Undecidable Word Problem in Subshift Automorphism Groups

Guillon, P.; Jeandel, Emmanuel; Kari, Jarkko; Vanier, Pascal

doi:10.1007/978-3-030-19955-5_16

Cited by 3 publications

(2 citation statements)

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“…Topological full groups (already on Z) are a well-known source of interesting examples of groups [10,6,9], and our groups can also be interpreted as subgroups of topological full groups of subshifts on torsion groups. In a symbolic dynamic context, [7] (independently) uses a similar construction to prove that the automorphism groups of multidimensional SFTs can have undecidable word problem.…”

Section: Some Relevant Existing Workmentioning

confidence: 99%

Four heads are better than three

Salo¹

2020

Preprint

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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 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 one, and they show the maximal height of a finite head hierarchy is indeed four.

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Section: Some Relevant Existing Workmentioning

confidence: 99%

Four heads are better than three

Salo¹

2020

Preprint

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“…In this section, we pinpoint the difficulty of the word problem for f.g. subgroups of automorphism groups of full shifts, since this complexity-theoretic statement does not seem to appear anywhere, and because the word problem is the first problem on Dehn's list whose second item our paper concerns. In [30], a similar method is used to show that the word problem of a finitely-generated subgroup of the automorphism group of a two-dimensional subshift of finite type can be undecidable.…”

Section: The Word Problemmentioning

confidence: 99%

Conjugacy of reversible cellular automata and one-head machines

Salo¹

2020

Preprint

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We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of f.g. groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every f.g. subgroup containing both has undecidable conjugacy problem. We say that conjugacy in these groups is eventually locally undecidable. We also prove that the Brin-Thompson group 2V and groups of reversible Turing machines have undecidable conjugacy problems, and show that the word problems of the automorphism group and the topological full group of every full shift are eventually locally co-NP-complete.

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Four Heads are Better than Three

Salo

2020

Cellular Automata and Discrete Complex Systems

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Undecidable Word Problem in Subshift Automorphism Groups

Abstract: This article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree.

Cited by 3 publications

References 11 publications

Four heads are better than three

Four heads are better than three

Conjugacy of reversible cellular automata and one-head machines

Four Heads are Better than Three

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