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

2020 ◽  
Vol 237 (1) ◽  
pp. 15-52 ◽  
Author(s):  
Daniele D’Angeli ◽  
Emanuele Rodaro ◽  
Jan Philipp Wächter
Keyword(s):  
Automaton Semigroups

scholarly journals Infinite automaton semigroups and groups have infinite orbits

2020 ◽  
Vol 553 ◽  
pp. 119-137 ◽  
Author(s):  
Daniele D'Angeli ◽  
Dominik Francoeur ◽  
Emanuele Rodaro ◽  
Jan Philipp Wächter
Keyword(s):  
Automaton Semigroups

scholarly journals On the structure theory of partial automaton semigroups

2020 ◽  
Vol 101 (1) ◽  
pp. 51-76
Author(s):  
Daniele D’Angeli ◽  
Emanuele Rodaro ◽  
Jan Philipp Wächter
Keyword(s):  
Structure Theory ◽  
Automaton Semigroups

scholarly journals THE FINITENESS PROBLEM FOR AUTOMATON SEMIGROUPS IS UNDECIDABLE

2014 ◽  
Vol 24 (01) ◽  
pp. 1-9 ◽  
Author(s):  
PIERRE GILLIBERT
Keyword(s):  
Cellular Automata ◽  
Mealy Automaton ◽  
Wang Tile ◽  
Positive Results ◽  
Automaton Semigroups

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 a Mealy automaton, such that the plane admits a valid Wang tiling if and only if the Mealy automaton generates a infinite 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 semigroups is undecidable.

scholarly journals Self-automaton semigroups

2014 ◽  
Vol 90 (1) ◽  
pp. 189-206 ◽  
Author(s):  
Alexander McLeman
Keyword(s):  
Automaton Semigroups

scholarly journals A New Hierarchy for Automaton Semigroups

2021 ◽  
pp. 1-21
Author(s):  
Laurent Bartholdi ◽  
Thibault Godin ◽  
Ines Klimann ◽  
Camille Noûs ◽  
Matthieu Picantin
Keyword(s):  
The State ◽  
Order Problem ◽  
Asymptotic Behaviors ◽  
State Activity ◽  
The Family ◽  
Exponential Part ◽  
Automaton Semigroups

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable whenever the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family. We extend the aforementioned hierarchy via a semi-norm making it more coarse but somehow more robust and we prove that the Order Problem is still decidable for the first two levels of this alternative hierarchy.

Automaton Semigroups: The Two-state Case

2014 ◽  
Vol 58 (4) ◽  
pp. 664-680 ◽  
Author(s):  
Ines Klimann
Keyword(s):  
State Case ◽  
Automaton Semigroups

scholarly journals Orbit expandability of automaton semigroups and groups

2020 ◽  
Vol 809 ◽  
pp. 418-429 ◽  
Author(s):  
Daniele D'Angeli ◽  
Emanuele Rodaro ◽  
Jan Philipp Wächter
Keyword(s):  
Automaton Semigroups

scholarly journals CAYLEY AUTOMATON SEMIGROUPS

2009 ◽  
Vol 19 (01) ◽  
pp. 79-95 ◽  
Author(s):  
VICTOR MALTCEV
Keyword(s):  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Automaton Semigroups

In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite.

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