Peter Dittrich - uni-jena.deusers.minet.uni-jena.de/~dittrich/p/main.pdf · On Artiﬂcial Chemistries Dissertation to receive the degree Dr. rer. nat. from the Department of Computer

On Artificial Chemistries

Dissertationzur Erlangung des Grades einesDoktors der Naturwissenschaften

der Universitat Dortmundam Fachbereich Informatik

von

Peter Dittrich

Dortmund

2001

Tag der mundlichen Prufung: 25.01.2001

Dekan: Prof. Dr. Bernd ReuschVorsitzender der Prufungskommission: Prof. Dr. Heinrich Muller1. Gutachter und Betreuer: Prof. Dr. Wolfgang Banzhaf2. Gutachterin: Prof. Dr. Susanne AlbersWissenschaftlicher Mitarbeiter in der Prufungskommission: Dr. Thomas Jansen

On Artificial Chemistries

Dissertationto receive the degree Dr. rer. nat. from the

Department of Computer ScienceUniversity of Dortmund

Germany

submitted by

Peter DittrichChair of Systems Analysis

Department of Computer ScienceUniversity of Dortmund

Germany

Dortmund

2001

submitted: November 27, 2000printed: March 7, 2001

Peter DittrichInformatik XIUniversity of DortmundD-44221 Dortmunddittrich@LS11.cs.uni-dortmund.dels11-www.cs.uni-dortmund.de/people/dittrich(+49) 231 9700 956

Oral examination: January 25, 2001

Dean: Prof. Dr. Bernd ReuschMembers of the board of examiners:Prof. Dr. Heinrich Muller (Chairman)Prof. Dr. Wolfgang Banzhaf (1. referee and supervisor)Prof. Dr. Susanne Albers (2. referee)Dr. Thomas Jansen

meinen Eltern

Contents

1 Introduction 13

1.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.1 What is Artificial Chemistry? . . . . . . . . . . . . . . . . . . 14

1.1.2 Definition of Molecules: Explicit or Implicit . . . . . . . . . . 16

1.1.3 Definition of Reaction Laws: Explicit or Implicit . . . . . . . . 17

1.1.4 Level of Abstraction: Analogous or Abstract . . . . . . . . . . 17

1.1.5 Dynamical Simulation of Artificial Chemistries . . . . . . . . . 18

1.1.6 Constructive Dynamical Systems . . . . . . . . . . . . . . . . 21

1.1.7 Random Chemistries . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.8 Models of Space . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.9 Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2 Three Artificial Chemistries . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.1 AC1 - Mathematical Expression as a Reaction Mechanism . . 23

1.2.2 AC2 - A Strange Artificial Chemistry . . . . . . . . . . . . . . 28

1.2.3 AC3 - Chaos and Order . . . . . . . . . . . . . . . . . . . . . 35

2 Motivation for Artificial Chemistry Research 39

2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.1 Artificial Chemistries as Models of Chemical Systems . . . . . 40

2.1.2 Artificial Chemistries as Models of Non-Chemical Systems . . 41

2.1.3 Artifical Chemistries as Sub-Systems of Complex Models . . . 42

2.1.4 Explaining General Phenomena and Mechanisms . . . . . . . 45

2.2 Information Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.1 Real Chemical Computing . . . . . . . . . . . . . . . . . . . . 47

2.2.2 Artificial Chemical Computing . . . . . . . . . . . . . . . . . . 48

2.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Discussion of a Potential Theory for Constructive Dynamical Systems 48

2.4.1 Current State . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.2 Components of a Theory . . . . . . . . . . . . . . . . . . . . . 49

2.5 Artificial Chemistry and Evolution Theory . . . . . . . . . . . . . . . 50

2.6 Strange Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 53

7

8 CONTENTS

3 Artifical Chemistry Approaches 553.1 Rewriting and Production Systems . . . . . . . . . . . . . . . . . . . 56

3.1.1 Lambda-Calculus (AlChemy) . . . . . . . . . . . . . . . . . . 563.1.2 The Chemical Abstract Machine (CHAM) . . . . . . . . . . . 583.1.3 The Chemical Rewriting System on Multisets (ARMS) . . . . 593.1.4 The Chemical Casting Model (CCM) . . . . . . . . . . . . . . 603.1.5 The Random Prolog Processor . . . . . . . . . . . . . . . . . . 61

3.2 Abstract Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.1 Artificial Molecular Machines . . . . . . . . . . . . . . . . . . 623.2.2 Polymer-Interaction by Turing Machines . . . . . . . . . . . . 633.2.3 Machine-Tape Interaction . . . . . . . . . . . . . . . . . . . . 643.2.4 Automata Reaction . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.1 Matrix-Multiplication Chemistry . . . . . . . . . . . . . . . . 673.3.2 Simple Arithmetic Operators . . . . . . . . . . . . . . . . . . 68

3.4 Assembler Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.1 Core War . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.2 Coreworld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.3 Tierra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4.4 Spontaneous Emergence of Self-Replicating Programs . . . . . 723.4.5 Avida . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Lattice Molecular Systems . . . . . . . . . . . . . . . . . . . . . . . . 733.5.1 Autopoietic System . . . . . . . . . . . . . . . . . . . . . . . . 733.5.2 Lattice Polymers . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.3 Lattice Molecular Automaton (LMA) . . . . . . . . . . . . . . 753.5.4 Self-Replicating Cell . . . . . . . . . . . . . . . . . . . . . . . 77

3.6 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6.1 Mechanical Artificial Chemistry . . . . . . . . . . . . . . . . . 773.6.2 The Chemical Metaphor in Cellular Automata . . . . . . . . . 783.6.3 Typogenetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Methods for Analysis and Visualization 834.1 Microscopic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1.1 Population Structure over Time . . . . . . . . . . . . . . . . . 844.1.2 Reaction Table . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.3 Monitoring Single Collisions . . . . . . . . . . . . . . . . . . . 85

4.2 Macroscopic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.1 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Distance Distribution Complexity . . . . . . . . . . . . . . . . 864.2.3 Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.4 Innovativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Mesoscopic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.1 Compound Objects . . . . . . . . . . . . . . . . . . . . . . . . 894.3.2 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

CONTENTS 9

4.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.4 Quantitative Description of Evolution . . . . . . . . . . . . . . . . . . 101

4.4.1 Evolutionary Activity . . . . . . . . . . . . . . . . . . . . . . . 101

5 Matrix Multiplication Chemistry 1055.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2 Toolbox of Fundamental Operations . . . . . . . . . . . . . . . . . . . 106

5.2.1 Random Strings . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.2 Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2.3 String-String Multiplication . . . . . . . . . . . . . . . . . . . 1105.2.4 Theta Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.5 Matrix-String Multiplication . . . . . . . . . . . . . . . . . . . 1125.2.6 Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.7 Parameter Extraction . . . . . . . . . . . . . . . . . . . . . . . 1135.2.8 Filter Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2.9 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.10 Toolbox Overview . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Matrix Reactions with Fixed Length 1176.1 General Parameter Setting and Restrictions . . . . . . . . . . . . . . 1176.2 The M196B Matrix Reaction . . . . . . . . . . . . . . . . . . . . . . . 118

6.2.1 Simulation of M196B . . . . . . . . . . . . . . . . . . . . . . . 1196.3 The M196A Matrix Reaction . . . . . . . . . . . . . . . . . . . . . . . 126

