Table Of ContentLINDENMAYER SYSTEMS: STRUCTURE,
LANGUAGES, AND GROWTH FUNCTIONS
P.M.B. VITANYI
LINDENMAYER SYSTEMS: STRUCTURE,
LANGUAGES, AND GROWTH FUNCTIONS
VRIJE UNIVERSITEIT TE AMSTERDAM
LINDENMAYER SYSTEMS: STRUCTURE,
LANGUAGES, AND GROWTH FUNCTIONS
ACADEMISCH PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN
DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN
AAN DE VRIJE UNIVERSITEIT TE AMSTERDAM,
OP GEZAG VAN DE RECTOR MAGNIFICUS
DR. D.M. SCHENKEVELD,
HOOGLERAAR IN DE FACULTEIT DER LETTEREN,
IN HET OPENBAAR TE VERDEDIGEN
OP VRIJDAG 28 APRIL 1978 TE 13.30 UUR
IN HET HOOFDGEBOUW DER UNIVERSITEIT,
DE BOELELAAN 1105
DOOR
PAUL MICHAEL B^LA VITANYI
GEBOREN TE BUDAPEST
1978
MATHEMATISCH CENTRUM, AMSTERDAM
PROMOTOR : PROF. DR. J.W. DE BARKER
COREFERENT: PROF. DR. A. SALOMAA
To.- THE L JUNGLE
dUcovened by ARJSTJV
containing many quaint life FORMS
in the WOODS
tike. TARZAW
and hii feiendly goKitia BOLGANI
and otheA GROWTH.
"But", I said, "Euler showed that hexagons
alone cannot enclose a volume"• To which
the innominate biologist retorted, "ttiat
proves the superiority of God over mathematics".
D'Arcy Thompson as quoted by
W. McCulloch in: Mysterium
Inequitatis of Sinful Man
Aspiring to the Place of God.
PREFACE
Usually, a Ph.D. Thesis reports on some scientific results, and, after
it has accomplished its purpose of being a catalyst in the transformation
of its author to a Ph.D., it is carved up into one or more pieces which are
presented to the scientific community in media with a wider circulation.
Here, I have followed the converse course. This work wants to present a
unified treatment of research, done by its author, most of which has been published
previously in reports, journals and conference proceedings. Where it was
necessary to my purpose I have drawn from the work of other investigators.
A bibliographical comment accompanies each chapter, disclosing its sources.
Whereas it has not been my contention to give a complete account of the
mathematical theory of L systems, part of the field seems reasonably
covered. The treatment of the subject is self-contained and, hopefully, easy
to follow, but it is obvious that a rudimentary knowledge of formal lan
guage theory is more or less required from the reader. For instance, a
glancing acquaintance with HOPCROFT and ULLMAN [1969], or SALOMAA [1973a],
will be helpful. Thus, Section 2.1 on formal grammars is intended as a
review of some elementary concepts, and to ensure uniform notation, but not
as a substitute for the required background.
In my investigations in L theory I have been helped along by
J.W. de Bakker, P.G. Doucet, G.T. Herman, J. van Leeuwen, A. Lindenmayer,
H. and J. Luck, G. Rozenberg, A. Salomaa, and W.J. Savitch.
Views expressed on the biological applicability of L systems, and the
merits of several attempts in that direction, are for the author's sole re
sponsibility, as are the views on the mathematical and the computer science
aspects of the same.
I wish to thank especially Aristid Lindenmayer for his encouragement
at the outset of my scientific work; my promotor Jaco de Bakker who is
also the head of the Computer Science Department at the Mathematical Centre?
and my coreferent Arto Salomaa. Prof. A. van Wijngaarden, director of the
Mathematical Centre, gave me the working environment in which this research
could take place.
The technical realization of this monograph was made possible by the
speedy and excellent typing of the manuscript by Linda Brown and Lenie Rijs,
the drawing of figures by Tobias Baanders, and the printing by Dick Zwarst,
Jan Schipper, Jan Suiker, Etienne Michel and Jos van der Werf; all at the
Mathematical Centre.
Hie front cover was designed by Tobias Baanders, using a fragment of
Botticelli's allegory La Primavera depicting Spring escaping from Winter.
Hie fragment symbolizes fertility and the beginning of growth of plants
which is associated with that season.
CONTENTS
Chapter 1 INTRODUCTION 1
Chapter 2 DEFINITIONS AND PRELIMINARIES 7
2.1 Formal grammars 7
2.2 Lindenmayer systems 10
2.3 Bibliographical comments 13
Chapter 3 L SYSTEMS, SEQUENCES, AND LANGUAGES 15
3.1 DOL systems 16
3.1.1 DOL languages 16
3.1.1.1 Functions which relate size of
language with size of alphabet 24
3.1.1.2 Asymptotic approximations of
S and P 29
3.1.1.3 Classification and closure
properties 33
3.1.2 Structure of DOL systems with applications
to growth functions, local catenativeness,
and characterizations 34
3.1.2.1 Growth functions 38
3.1.2.2 The locally catenative property 45
3.1.2.3 Regularity and context freeness 51
3.1.2.4 Biological interpretation 52
3.2 Deterministic context sensitive Lindenmayer
systems without tables 54
3.2.1 Lindenmayer systems and Turing machines 57
3.2.2 Nonrecursive L languages 59
3.2.3 Deterministic L languages and the Chomsky
hierarchy 62
3.2.4 Extensions and homomorphic closures of
deterministic L languages 65
3.2.5 Extensions and homomorphic closures of
propagating deterministic L languages 77
3.2.6 Combining the results of Section 3.2 86
3.3 Context sensitive table Lindenmayer systems
and a trade-off equivalent to the LBA problem 92
3.4 Stable string languages of L systems 100
3.4.1 Stable string languages of L systems
without tables 102
3.4.2 Stable string languages of table L
systems 105
3.4.3 Stable string languages of deterministic
table L systems 113
3.4.4 Relevance to theoretical biology and
formal language theory 115
3.5 Context variable L systems and some simple
regenerating structures 116
3.5.1 The extended French Flag problem 122
3.6 Bibliographical comments 126
Chapter 4 GROWTH FUNCTIONS 127
4.1 DOL growth functions: analytical approach 130
4.2 DOL growth functions: combinatorial approach 140
4.3 Growth functions of context sensitive L systems 144
4.3.1 Bounds on unbounded growth 146
4.3.2 Synthesis of context sensitive growth
functions 150
4.3.3 The hierarchy 154
4.3.4 Decision problems 160
4.4 Bibliographical comments 166
Chapter 5 PHYSICAL TIME GROWTH FUNCTIONS ASSOCIATED WITH
LINDENMAYER SYSTEMS OPERATING IN PHYSIOLOGICAL TIME 169
5.1 Sigmoidal growth functions of Lindenmayer systems
operating in physiological time 174
5.2 Some possible extensions and an interpretation
in terms of table L systems 182
5.3 Final remarks 184
5.4 Bibliographical comments 185
Chapter 6 EPILOGUE: EVALUATION OF RESULTS 187
Bibliography 195
Samenvatting 207
