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Register a new userAn automaton group with undecidable order and Engel problems
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Authors
Pierre Gillibert
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For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and only if, this element is of finite order.If the Turing machine is universal, the corresponding automaton group has an undecidable order problem. This solves a problem raised by Grigorchuk.The above group also has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-iterated commutator [...[[g,h],h],...,h]$ is the identity or not. This solves a problem raised by Bartholdi.
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The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However we prove that, in the most general case, the problem is undecidable. We study the case of automaton semigroups. Given a NW-deterministic Wang tile set, we construct an Mealy automaton, such that the plane admit a valid Wang tiling if and only if the Mealy automaton generates a finite semigroup. The construction is similar to a construction by Kari for proving that the nilpotency problem for cellular automata is unsolvable. Moreover Kari proves that the tiling of the plane is undecidable for NW-deterministic Wang tile set. It follows that the finiteness problem for automaton semigroup is undecidable.
Let $G$ be a finite group of odd order admitting an involutory automorphism $phi$. We obtain two results bounding the exponent of $[G,phi]$. Denote by $G_{-phi}$ the set ${[g,phi],vert, gin G}$ and by $G_{phi}$ the centralizer of $phi$, that is, the subgroup of fixed points of $phi$. The obtained results are as follows.1. Assume that the subgroup $langle x,yrangle$ has derived length at most $d$ and $x^e=1$ for every $x,yin G_{-phi}$. Suppose that $G_phi$ is nilpotent of class $c$. Then the exponent of $[G,phi]$ is $(c,d,e)$-bounded.2. Assume that $G_phi$ has rank $r$ and $x^e=1$ for each $xin G_{-phi}$. Then the exponent of $[G,phi]$ is $(e,r)$-bounded.
We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS2003, LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challenges in Timed Languages, From Applied Theory to Basic Theory, Bulletin of the EATCS, Volume 83, p. 106-120, 2004 ]. In particular, we show that one cannot decide whether a given timed automaton is determinizable or whether the complement of a timed regular language is timed regular. We show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. It is also undecidable whether the shuffle of two timed regular languages is timed regular. We show that in the case of timed Buchi automata accepting infinite timed words some of these problems are Pi^1_1-hard, hence highly undecidable (located beyond the arithmetical hierarchy).
The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of free product followed by adjoining an identity. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an automaton semigroup, and show that a wreath product of an automaton monoid and a finite monoid arises as a generalized automaton semigroup in both senses. We also suggest a potential counterexample that would show that a wreath product of an automaton monoid and a finite monoid is not a necessarily an automaton monoid in the usual sense.
An improvement on earlier results on free products of automaton semigroups; showing that a free product of two automaton semigroups is again an automaton semigroup providing there exists a homomorphism from one of the base semigroups to the other. The result is extended by induction to give a condition for a free product of finitely many automaton semigroups to be an automaton semigroup.
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