6.3.1 Simulation of M196A . . . . . . . . . . . . . . . . . . . . . . . 1276.4 Multilevel Syntactical Similarity . . . . . . . . . . . . . . . . . . . . . 127

6.4.1 Quantitative Analysis - The Similarity Characteristics . . . . . 1276.4.2 Syntactical and Semantic Closure in the M196B Reaction . . . 133

6.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Matrix Reaction with Variable Length 1437.1 The MV Family of Variable Length Matrix Reactions . . . . . . . . . 144

7.1.1 The MV1 Reaction . . . . . . . . . . . . . . . . . . . . . . . . 1457.1.2 Macroscopic Behavior of the MV1 Reaction . . . . . . . . . . 1477.1.3 The MV2 Reaction . . . . . . . . . . . . . . . . . . . . . . . . 1577.1.4 Macroscopic Behavior of the MV2 Reaction . . . . . . . . . . 1587.1.5 Syntactical Structure for Horizontal Folding (MV2) . . . . . . 1657.1.6 Syntactical Structure for Vertical Folding (MV2NTopV) . . . 1687.1.7 Similarity Characteristics . . . . . . . . . . . . . . . . . . . . . 1717.1.8 Self-Evolution in Simulations with the MV2 Reaction . . . . . 177

7.2 Constructive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 1787.2.1 Stability of Constructive Dynamical Systems . . . . . . . . . . 1797.2.2 Stability of the Constructive Equilibrium . . . . . . . . . . . . 1807.2.3 An ODE Model for the Constructive Equilibrium . . . . . . . 180

7.3 Matrix Reactions with Binding Position . . . . . . . . . . . . . . . . 183

10 CONTENTS

7.3.1 The MVB1 Reaction . . . . . . . . . . . . . . . . . . . . . . . 183

7.3.2 The MVB2 Reaction . . . . . . . . . . . . . . . . . . . . . . . 184

7.3.3 The MVB3 Reaction . . . . . . . . . . . . . . . . . . . . . . . 184

7.3.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3.5 Behavior of the MVB1 and MVB2 Reaction . . . . . . . . . . 186

7.3.6 Behavior of the MVB3 Reaction . . . . . . . . . . . . . . . . . 189

7.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 201

8 The Seceder Model 205

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.2 The Fundamental Seceder Model . . . . . . . . . . . . . . . . . . . . 207

8.3 Behavior of the Fundamental Seceder Model . . . . . . . . . . . . . . 208

8.4 ODE Model for the Seceder Model . . . . . . . . . . . . . . . . . . . 211

8.5 Complexity of the Population Structure . . . . . . . . . . . . . . . . . 213

8.6 Fitness Landscape in the Seceder Model . . . . . . . . . . . . . . . . 215

8.7 Variant: Effect of Tournament Size . . . . . . . . . . . . . . . . . . . 217

8.7.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.8 Variant: Bounded Space . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.8.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.9 Variant: Explicit Fitness . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.9.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.10 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 226

9 The Seceder Effect in Second Order Reaction Systems 229

9.1 The Fundamental HR-Model . . . . . . . . . . . . . . . . . . . . . . . 231

9.2 A Differential Equation Model for the HR-Model . . . . . . . . . . . 233

9.2.1 Example: One Group . . . . . . . . . . . . . . . . . . . . . . . 233

9.2.2 Example: Two Groups . . . . . . . . . . . . . . . . . . . . . . 234

9.3 Porperties of the HR-Model . . . . . . . . . . . . . . . . . . . . . . . 235

9.3.1 Dependence on Threshold . . . . . . . . . . . . . . . . . . . . 236

9.3.2 Dependence on Replication Rate . . . . . . . . . . . . . . . . 237

9.3.3 Dependence on Population Size . . . . . . . . . . . . . . . . . 237

9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

CONTENTS 11

10 Self-Organizing Topology 24510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24610.2 Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.3 Generating Topology through Hashing . . . . . . . . . . . . . . . . . 24810.4 An Artificial Chemistry with Hash-Topology . . . . . . . . . . . . . . 24910.5 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

10.5.1 Macroscopic Measurements . . . . . . . . . . . . . . . . . . . 25110.5.2 Visualization of the Topology . . . . . . . . . . . . . . . . . . 25110.5.3 Spatial Complexity . . . . . . . . . . . . . . . . . . . . . . . . 252

10.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25210.6.1 A Well-stirred Tank Reactor . . . . . . . . . . . . . . . . . . . 25210.6.2 Hash-Topology 1: Hashing the Reaction Product . . . . . . . 25310.6.3 Hash-Topology 2: Hashing Operand and Product . . . . . . . 255

10.7 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 256

11 Summary and Outlook 26111.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.2 Conflicting Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26411.3 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

11.3.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26911.5 Deutsche Zusammenfassung (Summary in German) . . . . . . . . . . 270

12 CONTENTS

Summary

This thesis reports on the following: (1) A categorization scheme for artificial chem-istry approaches is derived which is based on a systematic review of existing meth-ods. (2) Methods for analysis are introduced and their properties and applicabilitydemonstrated. Noteworthy is the mesoscopic analysis which allows to visualize thedevelopment of groups in a population. (3) Existing methods using matrix mul-tiplication as a mechanism for defining an artificial chemistry are extended and asystematic framework for matrix multiplication chemistries is presented. (4) Theproperties of different matrix multiplication chemistries are demonstrated. A note-worthy phenomenon that has been discovered by simulations of matrix multiplica-tion chemistries is the so-called constructive equilibrium. A state where a systemis stable on a macroscopic level while on a microscopic level ongoing new compo-nents are generated. (5) Along the seceder model it is demonstrated how a simplethird-order collision rule gives rise to complex group formation. The property of theseceder model and its variants are investigated by simulation and analytically. (6)The relation of the seceder model to real bio-molecular systems is demonstrated byintroducing a simple model of replicating and hybridizing molecules. This so-calledHR-model exhibits a similar dynamics like the seceder model, but employs realisticmolecular reaction mechanisms. (7) A self-organizing topology based on hashingis introduced where the topological structure of the reaction vessel depends on themolecules in the vessel. Preliminary experimental results show the properties ofsome systems with self-organizing hash-topology.

Chapter 1

Introduction

1.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . 14

1.1.1 What is Artificial Chemistry? . . . . . . . . . . . . . . . . 14

1.1.2 Definition of Molecules: Explicit or Implicit . . . . . . . . 16

1.1.3 Definition of Reaction Laws: Explicit or Implicit . . . . . 17

1.1.4 Level of Abstraction: Analogous or Abstract . . . . . . . 17

1.1.5 Dynamical Simulation of Artificial Chemistries . . . . . . 18

1.1.6 Constructive Dynamical Systems . . . . . . . . . . . . . . 21

1.1.7 Random Chemistries . . . . . . . . . . . . . . . . . . . . . 22

1.1.8 Models of Space . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.9 Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . 22

1.2 Three Artificial Chemistries . . . . . . . . . . . . . . . . . 23

1.2.1 AC1 - Mathematical Expression as a Reaction Mechanism 23

1.2.2 AC2 - A Strange Artificial Chemistry . . . . . . . . . . . 28

1.2.3 AC3 - Chaos and Order . . . . . . . . . . . . . . . . . . . 35

Several authors have recognized that there is a class of systems where classicalmethods, such as dynamical systems theory or automata theory, have difficultiesin describing, explaining, and predicting their behavior (Kampis 1991; Rosen 1991;Fontana 1992). These systems here called constructive dynamical systems(Fontana 1992) are characterized by the fact that they create new components.The dynamics of constructive systems is often governed by strange feedback loops(Hofstadter 1979) where a system is able to inspect and self-modify its own compo-nents. Examples of these so-called strange systems are self-modifying computerprograms or the molecular machinery of a biological cell. Strange systems are of

13

14 CHAPTER 1. INTRODUCTION

particular interest because their underlying mechanism plays a fundamental role inliving systems. Strange feedback loops can be found on all levels of life, from themolecular level up to cognitive and social processes. They can also appear in lan-guage, laws, or music (Hofstadter 1979). But also man-made information processingsystems allow to instantiate constructive systems which may also be governed bystrange feedback loops (Dewdney 1984).

