We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates; and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines. We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely-generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb{Z}^d$-shift on a non-trivial alphabet if and only if $d \geq 2$.

翻译：本文研究可逆图灵机构成的一个抽象群。在我们的模型中，每台图灵机被解释为某个空间上的同胚映射，该空间表示一条带有符号的纸带和一个携带状态的头。这些同胚映射只能在头周围有界距离内修改纸带、改变状态并以有界方式移动头。我们研究了该模型中产生的三个自然子群：有限状态自动机群，它推广了拓扑动力学和轨道等价理论中研究的拓扑全群；遗忘图灵机群，其运动独立于纸带内容，推广了点灯者群并与通用可逆逻辑门研究相关；以及基本图灵机群，即通过组合有限状态自动机和遗忘图灵机得到的图灵机构成的群。我们证明了遗忘图灵机群和基本图灵机群都是有限生成的，而有限状态自动机群和可逆图灵机群则不是。我们进一步证明基本图灵机群的挠性问题不可判定。由此，我们还得出细胞自动机群（更一般地，任何不可数一维软性子移位自同构群）包含一个具有不可判定挠性问题的有限生成子群。我们还证明，在非平凡字母表上的全$\mathbb{Z}^d$移位的拓扑全群中，挠性问题不可判定当且仅当$d \geq 2$。