Abelian networks IV. Dynamics of nonhalting networks

An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we extend the theory of abelian networks that halt o...

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Bibliographic Details
Main Authors:	Chan, Swee Hong, Levine, Lionel
Format:	eBook
Language:	English
Published:	Providence, Rhode Island American Mathematical Society 2022
Edition:	1
Series:	Memoirs of the American Mathematical Society
Subjects:	Abelian groups Machine theory Grothendieck group sandpile group injective action abelian mobile agents confluence exchange lemma burning algorithm cycle-rooted spanning forest Abelian distributed processors rotor walk atemporal dynamics chip-firing removal lemma commutative monoid action Eulerian walkers critical group
ISBN:	1470451417, 9781470451417
ISSN:	0065-9266, 1947-6221
Online Access:	Get full text
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Table of Contents:

Introduction -- Commutative Monoid Actions -- Review of Abelian Networks -- The Torsion Group of an Abelian Network -- Critical Networks: Recurrence -- Critical Networks: Dynamics -- Rotor and Agent Networks -- Concluding Remarks
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Flashback -- 1.2. Atemporal dynamics -- 1.3. Relating atemporal dynamics to traditional dynamics -- 1.4. Computational questions -- 1.5. The torsion group of a nonhalting abelian network -- 1.6. Critical networks -- 1.7. Example: Rotor networks and abelian mobile agents -- 1.8. Proof ideas -- 1.9. Summary of notation -- Chapter 2. Commutative Monoid Actions -- 2.1. Injective actions and Grothendieck group -- 2.2. The case of finite commutative monoids -- Chapter 3. Review of Abelian Networks -- 3.1. Definition of abelian networks -- 3.2. Legal and complete executions -- 3.3. Locally recurrent states -- 3.4. The production matrix -- 3.5. Subcritical, critical, and supercritical abelian networks -- 3.6. Examples: sandpiles, rotor-routing, toppling, etc -- Chapter 4. The Torsion Group of an Abelian Network -- 4.1. The removal lemma -- 4.2. Recurrent components -- 4.3. Construction of the torsion group -- 4.4. Relations to the critical group in the halting case -- Chapter 5. Critical Networks: Recurrence -- 5.1. Recurrent configurations and the burning test -- 5.2. Thief networks of a critical network -- 5.3. The capacity and the level of a configuration -- 5.4. Stoppable levels: When does the torsion group act transitively? -- Chapter 6. Critical Networks: Dynamics -- 6.1. Activity as a component invariant -- 6.2. Near uniqueness of legal executions -- Chapter 7. Rotor and Agent Networks -- 7.1. The cycle test for recurrence -- 7.2. Counting recurrent components -- 7.3. Determinantal generating functions for recurrent configurations -- Chapter 8. Concluding Remarks -- 8.1. A unified notion of recurrence and burning test -- 8.2. Forbidden subconfiguration test for recurrence -- 8.3. Number of recurrent configurations in a recurrent component -- Acknowledgement -- Bibliography -- Back Cover