In this study I shall investigate constructive and strange systems and introducemethods for their design and analysis. The aim is to contribute to the basis whichshould allow to formulate a theoretical framework for constructive systems in thefuture. Such a theory would give insight into fundamental questions raised in biologyand would enhance our understanding of complex information processing systems.One such question - to give an example - is the question of the origin of informationand information processing systems.

The conceptual framework in this thesis will be the metaphor of artificial chem-istry. An artificial chemistry consists of abstract molecules and their interactionlaws. This framework allows to define constructive and strange systems in an elegantand straightforward way. For the analysis and description of these systems we canpartly rely on methods taken from chemistry, but in addition, we have to developnew ones. Although chemistry describes a low level in the hierarchy of living systems(Miller 1978), the basic dynamical mechanisms can be found on all levels even on asocial scale (Hofbauer and Sigmund 1988). Therefore, the experience and method-ology that we gain from working with artificial chemistries is not restricted to thechemical domain, but can be transferred and applied to totally different problemdomains.

1.1 Fundamental Concepts

In this section fundamental concepts and terms are introduced and when possibledefined formally. The next section will demonstrate the concepts by introducingand discussing three concrete artificial chemistries as examples.

1.1.1 What is Artificial Chemistry?

The term artificial chemistry may either denote a research field or a concreteman-made, chemical-like system. Usually the term is used in the second sense. Inthat case, the plural artificial chemistries is meaningful. A broad definition mightread: An artificial chemistry is a man-made system which is similar to a realchemical system. This definition has been kept as general as possible in order notto exclude any relevant work. The drawback is that the definition includes workthat should not be considered as artificial chemistry, such as models of chemicalsystems which can be found in any textbook on chemistry. To exclude these classicmodels the definition can be refined in the following way: An artificial chemistry

1.1. FUNDAMENTAL CONCEPTS 15

is a man-made system which is similar to a real chemical system and where thereis no one-to-one relationship between molecules of the artifical chemistry and realmolecules. Most works on artificial chemistries would accept this definition. Butwhen using this restricted definition we have to be careful not to overlook importantapproaches that aim at a realistic representation of the molecular structure, e.g.,(Mayer and Rasmussen 1998a; Bersini 2000).

When, in the following, the definition becomes more precise, one should keep inmind that not all AC approaches can be subsumed under the following conceptualframework. In this framework, an artificial chemistry (AC) is given by a triple(S,R,A), where S is the set of all possible molecules, R is a set of collision rulesrepresenting the interaction among the molecules, and A is an algorithm describingthe reaction vessel and how the rules are applied to the molecules inside the vessel.This separation into three parts will be used throughout this thesis. Chapter 3demonstrates that this separation is also helpful to describe other work in a coherentmanner.

The Set of Molecules S

The set S describes all valid molecules that may appear in an AC. A vast variety ofmolecule definitions can be found in different approaches. Molecules may be abstractsymbols (Eigen and Schuster 1977), character sequences (Farmer, Kauffman, andPackard 1986; Bagley and Farmer 1992; Kauffman 1993; McCaskill, Chorongiewski,Mekelburg, Tangen, and Gemm 1994), lambda-expressions (Fontana 1992), binarystrings (McCaskill 1988; Thurk 1993; Banzhaf 1993b; Dittrich and Banzhaf 1998),numbers (Banzhaf, Dittrich, and Rauhe 1996), or proofs (Fontana and Buss 1996). Amolecules representation is often referred to as its structure in contrast to its func-tion which is given by the reaction rules R. The description of the valid moleculesand their structure is usually the first step when an AC is defined. This is analogousto a sub-field of chemistry which describes what kind of atom configurations formstable molecules and how these molecules appear.

The Set of Rules R

The set of reaction rules R describes the interactions between molecules si S. Arule r R can be written according to the chemical notation of reaction rules inthe form

s1 + s2 + + sn s1 + s2 + + sm. (1.1)A reaction rule implies that the n components (objects, molecules) on the left handside can react and therefore can be replaced by the m components on the right handside. n can be called the order of the reaction1. Note that the sign + is not anoperator here, but should only separate the components on either side.

1Note that according to chemistry this is a simplification.


A rule is then applicable if certain conditions are fulfilled. The major conditionchecks of course whether all of the left hand side components are available. Thiscondition can easily be broadened to other parameters such as a neighborhood, rateconstants, probability for reaction, or energy consumption. In this case a reactionrule may also contain additional information or parameters. Whether or not theseadditional predicates are taken into consideration depends on the objectives of theAC. If it is meant to simulate real chemistry as accurately as possible, then it isnecessary to integrate these parameters into the simulation. If the goal is to buildan abstract model, then these parameters may be omitted.

The Algorithm A Describing the Dynamics

The third component of an artificial chemistry is an algorithm denoted by A which describes the dynamics. It determines how the rules are applied to a col-lection of molecules P , called reactor, soup, reaction vessel or population2.The algorithm depends on the representation of P . In the most simple case, wherethere is no spatial structure in P , the population can be represented explicitly as amultiset3 or implicitly as a concentration vector.

The previous introduction has already indicated that there is a huge variety amongartificial chemistries. In order to describe and compare them we will now introducea few important dimensions along which artificial chemistries can be characterized.With the aid of these characteristics we are able to classify artificial chemistries.They also help to describe quickly important properties of an artificial chemistry.

1.1.2 Definition of Molecules: Explicit or Implicit

In an artificial chemistry {S,R,A} the set S represents the objects or moleculeswhich interact according to certain rules defined in R. The set S can be definedexplicitly or implicitly.

Molecules are explicitly defined if the set S is given as an enumeration of symbols.For example, S = {A,B,C}. An implicit definition is a description of how toconstruct a molecule. This description may be a grammar. Examples for implicitdefinitions are: S = {0, 1}, the set of all binary strings; or S = {1, 2, 3, . . . }, thenatural numbers.

To build constructive dynamical systems (Sec. 1.1.6) it is convenient to definemolecules implicitly. Typical implicitly defined molecules are character sequences(e.g., sequences like abbaab), mathematical objects (e.g. numbers), or compoundobjects which consist of different elements. Compound objects can be represented

2Here, the term population is used as a technical term according to its meaning in the field ofevolutionary computation and artificial life. It refers to a data structure which holds all individualsduring a simulation. It should not be confused with the technical term population used in biologywhich refers to a group of similar, interbreeding organisms that live in a particular area.

3A multiset is like a set, but the same object may appear in the multiset several times.


by classical data structures (Mayer and Rasmussen 1998a) or in an object orientedway (Bersini 2000). Here, the representation of an molecule is also called structure.

