四个脑袋总比三个好
Four heads are better than three
摘要：
构造了具有有界挠率且其字问题与给定的递归可枚举集合取等价（特别是正的和图灵等价的）的递归表示有限生成挠群.这些群可以被解释为有限状态机的群或拓扑完全群的子群，在其他扭群上的有效子移位上。我们定义了一组自然数的递归理论属性，称为不可预测性。它粗略地说，一个图灵机可以枚举数字，使得每一个图灵机偶尔错误地猜测（通过停止或不停止）它们是否在集合中，即使给定一个集合前缀的预言。我们证明了不可预测递归可预测集的存在。结合这些构造并稍微修改[Salo and Törmä，2017]的结果，我们得到四头群行走有限状态自动机可以在包含整数副本的群上定义比三头自动机严格更多的子移位，证实了[Salo and Törmä，2017]的猜想。这是第一个四个人头比三个人头好的群体的例子，它们表明有限人头等级的最大高度确实是四。
英文摘要：
We construct recursively-presented finitely-generated torsion groups which have bounded torsion and whose word problem is conjunctive equivalent (in particular positive and Turing equivalent) to a given recursively enumerable set. These groups can be interpreted as groups of finite state machines or as subgroups of topological full groups, on effective subshifts over other torsion groups. We define a recursion-theoretic property of a set of natural numbers, called impredictability. It roughly states that a Turing machine can enumerate numbers such that every Turing machine occasionally incorrectly guesses (by either halting or not) whether they are in the set, even given an oracle for a prefix of the set. We prove that impredictable recursively enumerable sets exist. Combining these constructions and slightly adapting a result of [Salo and Törmä, 2017], we obtain that four-headed group-walking finite-state automata can define strictly more subshifts than three-headed automata on a group containing a copy of the integers, confirming a conjecture of [Salo and Törmä, 2017]. These are the first examples of groups where four heads are better than three, and they show the maximal height of a finite head hierarchy is indeed four.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
