Cellular Automata and Groups

460 Pages·2010·2.95 MB·English

by  Tullio Ceccherini-Silberstein

Springer Monographs in Mathematics
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Tullio Ceccherini-Silberstein (cid:2) Michel Coornaert
Cellular Automata
and Groups
TullioCeccherini-Silberstein MichelCoornaert
DipartimentodiIngegneria InstitutdeRechercheMathématiqueAvancée
UniversitàdelSannio UniversitédeStrasbourg
C.soGaribaldi107 7rueRené-Descartes
82100Benevento 67084StrasbourgCedex
Italy France
[e-mail] [e-mail]
ISSN1439-7382
ISBN978-3-642-14033-4 e-ISBN978-3-642-14034-1
DOI10.1007/978-3-642-14034-1
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010934641
MathematicsSubjectClassification(2010):37B15,68Q80,20F65,43A07,16S34,20C07
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Preface
Twoseeminglyunrelatedmathematicalnotions,namelythatofanamenable
group and that of a cellular automaton, were both introduced by John von
Neumann in the first half of the last century. Amenability, which originated
from the study of the Banach-Tarski paradox, is a property of groups gener-
alizing both commutativity and finiteness. Nowadays, it plays an important
role in many areas of mathematics such as representation theory, harmonic
analysis, ergodic theory, geometric group theory, probability theory, and dy-
namicalsystems.VonNeumannusedcellularautomatatoserveastheoretical
models for self-reproducing machines. About twenty years later, the famous
cellular automaton associated with the Game of Life was invented by John
Horton Conway and popularized by Martin Gardner. The theory of cellular
automata flourished as one of the main branches of computer science. Deep
connections with complexity theory and logic emerged from the discovery
that some cellular automata are universal Turing machines.
A group G is said to be amenable (as a discrete group) if the set of all
subsets of G admits a right-invariant finitely additive probability measure.
All finite groups, all solvable groups (and therefore all abelian groups), and
all finitely generated groups of subexponential growth are amenable. Von
Neumann observed that the class of amenable groups is closed under the
operationof taking subgroups and thatthe free groupof rank twoF isnon-
2
amenable. It follows that a group which contains a subgroup isomorphic to
F is non-amenable. However, there are examples of groups which are non-
2
amenable and contain no subgroups isomorphic to F (the first examples
2
of such groups were discovered by Alexander Y. Ol’shanskii and by Sergei
I. Adyan).
Loosely speaking, a general cellular automaton can be described as fol-
lows. A configuration is a map from a set called the universe into another
set called the alphabet. The elements of the universe are called cells and the
elements of the alphabet are called states. A cellular automaton is then a
mapfromthesetofallconfigurationsintoitselfsatisfyingthefollowinglocal
property:thestateoftheimageconfigurationatagivencellonlydependson
vii
viii Preface
the states of the initial configuration on a finite neighborhood of the given
cell.Intheclassicalsetting,forinstanceinthecellularautomataconstructed
by von Neumann and the one associated with Conway’s Game of Life, the
alphabet is finite, the universe is the two dimensional infinite square lattice,
andtheneighborhoodofacellconsistsofthecellitselfanditseightadjacent
cells.Byiteratingacellularautomatononegetsadiscretedynamicalsystem.
Such dynamical systems have proved very useful to model complex systems
arising from natural sciences, in particular physics, biology, chemistry, and
population dynamics.
∗
∗ ∗
Inthisbook,theuniversewillalwaysbeagroupG(intheclassicalsetting
the corresponding group was G = Z2) and the alphabet may be finite or
infinite. The left multiplication in G induces a natural action of G on the set
of configurations which is called the G-shift and all cellular automata will be
required to commute with the shift.
It was soon realized that the question whether a given cellular automaton
is surjective or not needs a special attention. From the dynamical viewpoint,
surjectivity means that each configuration may be reached at any time. The
firstimportantresultinthisdirectionisthecelebratedtheoremofMooreand
Myhill which gives a necessary and sufficient condition for the surjectivity of
a cellular automaton with finite alphabet over the group G=Z2. Edward F.
MooreandJohnR.Myhillprovedthatsuchacellularautomatonissurjective
if and only if it is pre-injective. As the term suggests it, pre-injectivity is a
weakernotionthaninjectivity.Moreprecisely,acellularautomatonissaidto