In some chemistries, the structure of a molecule is not defined a priori. The arrivalof molecules is then an emergent phenomenon, and it is only possible to interpreta structure as a molecule a posteriori as, e.g., in Coreworld (Rasmussen, Knudsen,Feldberg, and Hindsholm 1990), or in cellular automata models (Sayama 2000).Therefore, in certain cases, it is not even necessary to define a molecule, neitherexplicitely nor implicitly.

1.1.3 Definition of Reaction Laws: Explicit or Implicit

The reactions can be defined, analogously to the molecules, in two different ways.

Explicitly: An explicit definition of the interaction between molecules is indepen-dent of the molecules structure. It requires an enumeration of explicit reactionrules, where molecules are represented by abstract exchangeable symbols. The totalnumber of possible elements of the AC remains fixed. All possible elements andtheir behavior are known before and their interaction rules do not change duringthe experiments.

Implicitly: An implicit definition of the interaction between molecules must referto the structure of the interacting molecules. The number of possible molecules canbe infinite, because there is no need for an explicitly defined interaction scheme ifit can be derived from the objects structure. Examples for artificial chemistrieswith implicit reaction rules are AC1-AC3 (see Sec. 1.2). An artificial chemistry withan implicit reaction scheme allows to derive the outcome of a collision from thestructure of the colliding molecules. Implicitly defined reactions are commonly usedfor constructive artificial chemistries.

1.1.4 Level of Abstraction: Analogous or Abstract

Another way to characterize artificial chemistries is according to their level of ab-straction. If there is a relation (isomorphism) of each molecule or reaction of theAC to a molecule or reaction in Chemistry, respectively, the AC can be called anal-ogous otherwise abstract. What makes an abstract AC a model of Chemistryare statistical or qualitative features of the reaction laws. The level of abstractiondepends on the aim underlying the models design.

Examples for analogous artificial chemistries are (with increasing level of abstrac-tion4) DNA computing models (see Sec. 2.2.1), Bersinis (2000) OO chemistry,and the lattice molecular automaton by Mayer and Rasmussen (1998a). Examples

4We can even become more precise when we take a look at the level of abstraction separatelyfor molecules and reactions. For example in Bersinis (2000) OO chemistry the reactions aremore abstract than in the lattice molecular automaton (LMA) by Mayer and Rasmussen (1998a).Whereas in the LMA the molecules are more abstract than in the OO chemistry.


for abstract artificial chemistries are (with increasing abstraction) the autopoieticmodel by Varela, Maturana, and Uribe (1974), polymer chemistries (Farmer, Kauff-man, and Packard 1986; Bagley and Farmer 1992), Typogenetic (Hofstadter 1979),Fontanas (1992) AlChemy, and the artificial chemistries AC1-AC3 (Sec. 1.2).

1.1.5 Dynamical Simulation of Artificial Chemistries

This section summarizes how the dynamics (denoted by A in Sec. 1.1.1) of a reactionvessel can be modeled and simulated. The approaches can be roughly characterizedby whether each molecule is treated explicitly or whether all molecules of one typeare represented by a number denoting the frequency or concentration of that type.

(1) Stochastic molecular collisions: In this approach every molecule and everysingle reaction is explicitly simulated like in the forthcoming examples in Sec. 1.2.The population can be represented as a multiset P . A typical algorithm drawsrandomly a sample of molecules from the population P and checks whether a rule r R can be applied. If so, the molecules are replaced by the right hand side moleculesgiven by r. The algorithm is not necessarily restricted to be so simple. Furtherparameters such as rate constants, energy, spatial information, or temperature canbe introduced for the chemistry to become more realistic.

The following example is an algorithm usually used for an AC where only second-order reactions are allowed:

Algorithm 1.1 (reactor algorithm 1)while terminate() do

(s1, P ) := draw(P );(s2, P ) := draw(P );if (s1 + s2 s1 + s2 + + sm) R

thenP := insert(P, s1, s

2, . . . , s

m); reactive collision

elseP := insert(P, s1, s2); elastic collision

fit := t + 1/size(P ); time increment

od

The non-deterministic function draw returns a randomly chosen molecule from Pand removes it from P such that P, s P, (s, P ) = draw(P ), insert(P , s) = P .The probability that a specific type s is returned is proportional to the concentrationof this type in P . The above algorithm does not simulate an influx or dilution flux.Nevertheless an influx of dilution flux can easily be added.

Measuring time: One step of the reactor algorithm (Alg. 1.1) can be interpretedas a collision of molecules. The simulated time is proportional to the number of col-lisions divided by the reactor size M = size(P ). It is common to measure the sim-


ulated time in generations, where one generation are M collisions, independentlyof whether a collision causes a reaction or not. Using M collisions (a generation) asa unit of time is realistic because otherwise an increase of the reactor size M wouldresult in a slow down of development speed. The underlying assumption is: whenthe reactor size is increased by a factor , then the number of collisions per timeunit should also increase by the same factor. The method for the time increment inAlg. 1.1 can be used even for varying population sizes.

The above concept measures time as a quantity which is proportional to the numberof collision per volume. The concept has to be refined if we introduce higher physicaldetails such as temperature or spatial extension of molecules. In this case we canemploy kinetic gas theory which provides a formula for the frequency of collisions:

z =

2N

Vd2

8kT

m(1.2)

where N is the number of molecules per volume V , k the Bolzmann constant, d thediameter of a molecule, and m its mass ((Vemulapalli 1993), p. 659). The formulaassumes that velocities of the molecules are Bolzmann distributed.

The memory consumption of Alg. 1.1 is in general (M) (that is, the memory neededto store all molecules is linearly dependent on M , the total number of molecules inthe reaction vessel). The explicit simulation of every collision is very realistic andcircumvents some artifacts of the numerical integration described in the next para-graph. But there are certain disadvantages of the explicit simulation of collisions.The simulation of a system, where the rate constants or the concentrations differ byseveral orders of magnitude is not efficient enough. If the total number of differentmolecules is low or the population is large, then the explicit simulation is slow.

(2) Continuous differential or discrete difference equations: A commonapproach to describe the dynamics of a chemical system is to use ordinary differentialequations (ODE) which reflect the development of the concentrations xi of moleculartype si. Let N = |S| be the total number of different species in S. A reaction r canbe written as

r : a1s1 + a2s2 + + aNsN b1s1 + b2s2 + + bNsN .

where the coefficients ai, bi are the stoichiometric factors of the reaction. They areequal to zero if the molecular type si does not participate in the reaction (ai = 0 ifsi is not a reactant and bi = 0 if it is not a product of the reaction). According tothe law of mass action kinetics5, the reaction r causes the following change to the

5The law of mass action kinetics has been formulated in the second half of the 19th centuryby Cato M. Guldberg and Peter Waage. The law states that the rate of any simple chemicalreaction is proportional to the product of the masses of the reacting substances, each raised to apower. The exponent of each mass is equal to the corresponding number of molecules taking partin the reaction. The law is valid only in certain ideal situations, e.g., in low concentrated solutions.


concentration xi of the molecular type i:

xi =N

j=1

bjxj N

j=1

ajxj , i = 1, . . . , N. (1.3)

To take every reaction r R into account, Eq. (1.3) becomes

xi =rR

[N

j=1

b(r)j xj

Nj=1

a(r)k xj

], i = 1, . . . , N. (1.4)

Note that this is a continuous model for a discrete system. If Eq. (1.4) is solvedwith a numerical integrator, the number of molecules of a certain type si is usually afloating point value and thus an approximation. The modeling and simulation of thedynamics by differential or difference equations allows to simulate a reaction vesselwith a huge number of molecules. This may result in a very efficient simulation ofthe dynamics.

