An automaton group with undecidable order and Engel problems

Gillibert, Pierre

doi:10.1016/j.jalgebra.2017.11.049

Cited by 27 publications

(24 citation statements)

References 45 publications

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“…For this algorithm, one can use the naïve approach of enumerating all state sequences (in order for ascending length) until no further new group elements are found. The result closest to proving undecidability of Group Finiteness is due to Gillibert, who showed that the finiteness problem for automaton semigroups is undecidable [20], and the recent result on the undecidability of the order problem for automaton groups (checking whether a group element has finite order) [21].…”

Section: The Finiteness Problem For Invertible Bi-reversible Automatamentioning

confidence: 99%

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

2020

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In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary.First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i. e. a relation * The first author was supported by the Austrian Science Fund projects FWF P24028-N18 and FWF P29355-N35. † The second author thanks the project INDAM-GNSAGA. IntroductionAutomaton groups, i. e. groups generated by synchronous Mealy automata, are a quite intriguing class of groups. They have deep connections with many areas of mathematics, from the theory of profinite groups to complex dynamics and theoretical computer science, and they serve as a source of examples or counterexamples for many important group theoretic problems (see e. g. [40] for an introduction). Despite these connections and the many surprising and interesting consequences, knowledge about the class of automaton groups from the algebraic, algorithmic and dynamical perspective is still rather limited. From the algorithmic point of view, the word problem for automaton groups is decidable 1 while many other problems are suspected (and sometimes proven) to be undecidable. In this regard, the most studied problems in the literature are the finiteness problem, the freeness problem and the conjugacy problem. The latter has been proven to be undecidable by Šunić and Ventura [38]. However, if this problem is restricted to the contracting case, then it turns out to be decidable, see [5]. Decidability of the finiteness problem for automaton groups is still open. However, some partial results are known when certain properties of the generating automaton are relaxed: Belk and Bleak showed undecidability of the finiteness problem for groups generated by asynchronous automata [3] and Gillibert showed undecidability of the finiteness problem for automaton semigroups [20], which are generated by synchronous but not necessarily invertible Mealy automata. Other results worth mentioning in this respect are the recent proof by Gillibert showing undecidability of the order problem of automaton groups [21] and the result by Bartholdi and Mitrofanov that the order problem is already undecidable for contracting automaton groups [2]. On the other side, Klimann showed that the finiteness problem is solvable for reversible, invertible automata with two states or two letters and that so is the freeness problem for semigroups generated by two state invertible, reversible automata [28]. Obviously, it is usually easier to show undecidability results for automaton semigroups than it is for automaton groups. This might be one of the reasons why the less studied class of automaton semigroups seems to stir up more interest lately. There is, for example, the semigroup theoretic work of Cain [9], and Brough and Cain [6,7], approaches to semigroups via duals ...

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Section: The Finiteness Problem For Invertible Bi-reversible Automatamentioning

confidence: 99%

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

2020

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“…a G-automaton T = (Q, Σ, δ) and some q ∈ Q * Question: has q finite order in G (T )? is undecidable [3]. It is a special case of Free Subgroup Presentation and, therefore, the undecidability of Free Subgroup Presentation follows directly (and independently from the undecidability of IICP) from Gillibert's result.…”

Section: Inputmentioning

confidence: 90%

“…Again, the corrected version of Theorem 3.12 also follows from the undecidability of the order problem for automaton groups [3] (and the corrected version of Corollary 3.13 even follows from the undecidability of the order/torsion problem for automaton semigroups implied by [2, Lemma 3.11 and 3.12]).…”

Section: Inputmentioning

confidence: 99%

Erratum to “Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness”

2021

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“…For automaton semigroups, the order problem could be proved to be undecidable [13,Corollary 3.14]. Recently, this could be extended to automaton groups [14] (see also [4]). On the other hand, the undecidability result for the finiteness problem for automaton semigroups [13,Theorem 3.13] could not be lifted to automaton groups so far.…”

Section: Introductionmentioning

confidence: 99%

An Automaton Group with PSPACE-Complete Word Problem

Wächter,

Weiß

2019

Preprint

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We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D'Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a PSPACE-complete word problem and the second one is to utilize a construction used by Barrington to simulate circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

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An automaton group with undecidable order and Engel problems

Cited by 27 publications

References 45 publications

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

Erratum to “Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness”

An Automaton Group with PSPACE-Complete Word Problem

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