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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Summary: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 on all inputs to networks that can run forever. A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as
ISBN:1470451417
9781470451417
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1358