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Four heads are better than three
arXiv - CS - Formal Languages and Automata Theory Pub Date : 2020-03-12 , DOI: arxiv-2003.05706
Ville Salo

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, which roughly states that a Turing machine can enumerate numbers such that every Turing machine occasionally "correctly guesses" whether they are in the language (by halting on them or not), even if trying not to, and given an oracle for shorter identities. We prove that impredictable recursively enumerable sets exist. Combining these constructions and slightly adapting a result of [Salo and T\"orm\"a, 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 T\"orm\"a, 2017]. These are the first examples of groups where four heads are better than one, and they show the maximal height of a finite head hierarchy is indeed four.

中文翻译:

四头胜于三头

我们构造了递归表示的有限生成扭力群,这些扭力群具有有界扭力并且其词问题是合取等价的(特别是正的和图灵等价的)给定的递归可枚举集。这些群可以解释为有限状态机群或拓扑完整群的子群,对其他扭转群进行有效子位移。我们定义了一组自然数的递归理论属性,称为不可预测性,它粗略地指出,图灵机可以枚举数字,这样每个图灵机偶尔会“正确猜测”它们是否在语言中(通过停止或不停止) ),即使尽量不这样做,并为较短的身份提供了一个预言。我们证明了不可预测的递归可枚举集存在。结合这些构造并稍微调整 [Salo and T\"orm\"a, 2017] 的结果,我们得到四头群行走有限状态自动机可以比三头自动机在包含整数的副本,证实了 [Salo and T\"orm\"a, 2017] 的猜想。这些是四个头比一个头更好的组的第一个例子,它们表明有限头层次结构的最大高度确实是四个。
更新日期:2020-03-16
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