be pre-injective if two configurations are equal whenever they have the same
image and coincide outside a finite subset of the group. Moore proved the
“surjective⇒pre-injective”partandMyhillprovedtheconverseimplication
shortlyafter.OneoftenreferstothisresultastotheGardenofEdentheorem.
This biblical terminology is motivated by the fact that, regarding a cellular
automaton as a dynamical system with discrete time, a configuration which
is not in the image of the cellular automaton may only appear as an initial
configuration, that is, at time t=0.
Thesurprisingconnectionbetweenamenabilityandcellularautomatawas
established in 1997 when Antonio Mach`ı, Fabio Scarabotti and the first au-
thor proved the Garden of Eden theorem for cellular automata with finite
alphabets over amenable groups. At the same time, and completely inde-
pendently, Misha Gromov, using a notion of spacial entropy, presented a
more general form of the Garden of Eden theorem where the universe is
an amenable graph with a dense holonomy and cellular automata are called
maps of bounded propagation. Mach`ı, Scarabotti and the first author also
showed that both implications in the Garden of Eden theorem become false
if the underlying group contains a subgroup isomorphic to F . The question
2
whether the Garden of Eden theorem could be extended beyond the class of
Preface ix
amenable groups remained open until 2008 when Laurent Bartholdi proved
thattheMooreimplicationfailstoholdfornon-amenablegroups.Asaconse-
quence, the whole Garden of Eden theorem only holds for amenable groups.
Thisgivesanewcharacterizationofamenablegroupsintermsofcellularau-
tomata.Letusmentionthat,uptonow,thevalidityoftheMyhillimplication
for non-amenable groups is still an open problem.
FollowingWalterH.Gottschalk,agroupGissaidtobesurjunctiveifevery
injective cellular automaton with finite alphabet over G is surjective. Wayne
Lawton proved that all residually finite groups are surjunctive and that ev-
ery subgroup of a surjunctive group is surjunctive. Since injectivity implies
pre-injectivity,animmediateconsequenceoftheGardenofEdentheoremfor
amenable groups is that every amenable group is surjunctive. Gromov and
Benjamin Weiss introduced a class of groups, called sofic groups, which in-
cludes all residually finite groups and all amenable groups, and proved that
everysoficgroupissurjunctive.Soficgroupscanbedefinedinthreeequivalent
ways: in terms of local approximation by finite symmetric groups equipped
withtheirHammingdistance,intermsoflocalapproximationoftheirCayley
graphs by finite labelled graphs, and, finally, as being the groups that can
be embedded into ultraproducts of finite symmetric groups (this last charac-
terization is due to Ga´bor Elek and Endre Szabo´). The class of sofic groups
is the largest known class of surjunctive groups. It is not known, up to now,
whether all groups are surjunctive (resp. sofic) or not.
Stimulated by Gromov ideas, we considered cellular automata whose al-
phabets are vector spaces. In this framework, the space of configurations has
a natural structure of a vector space and cellular automata are required to
be linear. An analogue of the Garden of Eden theorem was proved for linear
cellularautomatawithfinitedimensionalalphabetsoveramenablegroups.In
the proof, the role of entropy, used in the finite alphabet case, is now played
by the mean dimension, a notion introduced by Gromov. Also, examples of
linear cellular automata with finite dimensional alphabets over groups con-
taining F showing that the linear version of the Garden of Eden theorem
2
may fail to hold in this case, were provided. It is not known, up to now, if
the Garden of Eden theorem for linear cellular automata with finite dimen-
sional alphabet only holds for amenable groups or not. We also introduced
the notion of linear surjunctivity: a groupG is said to be L-surjunctive if ev-
ery injective linear cellular automaton with finite dimensional alphabet over
G is surjective. We proved that every sofic group is L-surjunctive. Linear
cellular automata over a group G with alphabet of finite dimension d over
a field K may be represented by d×d matrices with entries in the group
ring K[G]. This leads to the following characterization of L-surjunctivity: a
groupisL-surjunctiveifandonlyifitsatisfiesKaplansky’sconjectureonthe
stable finiteness of group rings (a ring is said to be stably finite if one-sided
invertiblefinitedimensionalsquarematriceswithcoefficientsinthatringare
in fact two-sided invertible). As a corollary, one has that group rings of sofic
groups are stably finite, a result previously established by Elek and Szabo´