Small Universal Cellular Automata in Hyperbolic Spaces: A Collection of Jewels

327 Pages·2013·45.519 MB·English

by  Maurice Margenstern (auth.)

Maurice Margenstern
EMERGENCE, C
COMPLEXITY
C
AND
COMPUTATION E
Small Universal
Cellular Automata
in Hyperbolic Spaces
A Collection of Jewels
123
Emergence, Complexity and Computation 4
SeriesEditors
Prof.IvanZelinka Prof.AndrewAdamatzky
TechnicalUniversityofOstrava UnconventionalComputingCentre
CzechRepublic UniversityoftheWestofEngland
Bristol
Prof.GuanrongChen UnitedKingdom
CityUniversityofHongKong
EditorialBoard
OttoRössler JuanC.Burguillo
InstituteofPhysicalandTheoretical UniversityofVigo,
Chemistry,Tübingen, Spain
Germany SergejCˇelikovský
EdwardOtt AcademyofSciencesofthe
UniversityofMaryland, CzechRepublic,
USA CzechRepublic
AjithAbraham DonaldDavendra
MirLabs, TechnicalUniversityofOstrava,
USA CzechRepublic
VaclavSnasel JuanA.Rodriguez-Aguilar
TechnicalUniversityofOstrava, IIIA-CSIC,
CzechRepublic Spain
AnaLuciaC.Bazzan
JouniLampinen
UniversidadeFederaldoRioGrandedoSul,
UniversityofVaasa,
PortoAlegreRSBrasil
Finland
GheorghePa˘un
EmilioCorchado
RomanianAcademy,Bucharest,
UniversityofSalamanca,
Romania
Spain
LinqiangPan
HendrikRichter
HuazhongUniversityofScienceand
HTWKLeipzigUniversityof
Technology,Wuhan,
AppliedSciences,
China
Germany
IvoVondrák
MartinMiddendorf
TechnicalUniversityofOstrava,
UniversityofLeipzig,
CzechRepublic
Germany
AndrewIlachinski
MohammedChadli
CenterforNavalAnalyses,
UniversityofJulesVerne,
USA
France
Forfurthervolumes:
http://www.springer.com/series/10624
Maurice Margenstern
Small Universal
Cellular Automata
in Hyperbolic Spaces
A Collection of Jewels
ABC
Author
MauriceMargenstern
UniversitédeLorraine,LITA
CampusduSaulcy
METZ
France
ISSN2194-7287 e-ISSN2194-7295
ISBN978-3-642-36662-8 e-ISBN978-3-642-36663-5
DOI10.1007/978-3-642-36663-5
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013932465
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
Thisbookisaboutsmalluniversalcellularautomatainhyperbolicspaces.Asthe
subtitlesuggests,ithasanaestheticsidewhich,Ihope,thereaderwillappreciate.
This aesthetic aspectwasone partofthe motivations for me to tacklethis topic
and to persevere in the task.
I remember that at the beginning of my carrier, more than forty years ago,
I was giving a temporary lecture in a school specially devoted to the training
of future teachers in primary schools. At that time, it was still a school for
womenonly1 andthe classroomwas full of many younggirls. I decided to teach
them a short introduction to hyperbolic geometry. I remember how astonished
my students were andalthoughmost ofthem claimed they understandnothing,
they were fascinated as I was myself by the beauty of the topic. For various
reasons,Inevertaughtthelecturesanymore,noteventhenextyearwhereIwas
still giving lectures in the same school. However,I told myselfthat, sometime, I
shouldgobacktothistopic.Icoulddothatindeed,somethingliketwentyseven
years later...
If somebody would tell me at that time that my return to this topic would
not only be a pleasure for me but also a successful adventure, I certainly would
not believe in such a flattering prediction. But let me deepen the topic with a
personalanecdote.Inanofficialreportin2008totheMinisterialFrenchAgency
supervising the research teams of the French universities, about my personal
perspective for 2009-2012, I wrote: ”I do not know what I shall find tomorrow,
all the more for 2009 and for what is 2012, it is not worth to say anything”.
A few months later, a committee of this agency visited my department for an
evaluation.Afterapresentationofmyteam,itwasapublicpresentationinfront
ofthemembersofthedepartmentandthegraduatedstudents.Thefirstquestion
I was asked was about the above words: ”Did you actually wrote these words”.
I said ”yes”. ”Do you confirm them?” Again, I said ”yes”. ”How can you say
so?” I justified my answer as follows. ”When I turned to the subject in 1997, I
certainlywouldbeastonishedifIwouldbetoldthatayearlater,Iwouldopena
1 Feminists may think that, happily, such schools no more exist. The question was a
longtimediscussedinFrance:theexistenceofsuchschoolswasindeedanimportant
contributiontothepromotionofmanywomenwho,withoutsuchinstitutionswould
certainly not reach the level the schools permitted. Also, the jobs offered after the
school was probably theonly placein theFrenchsociety wherewomen received the
same salary as men while performing thesame job.
VI Preface
new pathinthe subject, confirmedby a definite resultafter lessthantwo years.