But there are disadvantages: Due to the properties of the numerical solution, allconcentrations are floating point values and therefore, for very low concentrationsof a species si, the concentration value may be below the threshold that indicates asingle molecule in the reactor. Thus, for the ODE approach the number of differentmolecules N should be low, because the memory consumption, now independentfrom M , depends on the number of different species N involved.

(3) Metadynamics: In this approach the reaction system is modeled by a differ-ential equation system, where the equations may change over time (Farmer, Kauff-man, and Packard 1986; Bagley and Farmer 1992). The equations at a given timerepresent the dominant molecular types. A molecular type is dominant if its con-centration is above a certain threshold. As concentrations change the dominanttypes also may change. In this case the differential equation system is modifiedby adding and/or removing equations. Bagley, Farmer, and Fontana (1992) distin-guish between deterministic and stochastic metadynamics: In the deterministicmetadynamics the sequence of graphs equivalent with the sequence of ODEsystems is purely deterministic. The graph changes relatively to a concentrationthreshold.The method explores only the internally catalyzed pathways and doesnot considers random fluctuations. The stochastic metadynamics allows randomfluctuations in the reaction network and thus increased physical accuracy combinedwith the speed of the deterministic metadynamics approach.

(4) Mixed approaches: There are also approaches where single macro moleculesare simulated explicitly and small molecules are represented by their concentrations(Zauner and Conrad 1996; Zauner and Conrad 1998).


(5) Symbolic analysis of the equations: If the differential equation system,Eq. (1.4), is simple enough, a symbolic analysis is possible, like the fixed pointanalysis performed for the cluster level of AC3 (see p. 37). The symbolic analysismay give the steady state behavior of the system (e.g. fixed points, limit cycles,chaotic attractors) by symbolic operations on the equations. This method is, in amathematical sense, the most exact one. It can be combined with the metadynamicsapproach to calculate the dynamical fixed point of the differential equations inone step. The method has been applied to simulate polymer reaction networks(Bagley and Farmer 1992; Bagley, Farmer, and Fontana 1992; Farmer, Kauffman,and Packard 1986). In these simulations the efficiency has been drastically increasedby assuming that the ODE has only one stable fixed point.

1.1.6 Constructive Dynamical Systems

In a constructive dynamical system new components can appear which maychange the dynamics of the system (Fontana 1992). In a conventional, non-constructive dynamical system all components and all interactions are given ex-plicitly at the beginning. If new components are generated randomly a constructivesystem is called weakly constructive (Fontana, Wagner, and Buss 1994). It iscalled strongly constructive if new components are generated through the ac-tion of other components. Chemistry is considered a strongly constructive system(Fontana, Wagner, and Buss 1994).

Thinking in the lines of differential equations, one can imagine a constructive sys-tem, e.g., as an ODE system where equations are modified, added and/or removeddynamically. Of course, a constructive dynamical system can also be modeled by afixed ODE system with a large or infinite state space and there are also mathemat-ical techniques available to handle such systems; but this approach is not intuitiveand has many additional drawbacks.

Discussion: Note that intentionally the term constructive dynamical system is notdefined formally here. I hope that the explanation given above, the examples inthis thesis, and the pointers to literature are sufficient for a precise description ofthe concept behind the term. A formal definition would narrow our view and wouldlead us to belive in having achieved sufficient comprehension. Yet, there are manysystems where it is difficult to decide whether they are constructive or not. In fact asystem may be non-constructive on one level. But on a higher level components ofthe lower level form large agglomerates. Thus the system may become constructiveon the level of these agglomerates. An example for this type is given by Suzuki andTanaka (2000). In their artificial chemistry the set of molecules consists only of a fewsymbols and the reaction rules are also given explicitly. Thus on a molecular levelthe system is not constructive. But molecules may form cells which are multisets ofmolecules and may also contain other cells. Thus the system is constructive on thecellular level.


1.1.7 Random Chemistries

Artificial chemistries can be generated randomly by drawing and storing randomnumbers for rate constants or generating explicit reaction laws randomly. The ran-dom autocatalytic systems investigated by Stadler, Fontana, and Miller (1993) orthe GARD by Segre, Lancet, Kedem, and Pilpel (1998) are examples for non-constructive, random chemistries. But it is also possible to build constructivechemistries randomly by drawing parameters for the reaction laws on the fly whenthey are needed, as demonstrated by Bagley and Farmer (1992). The advantage ofrandom chemistries is that the statistical characteristics of the reaction mechanismcan be specified arbitrarily and easily (Lancet, Sadovsky, and Seidemann 1993).The disadvantage is the increasing memory consumption when new molecular typesappear, because every randomly generated reaction law has to be stored explicitlyfor the whole simulation6.

1.1.8 Models of Space

The spatial structure of the reactor is described by the algorithm A of the AC{S,R,A}. The description can be independently of S and R, except for approacheswhere molecules grow and occupy more than one point in the reactor space (Sec. 3.5).In many cases, the reactor is modeled as a well stirred tank reactor often with acontinuous in- and outflow of substances, e.g., (Bagley and Farmer 1992; Fontana1992; Dittrich and Banzhaf 1998).

In a well stirred tank reactor the probability for a molecule si to participate in a reac-tion r is independent from its position inside the reactor. In a reactor with topology,this probability depends on the neighborhood of si. This neighborhood can be de-fined as the vicinity (maybe depending on further parameters) of si in an Euclideanspace. This space may be two- or more-dimensional,e.g., three-dimensional as in(Zauner and Conrad 1996). The neighborhood can be defined like the neighborhoodin cellular automata (Lugowski 1989; Varela, Maturana, and Uribe 1974; Banzhaf,Dittrich, and Eller 1999)); or as a self-organizing associative space as described inChapter 10 (Dittrich and Banzhaf 1997).

1.1.9 Pattern Matching

Pattern matching is a method widely used in artificial chemistries and other artificiallife systems. A pattern can be regarded as a means to identify the semantics of alocation or subcomponent. It allows to refer to parts of a system in an associativeway independently from the absolute position of these parts. Koza (1994) used theshape of trees to address subtrees and to select reaction partners. In the field ofDNA computing (Adleman 1994), pattern matching is a central mechanism in the

6This can be circumvented by using a deterministic pseudo random number generator but withthe expense of computing time.

1.2. THREE ARTIFICIAL CHEMISTRIES 23

alignment of two DNA strands. In this case, a pattern is a sequence of nucleotidessuch as CGATTGAGGGA. In TIERRA a pattern is given by sequence of NOP0and NOP1 operations in the genome of an individual and is used to direct jumps ofthe programm counter (Ray 1992).

The accuracy of a match can be used to calculate the probability of a reaction or rateconstants, as suggested by Bagley and Farmer (1992). To compute the accuracy, adistance measure is needed that computes the similarity of molecules or moleculardomains. A convenient distance measure for variable length sequences is the stringedit distance. The distance between strings of the same length is usually computedby the Hamming-distance. For more details see (Gusfield 1997; Watermann 1995).

