Lifts, derandomization, and diameters of Schreier graphs of Mealy automata

We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow’s rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion. This gives a partial answer to a question of Benjamini and Duminil-Copin.

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Supervisory Control of Discrete Event Systems Modeled by Mealy Automata with Nondeterministic Output Functions

Transactions of the Institute of Systems Control and Information Engineers ◽

10.5687/iscie.22.154 ◽

2009 ◽

Vol 22 (4) ◽

pp. 154-160 ◽

Cited By ~ 1

Author(s):

Toshimitsu Ushio ◽

Shigemasa Takai

Keyword(s):

Supervisory Control ◽

Discrete Event Systems ◽

Discrete Event ◽

Event Systems ◽

Mealy Automata

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ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

International Journal of Algebra and Computation ◽

10.1142/s021819671250052x ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250052 ◽

Cited By ~ 22

Author(s):

ALI AKHAVI ◽

INES KLIMANN ◽

SYLVAIN LOMBARDY ◽

JEAN MAIRESSE ◽

MATTHIEU PICANTIN

Keyword(s):

Decision Problem ◽

Decision Procedure ◽

Necessary Condition ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Mealy Automata ◽

Necessary And Sufficient ◽

New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

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Algebraic Specification and Coalgebraic Synthesis of Mealy Automata

Electronic Notes in Theoretical Computer Science ◽

10.1016/j.entcs.2006.05.030 ◽

2006 ◽

Vol 160 ◽

pp. 305-319 ◽

Cited By ~ 11

Author(s):

J.J.M.M. Rutten

Keyword(s):

Algebraic Specification ◽

Mealy Automata

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Numerical upper bounds on growth of automaton groups

International Journal of Algebra and Computation ◽

10.1142/s0218196722500072 ◽

2021 ◽

pp. 1-33

Author(s):

Jérémie Brieussel ◽

Thibault Godin ◽

Bijan Mohammadi

Keyword(s):

Group Theory ◽

Upper Bounds ◽

Geometric Invariant ◽

Finitely Generated ◽

Grigorchuk Group ◽

Finitely Generated Group ◽

Intermediate Growth ◽

Mealy Automata ◽

Optimal Bounds ◽

Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

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