If I was told in 2000, after that result, that seven years later I would solve a
more than, at that time, thirty years old conjecture, I would certainly laugh at
itandraisethe shoulders.Thisis whyIcannotsaywhatI shallfindtomorrow”.
Two books, [15, 17] published in 2007, 2008 respectively, crowned this ad-
venture. It happened that the adventure was not yet finished. I could obtain in
2008 an important result and in 2012, I succeeded to give a partial but decisive
answer to a challenge that Donald Knuth put on me: to find out an equivalent
of the game of life in the hyperbolic plane. Chapter 5 is devoted to the solution
I could find in infinitely many tilings. It is not the game of life, it is a some-
how new implementation of a railway circuit. Accordingly, this book offers new
and important developments with respect to [15, 17]. Also, for the first time, a
chapter is devoted to the consideration of the programming tools I personally
developedfor my own researchand, in particular and not the least, to illustrate
my papers and this book. Although the visibility is a very difficult question in
hyperbolicspaces,IwouldprobablynotobtaintheresultsIcouldreachwithout
the explorationI could perform with the programmingof the tools presented in
Chapter 8. Especially with deterministic cellular automata, checking the non-
contradictionofasetofrulesisextremelytediouswithoutthehelpofacomputer
program.
I found it could be interesting for the reader to provide him/her with an
introductiontohyperbolicgeometry,includingashorthistoricalaccount.Ifound
it appropriate to start the book by a personal answer to this question: why
hyperbolic geometry?
Contents
1 Why Hyperbolic Geometry? 1
1.1 An Important Discovery . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Nineteen Centuries of Wandering . . . . . . . . . . . . . . 2
1.1.2 The Precursors . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 The Discovery . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Consequences of an Earthquake in the Minds . . . . . . . 6
1.2 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Reflections in a Line . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 The Sum of Angles in a Triangle . . . . . . . . . . . . . . 11
1.2.4 Three Points not on the Same Line . . . . . . . . . . . . . 12
1.2.5 Ends, Horocycles and Equidistant Curves . . . . . . . . . 14
1.3 Poincar´e’s Disc Model and Its Generalizations . . . . . . . . . . . 17
1.3.1 Poincar´e’sDisc in the Hyperbolic Plane . . . . . . . . . . 17
1.3.2 Poincar´e’sBall in the Hyperbolic 3D-Space . . . . . . . . 22
1.4 Tilings of the Hyperbolic Plane . . . . . . . . . . . . . . . . . . . 23
2 Cellular Automata and the Railway Model 27
2.1 Cellular Automata in Hyperbolic Spaces . . . . . . . . . . . . . . 28
2.1.1 Coordinates in the Pentagrid . . . . . . . . . . . . . . . . 28
2.1.2 Coordinates in the Heptagrid . . . . . . . . . . . . . . . . 36
2.1.3 Coordinates in the Dodecagrid . . . . . . . . . . . . . . . 39
2.2 The Railway Model . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.2 Its Hyperbolic Implementations . . . . . . . . . . . . . . . 55
2.2.3 Universality and Weak Universality. . . . . . . . . . . . . 55
3 In the Pentagrid 57
3.1 The Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 The Backgroundand the Tracks . . . . . . . . . . . . . . 57
3.1.2 The Crossings. . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1.3 The Switches . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 The Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Rules for the Crossings . . . . . . . . . . . . . . . . . . . 71
3.2.2 Rules for the Switches . . . . . . . . . . . . . . . . . . . . 73
VIII Contents
4 In the Heptagrid 79
4.1 A Weakly Universal Cellular Automaton with Six States . . . . . 79
4.2 A Weakly Universal Cellular Automaton with Four States . . . . 84
4.2.1 Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.2 Crossings and Switches . . . . . . . . . . . . . . . . . . . 89
4.2.3 The Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.4 Rules for the Tracks . . . . . . . . . . . . . . . . . . . . . 100
4.2.5 Rules for the Crossings . . . . . . . . . . . . . . . . . . . 101
4.2.6 Rules for the Switches . . . . . . . . . . . . . . . . . . . . 107
5 In the Tilings {p,3} 115
5.1 Cellular Automata on the Grids {p,3} . . . . . . . . . . . . . . . 116
5.1.1 Coordinates in the Grids {p,3} . . . . . . . . . . . . . . . 116
5.1.2 Rules for Cellular Automata on the Grids {p,3} . . . . . 118
5.2 A Universal Cellular Automaton in the Grid {13,3} . . . . . . . 119
5.2.1 Representations in the Grid {13,3} . . . . . . . . . . . . . 119
5.2.2 The Scenario of the Present Implementation. . . . . . . . 120
5.2.3 The Implementation of the Tracks . . . . . . . . . . . . . 123
5.2.4 The Crossings. . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.5 The Switches . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 The Rules for a Weakly Universal Cellular Automaton on {13,3} 134