1.2 Three Artificial Chemistries

In this section three constructive artificial chemistries are introduced. One of them,AC2, contains a strange loop. These three chemistries provide a concrete basis towhich the discussion of fundamental concepts and motivations of artificial chemistryresearch can refer to. Thus only some exemplifying properties are shown. A deeperanalysis of artificial chemistries can be found in the following chapters.

As described previously, to introduce an artificial chemistry it is convenient to sepa-rate the description into three parts: (1) molecules, (2) reactions, and (3) dynamics.In the first part the set of all valid molecules or atoms is defined which may appearin the system. Usually this is done implicitly by specifying rules that define whatkind of molecular structures are valid or stable. The second part describes howmolecules interact when they collide or come close to each other. If the moleculesare unchanged, the collision is called elastic collision otherwise reactive. Thethird part describes the reaction vessel and how the molecules move and collide.The molecular dynamics may be explicit if single molecules are simulated or quiteabstract in the case of differential equations.

1.2.1 AC1 - Mathematical Expression as a Reaction Mechanism

The first example of an artificial chemistry shows that only a few simple rules aresufficient to define a constructive dynamical system with quite interesting properties.

Molecules: The molecules are just real numbers. The set of all valid molecules isdenoted by S = R.Reactions: We assume that molecules can interact with each other according tothe interaction scheme

s1 + s2 + s3 s1 + s2 + s4 (1.5)

where si S. In general, a reaction equation like this states that the moleculeson the left hand side can react to the molecules on the righthand side. For a


simulation this means that the molecules on the left hand side can be replaced bythe righthand side. Here, we can interprete the reaction equation in the followingway: The molecule s3 is transformed to s4 under the catalytic

7 influence of s1 ands2.

If we further assume that all reactions are deterministic so that the outcome of areaction depends only on the reacting molecules s1, s2, s3, then we can define thereaction by a function r : S S S S such that

s4 = r(s1, s2, s3). (1.6)

The reaction function r should be defined here as:

r(x, y, z) =1

9

(x + y 2 z)2(

1 + e(2 xyz) (yz)

3

) (1 + e

((xz) (x2 y+z))3

)+

(x 2 y + z)2(1 + e

((xy) (x+y2 z))3

) (1 + e

2 x y+y2+2 x zz23

)

+(2 x + y + z)2(

1 + e(xy) (x+y2 z)

3

) (1 + e3+

(2 x+y+z)29

) (1.7)

Note that r is defined implicitly by just using some simple arithmetic operatorssuch as addition, multiplication, and power. For now, its sufficient to regard r asa function composed of some well known arithmetic operators. The structure andsemantics of the expression will be explained later.

Dynamics: In order to simulate an artificial chemistry the reaction vessel and thedynamics inside this vessel have to be specified, because the definition of moleculesand reaction equations do not necessarily allow to derive the dynamics of the reac-tion system. Here, we assume a well-stirred tank reactor to be the reaction vessel.Because the vessel is well-stirred, the concentration of a substance is everywhere thesame and we need not to simulate the spatial structure. Thus, the vessel can be rep-resented by a multiset P where each element in P represents one concrete molecule.P is called population. The number of elements in P is called population sizeM . The population dynamics is given by the following algorithm:

7A catalyst is a substance or molecule which increases the speed of a reaction without beingused up.



(s1, P ) = draw(P );(s2, P ) = draw(P );(s3, P ) = draw(P );s4 = r(s1, s2, s3);P := insert(P, s1, s2, s4);

od

Because, here, the population size does not change over time, we can represent thepopulation by a fixed sized array over S. In this case a nearly8 equivalent algorithmreads:


i = randInt(1,M)P [i] := r(P [randInt(1,M)], P [randInt(1,M)], P [i])

od

To measure time we define a generation as M iterations of the while loop (see p. 18).It should also be noted that one iteration can be interpreted as a collision. Here,three molecules collide and react in a way such that one molecule is transformed.Therefore, every collision of three molecules is reactive and the rate of every reactionis the same and may be assumed equal to one. For a concrete simulation we initializethe population just with M zeros.

Discussion: We have now defined a dynamical system with only some formal ex-pressions which are composed of a few simple operators and data structures. Onepage is sufficient to fully specify the system in a precise way which becomes possiblebecause we rely on formalisms taken from mathematics and computer science. Butcan we predict the behavior without running the algorithm? The answer seems tobe no. In other words:

There is no general theory available to predict the behavior of AC1 or in general, ofconstructive systems in an efficient way when only the interaction rules are given.

This statement is of course quite loosely formulated and a statement saying thatsomething does not exists can not be proven, outside the realm of formal systems.What we are looking for is a theory of constructive dynamical systems which is alsoable to predict and describe the behavior of AC1. The form of that theory and apotential way towards it will be discussed in Sec. 2.4. For now we have to simulateAC1 in order to investigate its behavior. We use Alg. 1.3 for the simulation, becausein that case we can also perform simulations with M = 2.

Simulation for M = 2: We begin the simulation with a simple case where thepopulation consists only of two molecules, thus M = 2. In the language of classicaldynamical systems theory we can say, that the state space of the system is RM

8The algorithms are equivalent for large population size M .


(a)

-15

-10

-5

0

5

10

15

-10 -5 0 5 10 15

P[1

]

P[0]

(b)

-30-25-20-15-10

-505

10152025

-30 -20 -10 0 10 20 30 40

P[1

]

P[0]

(c)

-50-40-30-20-10

0102030405060

-60 -40 -20 0 20 40 60

P[1

]

P[0]

Figure 1.1: Simulation of AC1 for population size M = 2. Three trajectories areshown for different duration of total simulation time: (a) 10 generations, (b) 100generations, and (c) 1000 generations. A dot represents a state of the populationP = (P [0], P [1]).

and its dynamics is given by an iterated non-deterministic map. The dynamicbehavior is a trajectory in state space which can be visualized as shown in Fig. 1.1.The trajectories do not reveal any structure. It seems that the system performs arandom walk in state space.

Simulation for M = 500: The interesting and more realistic case is where thepopulation size is large. In this case we face the problem how to analyze andvisualize the simulation because a state space trajectory cannot be directly and aseasily depicted as in Fig. 1.1. For visualization a method has to be chosen whichprojects the high-dimensional state space to a low-dimensional space to be plotted.It should be clear that whether a certain characteristics of the dynamical behaviorbecomes visible or not depends on the projection method. What method is suitabledepends on the system under investigation.

Because the molecules are one-dimensional quantities and the reaction vessel is well-stirred we can visualize the complete population in a simple way as shown in Fig. 1.2.Each dot represents a single molecule at a certain point in time. The abscissa posi-tion is given by the molecules value which is in general called structure or genotypeof the molecule. The resulting picture of the evolving population reveals a surpris-ingly complex pattern. The population separates spontaneously into several groups.Dominating groups suddenly die out. The pattern is symmetric on a macroscopicscale.


-3000

-2000

-1000

0

1000

2000

3000

0 200 400 600 800 1000ge

noty

petime [generation]

Figure 1.2: Simulation of AC1 for population size M = 500. Population initializedwith molecular type 0 at t = 0. Only a random fraction of 10% of the populationis displayed every third generation.

-80

-60

-40

-20

0

20

40

60

80

0 200 400 600 800 1000

geno

type

time [generation]

Figure 1.3: Overlay of 25 simulation of AC1 for population size M = 2. The numberof displayed points is the same as in Fig. 1.2.

