Abelian Networks IV. Dynamics of Nonhalting Networks

• Autore: Chan Swee Hong
• Pubblicazione: Providence : , : American Mathematical Society, , 2022; ©2022
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (104 pages)
• Disciplina: 512/.25; 512.25
• Soggetto topico: Abelian groups; Combinatorics -- Graph theory -- Graphs and abstract algebra (groups, rings, fields, etc.); Group theory and generalizations -- Abelian groups -- Finite abelian groups; Group theory and generalizations -- Semigroups -- Commutative semigroups; Group theory and generalizations -- Semigroups -- Semigroups in automata theory, linguistics, etc; Dynamical systems and ergodic theory -- Topological dynamics -- Cellular automata; Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Combinatorial dynamics (types of periodic orbits)
• Classificazione: 05C2520K0120M1420M3537B1537E15
• Altri autori: LevineLionel
• Nota di contenuto: 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.
• Sommario/riassunto: "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 torsion group of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. 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 where n is the number of recurrent chip-and-rotor configurations with n chips; D is the diagonal matrix of outdegrees, and A is the adjacency matrix. A consequence is that the sequence (n)n1 completely determines the spectrum of the simple random walk on the network"--
• Titolo autorizzato: Abelian Networks IV. Dynamics of Nonhalting Networks
• ISBN: 9781470470241; 1470470241
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910965973803321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society