical dynamics. It further includes a metric-based proof that the hitherto analyzed maps are continuous with respect to the Stone topology. <br/><br/>Paper VI presents two characterization theorems for the existence of *reduction laws*, a common tool in obtaining complete dynamic logics. In the compact case, continuity in the Stone topology characterizes existence, while a strengthening is required in the non-compact case. The results allow the recasting of many logical dynamics of contemporary interest as topological dynamical systems.},
  author       = {Rendsvig, Rasmus Kraemmer},
  isbn         = {9789188473714},
  keyword      = {Formal epistemology,Modal logic,Dynamic epistemic logic,Dynamical systems,General topology},
  language     = {eng},
  pages        = {229},
  publisher    = {Media-Tryck, Lund University, Sweden},
  school       = {Lund University},
  title        = {Logical Dynamics and Dynamical Systems},
  year         = {2018},
}