But does this complex pattern become only visible because we have chosen a dif-ferent visualization method and a larger number of molecules so that the averagebehavior of two molecules becomes visible? Figure 1.3 shows that this is not thecase. In Fig 1.3, 25 runs for population size M = 2 are overlayed such that the num-ber of displayed points is the same as in Fig. 1.2. But no complex pattern appears.Therefore, the interaction between a large number of molecules is necessary to gen-erate these complex pattern, here. Phenomena of this kind are called emergentphenomena.

Notes: The state space RM is invariant to all permutation of its dimensions. Thismeans that if we exchange the contents of two variables we get isomorphic trajec-tories. This property stems from the assumption of a well-stirred reaction vessel.


1.2.2 AC2 - A Strange Artificial Chemistry

In the previous chemistry, AC1, the three molecules of a reaction are treated equallyin a symmetric way. The interaction among molecules is encoded directly by amathematical expression which codes for an operator r : S S S S. Now, inAC2, this operator will be decomposed into two separate steps. In the first step thefirst molecule s1 will be transformed to an operator O : SS S which is applied inthe second step to the remaining two molecules s2 and s3. The interesting propertyof AC2 is the dualism of structure and function: The same molecule can be passivedata (e.g., a number) or an active operator which operates on other molecules. Thisability of a structure (the molecule) to operate on itself9 is a prerequisite for astrange loop.

Molecules: The molecules are positive natural numbers. The number of digitsshould not exceed L. Formally: S = {0, 1, 2, . . . , 10L 1}. For the simulations inthis section we set L = 300.

Reactions: The reaction scheme of AC2 is the same as for AC1:

s1 + s2 + s3 s1 + s2 + s4. (1.8)

And again, s4 depends deterministically only on s1, s2,and s3. But now, the reactionis carried out in two steps: (1) Transform s1 to an operator O. This step is in generalcalled folding and can formally be written as O = F(s1). (2) Apply O to s2 ands3 to calculate s4 = O(s2, s3). The reaction function can now be written in adecomposed way:

r(s1, s2, s3) = F(s1)(s2, s3). (1.9)

To define the folding of a molecule s S to an operator O we map s to a math-ematical expression composed of addition, subtraction, multiplication and divisionoperators (Fig. 1.4). Let di be the i-th digit of s such that s =

n1i=0 10

idi. Thereare n digits called length of s and no leading zeros so that dn1 6= 0 for n > 0.A digit is interpreted according to Tab. 1.1 as an operator, a variable, or a con-stant. All operators have arity10 two. The division operator is protected such thatit returns its first argument if the second argument is zero.

To build the expression we consider it as a binary tree. This is possible because thelargest arity is 2. If we assign a coordinate to each node in the tree where l is thelevel and m is the m-th node on level l then we can easily assign to each node adigit di by

i = (2l + m) mod n, l = 0, 1, . . . , m = 0, 1, . . . (2l 1) (1.10)9To be more precise, a molecule does not operate on itself, but can operate on an instance which

is an exact copy and thus possesses exactly the same structure as itself.10The arity is the number of parameters of an operator.


digit symbol interpretation0 + addition operator1 - subtraction operator2 * multiplication operator3 x first variable4 x first variable5 y second variable6 y second variable7 / protected division operator8 2 constant9 3 constant

Table 1.1: Interpretation of a digit for the construction of an arithmetic expressionfor AC2. The protected division returns the first argument if the second argumentis zero.

genotype 321053902

*

+ 3

x y

genotype 0

++

+

1 1 1 1

level 0

level 1

level 2

Figure 1.4: Example of the interpretation of numbers in AC2. Left: The exampledemonstrates a case where not all digits are required to build an expression. Right:The example demonstrates that digits may be used many times to build an expres-sion. In this extreme case, there is only one digit, which codes for the + operator.Therefore, the expression consists only of + operators. Because there is a depthlimitation (Lmax = 3), the + operator is substituted by the constant 1 on thelowest level.


where the root node is at level 0. The first argument of the root node (m = 0, l = 0)would be node m = 0 at level l = 1 and the second argument would be node m = 1at level l = 1. To avoid trees with unmanageable size, a maximum depth lmax isintroduced. A node with l = lmax is a constant c. For the following simulations weset lmax = 4 and c = 1.

Dynamics: The dynamics is the same as for AC1.

Simulation: For a simulation we set the population size to M = 100 and initializethe population with numbers drawn randomly from {0, 1 . . . , 9999}. Figure 1.5visualizes a simulation in the same way as for AC1 in Fig. 1.2. Because somenumbers are very large the genotype axis has to be scaled logarithmically. We cansee that the pattern of the evolving population becomes regular. After a while thepopulation consists only of a few molecular types which are quite persistent. Newmolecular types do not appear any more11.

Another way of visualizing the dynamics of an evolving population is by measuringand displaying macroscopic quantities such as temperature, entropy, or diver-sity. A macroscopic quantity accumulates properties of a large number of objects ofthe system under investigation into a low dimensional value (e.g., a scalar) whichdescribes a certain property of the system (Dittrich, Ziegler, and Banzhaf 1998).Figure 1.5 shows as an example the diversity over time. Here, the diversity isdefined as the number of different molecules present in the population, or in otherwords, the number of molecular types present (Dittrich 1995). It is clear that thisquantitative definition is only one way to formalize the concept of diversity. Itslimitation can already be recognized if applied to AC1. More accurate and realisticmeasurements of diversity can be taken from the field of biodiversity research (seee.g. (Schwobbermeyer and Kim 1999) for a comparison).

Figure 1.5 indicates that the population has converged to a state where the di-versity and the composition of the population does not change very much. Toinvestigate the organization of the surviving molecules we have to take a closerlook at their structure. In Tab. 1.2 all molecular types are listed which are presentin the population at t = 120. This table is typical for simulations of AC2 withM = 100 and would not change notably if we continued the simulation. The table issorted according to the molecule structure, but it would be also convenient to sortit according to the concentration of the molecular types.

In Tab. 1.2 a syntactic similarity of the molecular structures is noticeable: The tailsof the numbers are similar. Except for the first molecule, all molecules end either on4, 6, or 32. If we take a look at the function of the molecules (Tab. 1.2) the reasonfor this syntactic similarity becomes clear. If we consider a reaction of the form

s1 + s2 + s3 s1 + s2 + s4 (1.11)then the reaction product s4 can only be 32, s2, s3, s

22 or s

52 depending on s1. Thus the

interaction among molecules is governed only by five simple functions. We will call11Because of the resolution this can of course not be inferred from Fig. 1.5. But if the resolution

allowed it the figure would show when new molecular types would appear.


1

1e+50

1e+100

1e+150

1e+200

1e+250

1e+300

0 20 40 60 80 100 120

geno

type

time [generation]

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120

divs

ersi

ty

time [generation]

Figure 1.5: Simulation of AC2 visualized in the same way as AC1 in Fig. 1.2.Population size M = 100. Note that the genotype axis is scaled logarithmically.


no number genotype / structureof mol.