5.3.1 The Computer Programand the Rules . . . . . . . . . . . 134
5.3.2 Implementation of the Configurations . . . . . . . . . . . 136
5.3.3 Rules for the Tracks . . . . . . . . . . . . . . . . . . . . . 137
5.3.4 Rules for the Crossings . . . . . . . . . . . . . . . . . . . 138
5.3.5 The Rules for the Switches . . . . . . . . . . . . . . . . . 146
5.4 The Rules for {p,3}, p≥17 . . . . . . . . . . . . . . . . . . . . . 171
5.4.1 Patterns and Rules for the Tracks . . . . . . . . . . . . . 172
5.4.2 Patterns and Rules for the Crossings . . . . . . . . . . . . 175
5.4.3 Patterns and Rules for the Fixed Switch . . . . . . . . . . 179
5.4.4 Patterns and Rules for the Flip-Flop . . . . . . . . . . . . 180
5.4.5 Patterns and Rules for the Memory Switch . . . . . . . . 183
6 In the Dodecagrid 191
6.1 A Weakly Universal Cellular Automaton with Three States . . . 192
6.1.1 Implementation of the Tracks . . . . . . . . . . . . . . . . 192
6.1.2 Implementation of the Switches . . . . . . . . . . . . . . . 199
6.1.3 Devising the Rules and Checking Their Correctness . . . 215
6.2 A Weakly Universal Cellular Automaton with Two States . . . . 223
6.2.1 A New Implementation in the Dodecagrid . . . . . . . . . 224
6.2.2 Implementing the Switches . . . . . . . . . . . . . . . . . 232
6.3 The Rules and the Computer Program . . . . . . . . . . . . . . . 242
6.3.1 The Rules for the Tracks . . . . . . . . . . . . . . . . . . 243
6.3.2 The Rules for the Flip-Flop Switch . . . . . . . . . . . . . 245
6.3.3 The Rules for the Memory Switch . . . . . . . . . . . . . 246
Contents IX
7 Strongly Universal Hyperbolic Cellular Automata 249
7.1 Embedding 1D-Cellular Automata . . . . . . . . . . . . . . . . . 249
7.1.1 Proof of Theorem 7 and Its Corollary . . . . . . . . . . . 250
7.1.2 Refining Theorem 7 . . . . . . . . . . . . . . . . . . . . . 253
7.2 Strong Universality and the Halting Problem . . . . . . . . . . . 260
7.2.1 Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.2.2 Implementation of the 1D-Structure . . . . . . . . . . . . 262
7.2.3 Continuing the Simulation . . . . . . . . . . . . . . . . . . 268
7.2.4 Reducing the Number of States of A . . . . . . . . . . . . 274
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
8 The Help of Programming 281
8.1 About the Computer Programs . . . . . . . . . . . . . . . . . . . 282
8.1.1 Programs for the Plane . . . . . . . . . . . . . . . . . . . 283
8.1.2 Programs for the Dodecagrid . . . . . . . . . . . . . . . . 291
8.2 The PostScript Programs . . . . . . . . . . . . . . . . . . . . . . 292
8.2.1 Preliminary Computations. . . . . . . . . . . . . . . . . . 293
8.2.2 A Quick Introduction to PostScript Programming . . . . 298
8.2.3 Procedures for the Pentagridand for the Heptagrid. . . . 304
9 Further Work 315
References 317
Index 319
Chapter 1
Why Hyperbolic Geometry?
Abstract. In this first chapter, I shall introduce the unfamiliar reader to hy-
perbolic geometry. Section 1.1 is a very short survey about the history of this
geometry.Thishistoryisaveryimportantstepinthegeneralhistoryofsciences.
But this would require almost a whole book. Another summary can be found
in [34] where much interesting information is given about hyperbolic geometry,
inparticularfromanaesthetic pointofview. InSection1.2, I givea few general
indications on this geometry. In Section 1.3, I introduce the Poincar´e’s model
which is intensively used in the other chapters of the book.
May be my first answer to the title of this chapter would be: why not?
I remember a talk I gave about a universal cellular automaton in hyperbolic
spaces at a conference. When the chairman asked about questions, a very well
knownparticipantaskedme:”Inwhatisthisuseful?”.Isaid:”Idon’tknow”.He
insisted: ”What is the reasonfor that?”. I thought: ”I don’t care”, but thinking
that it would not sound very pleasant I said: ”I like it. I like it very much. For
me it is a very good reason.” A large smile went over the audience and the
participant himself also smiled.
Hyperbolic geometry is not a very well known topic. Mathematicians heard
aboutitbutonlyafewofthemarefamiliarwithit.Probably,physicistsaremore
familiarwith this geometrythanmathematicians, especially astrophysicistsand
cosmologists and also people familiar with the theory of relativity. Of course,
it may be surprising that a computer scientist ventures in this domain which
seems to be a kind of chasse gard´ee of very few specialists. I remember that
at the beginning of my adventures in this land, I was asked my passport by
somebody who found very awkward that a computer scientist dared marching
on this protected area. Fortunately, during my wanderings in this wonderful
land,I metwithmanyopenmindedpeople.Itwasabigpleasureforme tohave
very interesting discussions with them.
M.Margenstern:SmallUniversalCellularAutomatainHyperbolicSpaces,ECC4,pp.1–26.
DOI:10.1007/978-3-642-36663-5_1 (cid:2)c Springer-VerlagBerlinHeidelberg2013