Refereed journal article or data article (A1)

Effective Projections on Group Shifts to Decide Properties of Group Cellular Automata




List of AuthorsBéaur Pierre, Kari Jarkko

PublisherWORLD SCIENTIFIC PUBL CO PTE LTD

Publication year2023

JournalInternational Journal of Foundations of Computer Science

Journal name in sourceINTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE

Journal acronymINT J FOUND COMPUT S

Number of pages24

ISSN0129-0541

eISSN1793-6373

DOIhttp://dx.doi.org/10.1142/S0129054123480040

URLhttps://doi.org/10.1142/S0129054123480040


Abstract

Many decision problems concerning cellular automata are known to be decidable in the case of algebraic cellular automata, that is, when the state set has an algebraic structure and the automaton acts as a morphism. The most studied cases include finite fields, finite commutative rings and finite commutative groups. In this paper, we provide methods to generalize these results to the broader case of group cellular automata, that is, the case where the state set is a finite (possibly non-commutative) finite group. The configuration space is not even necessarily the full shift but a subshift - called a group shift - that is a subgroup of the full shift on Zd, for any number d of dimensions. We show, in particular, that injectivity, surjectivity, equicontinuity, sensitivity and nilpotency are decidable for group cellular automata, and non-transitivity is semi-decidable. Injectivity always implies surjectivity, and jointly periodic points are dense in the limit set. The Moore direction of the Garden-of-Eden theorem holds for all group cellular automata, while the Myhill direction fails in some cases. The proofs are based on effective projection operations on group shifts that are, in particular, applied on the set of valid space-time diagrams of group cellular automata. This allows one to effectively construct the traces and the limit sets of group cellular automata. A preliminary version of this work was presented at the conference Mathematical Foundations of Computer Science 2020.


Last updated on 2023-07-12 at 13:10