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 Z^d, 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.

翻译：关于元胞自动机的许多判定问题在代数元胞自动机情形下已知是可判定的，即当状态集具有代数结构且自动机作为态射作用时。最常研究的情形包括有限域、有限交换环和有限交换群。本文提供方法将这些结果推广到更广泛的群元胞自动机情形，即状态集为有限（可能非交换）群的情形。构型空间甚至不必是完全平移，而是子平移——称为群平移——它是任意维数d下Z^d上完全平移的子群。我们特别证明了群元胞自动机的单射性、满射性、等连续性、敏感性和幂零性是可判定的，而非传递性是半可判定的。单射性总是蕴含满射性，且联合周期点在极限集中稠密。花园-伊甸定理的Moore方向对所有群元胞自动机成立，而Myhill方向在某些情形下失效。证明基于群平移上的有效投影操作，这些操作特别应用于群元胞自动机有效时空图集上。这使得能够有效构造群元胞自动机的迹和极限集。本工作的初版曾在2020年数学基础计算机科学会议上发表。