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.File | Dimensione | Formato | |
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