We focus on how the dynamical properties of any Linear CA over (Z/mZ)n are hidden inside the characteristic polynomial of its defining matrix, namely, a polynomial of degree n in the indeterminate t and with Laurent polynomials over Z/mZ as coefficients. In particular, as far as Linear CA over (Z/mZ)n are concerned, we review the mostly recent algebraic decidable characterizations of the following properties: injectivity, surjectivity, sensitivity to the initial conditions, equicontinuity, topological transitivity, and positive expansivity. These characterizations are easy to check, i.e., related decision algorithms can be designed is such a way that exponential terms in their computational complexity are avoided as much as possible. In particular, gcd operations are involved, while the prime factor decomposition of m is bypassed. Finally, we recall how such characterizations regarding Linear CA over (Z/mZ)n can be exploited to decide the above mentioned dynamical properties for the whole class of Additive CA over a finite abelian group.

Dennunzio, A. (2024). Easy to Check Algebraic Characterizations of Dynamical Properties for Linear CA and Additive CA over a Finite Abelian Group. In Cellular Automata and Discrete Complex Systems 30th IFIP WG 1.5 International Workshop, AUTOMATA 2024, Durham, UK, July 22–24, 2024, Proceedings (pp.23-34). Springer Science and Business Media Deutschland GmbH [10.1007/978-3-031-65887-7_2].

Easy to Check Algebraic Characterizations of Dynamical Properties for Linear CA and Additive CA over a Finite Abelian Group

Dennunzio A.
2024

Abstract

We focus on how the dynamical properties of any Linear CA over (Z/mZ)n are hidden inside the characteristic polynomial of its defining matrix, namely, a polynomial of degree n in the indeterminate t and with Laurent polynomials over Z/mZ as coefficients. In particular, as far as Linear CA over (Z/mZ)n are concerned, we review the mostly recent algebraic decidable characterizations of the following properties: injectivity, surjectivity, sensitivity to the initial conditions, equicontinuity, topological transitivity, and positive expansivity. These characterizations are easy to check, i.e., related decision algorithms can be designed is such a way that exponential terms in their computational complexity are avoided as much as possible. In particular, gcd operations are involved, while the prime factor decomposition of m is bypassed. Finally, we recall how such characterizations regarding Linear CA over (Z/mZ)n can be exploited to decide the above mentioned dynamical properties for the whole class of Additive CA over a finite abelian group.
paper
Additive Cellular Automata; Cellular Automata; Decidability; Discrete Dynamical Systems; Linear Cellular Automata;
English
30th IFIP WG 1.5 International Workshop, AUTOMATA 2024 - July 22–24, 2024
2024
Gadouleau, M; Castillo-Ramirez, A
Cellular Automata and Discrete Complex Systems 30th IFIP WG 1.5 International Workshop, AUTOMATA 2024, Durham, UK, July 22–24, 2024, Proceedings
9783031658860
15-lug-2024
2024
14782 LNCS
23
34
reserved
Dennunzio, A. (2024). Easy to Check Algebraic Characterizations of Dynamical Properties for Linear CA and Additive CA over a Finite Abelian Group. In Cellular Automata and Discrete Complex Systems 30th IFIP WG 1.5 International Workshop, AUTOMATA 2024, Durham, UK, July 22–24, 2024, Proceedings (pp.23-34). Springer Science and Business Media Deutschland GmbH [10.1007/978-3-031-65887-7_2].
File in questo prodotto:
File Dimensione Formato  
Dennunzio-2024-AUTOMATA-VoR.pdf

Solo gestori archivio

Tipologia di allegato: Publisher’s Version (Version of Record, VoR)
Licenza: Tutti i diritti riservati
Dimensione 268.9 kB
Formato Adobe PDF
268.9 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/547533
Citazioni
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 0
Social impact