0 19 01 26 322 10 10243 4 10485764 14 335544325 9 11258999068426246 1 12676506002282294014967032053767 8 425352958651173079329218259289710264328 4 1809251394333065..8131165247501236426506249 3 327339060789614187...25654588539305332852758937610 2 139234637988958594318....704661873320009853338386432no number operator

of mol.0 19 (+ (+ (+ (+ (+ 1 1) ... (+ 1 1)))))1 26 (* X (* X (* X (* X (* X 1)))))2 10 X3 4 Y4 14 (* X X)5 9 X6 1 Y7 8 (* X X)8 4 X9 3 Y10 2 (* X X)

Table 1.2: Organization of molecules taken from a simulation of AC2. In the uppertable the functional part of a molcule is printed in bold face. The rest of the moleculedoes not play any role if the molecule acts as an operator s1.

s2 / s1 0 1 2 3 4 5 6 7 8 9 100 1 0 0 9 0 0 9 0 0 9 01 1 4 1 9 2 1 9 2 1 9 22 1 5 2 9 3 2 9 3 2 9 33 1 6 3 9 * 3 9 * 3 9 *4 1 7 4 9 5 4 9 5 4 9 55 1 8 5 9 6 5 9 6 5 9 66 1 9 6 9 * 6 9 * 6 9 *7 1 10 7 9 8 7 9 8 7 9 88 1 0 8 9 9 8 9 9 8 9 99 1 0 9 9 0 9 9 0 9 9 010 1 0 10 9 0 10 9 0 10 9 0

Table 1.3: Example of a reaction table for fixed s3 = 32...589376 (molecular type 9of Tab. 1.2). The apearance of the table is typical in the sense that s3 can be setto any molecule taken from Tab. 1.2 and the resulting table would show the sameregularities and irregularities as the table displayed here.


s2 / s3 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 4 4 4 4 4 4 4 4 4 4 42 5 5 5 5 5 5 5 5 5 5 53 6 6 6 6 6 6 6 6 6 6 64 7 7 7 7 7 7 7 7 7 7 75 8 8 8 8 8 8 8 8 8 8 86 9 9 9 9 9 9 9 9 9 9 97 10 10 10 10 10 10 10 10 10 10 108 0 0 0 0 0 0 0 0 0 0 09 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0

Table 1.4: Example of a reaction table for fixed s1 = 32 (molecular type 1 ofTab. 1.2). The regularity which becomes visible in this table is typical for the casewhen s1 is fixed.

the set of surviving molecules and all molecules that can be generated by successivereactions among them an organization. In general an organization is defined asa closed and self-maintaining set (Fontana and Buss 1994a). The property closedmeans that no reaction among molecules in the organization creates a moleculenot element of the organization. A set is self-maintaining iff every molecule isproduced by at least one reaction among molecules of the set. As in our examplean organization does not need to be completely present in the population.

The surviving organization of the simulation shown in Fig. 1.5 consists of the fol-lowing molecules:

O = {0} {32n S|n = 5i2j, i N 0, j N 0}. (1.12)

This organization may be represented in a more abstract way as:

O = {0} {(i, j)|i N 0, j N 0, 32n S} (1.13)

so that the abstract molecule (i, j) O corresponds to 322i5j S. We can now lookat the function of an abstract molecule (i, j):

abstract molecule corresponding molecule function conditions O s O

0 0 32(0, 0) 32 s52(0, j) ...432 s22 j 1(1, j) ...4 s2(i, j) ...6 s3 i 2

The validity of the table can be easily proven by induction. From knowing thefunction of an abstract molecule (i, j) we can now define a reaction function r :


(2,2) (1,3)(3,1) (0,4)

(3,0) (2,1) (1,2) (0,3)

(2,0) (1,1) (0,2)

(1,0) (0,1)

(0,0)

0

0 00

00

(4,0)

Figure 1.6: Reaction network of an abstract organization of AC2. The dotted ar-row represents all reactions where the length limit is exceeded and thus the reac-tion product is 0. The arrows on the left-hand side represent reactions of the form0 + s2 + s3 0 + s2 + (0, 0). Recall that the corresponding function of 0 is aconstant function which returns 32. Arrows pointing left upwards are reactions cat-alyzed by s1 = (0, j), j 1. Arrows pointing right upwards are reactions catalyzedby s1 = (0, 0). Replicating and auto-catalytic reactions are not shown.

O O O O operating on abstract molecules which is isomorphic to the reactionfunction r : O O O O:

r((i1, j1), (i2, j2), (i3, j3)) =

(0, 0) if (i1, j1) = 0,(i2, j2 + 1) if (i1, j1) = (0, 0),(i2 + 1, j2) if i1 = 0, j1 1,(i2, j2) if i1 = 1,(i3, j3) if i1 2.

(1.14)

Note that Eq. (1.14) does not consider the length limitation given by L. To addthe length limitation we just have to define r(. . . ) = 0 iff the length limitation isexceeded. This definition can be easily added to Eq. (1.14) but has been omitted forclarity. Figure 1.7 illustrates Eq. (1.14). Figure 1.6 sketches the reaction network ofthe abstract molecules as a graph.

To summarize what we have done above: First, we have recognized a syntacticsimilarity among the molecules forming the surviving organization that allowed torepresent these molecules in a more abstract way. In the abstract representation amolecule of the organization is uniquely specified by a pair of numbers or the symbol0. This new abstract representation is simpler in the sense that a lower numberof characters is needed to print the structure of a molecule. In a second step theinteraction r between abstract molecules has been defined. Its definition need notrefer to the underlying more complex reaction mechanism of AC2.


(i, j) (0, 0)(i, j+1)

(i+1, j)

(i>0, j)

0

(0, j>0)

(0,0)

Figure 1.7: Illustration of Eq. (1.14)

00.20.40.60.8

1

0 2 4 6 8 10 12 14

dive

rsity

time [generation]

00.20.40.60.8

1

0 2 4 6 8 10 12 14

inno

vativ

ity

time [generation]

Figure 1.8: A typical run of AC3.

The property that emergent organizations can be described in a more abstract wayby a syntax and semantics (e.g., an algebra) which are independent of the underlyingchemistry has been discovered by Fontana and Buss (Fontana and Buss 1994a).They showed along various examples how the syntax of members of an emergentorganization can be described by a grammar and the semantics (reactions) can begiven by an algebra. The property which allows the simpler syntactic description ofthe members of an organization is called syntactic closure and the property whichallows a semantic description is called semantic closure by Fontana and Buss.

1.2.3 AC3 - Chaos and Order

The following example shows that a population does not necessarily become ordered.As for example in AC1 where spontaneously a hierarchical group structure appears oras in AC2 where the diversity reduces and a self-maintaining organization appeared.The artificial chemistry AC3 demonstrates that even a simple reaction mechanismcan lead to disorder.


conc. molecule372 00001239 00002145 0000469 0000640 0000735 0000827 0000519 0000a15 0000913 0000c9 0000b4 0000d3 000113 0000f2 000102 0000e2 000121 00018

conc. molecule3 0022c2 002d02 005312 001262 0025a2 000702 011ce2 00d462 000c02 0029f2 00b342 004c72 00fe22 00c7b2 00aee2 010e02 0032a

. . . . . .

conc. molecule2 89a981 13fcb1 d59cf1 238791 c55f91 8c9341 66df41 210301 6b4071 d21351 cab6a1 dbc8c1 eaa621 cf26d1 b14c61 8a3171 84108

. . . . . .

Figure 1.9: Sorted list of molecules present in the population of AC3 for t = 1, t = 8,and t = 14. The population is initialized at t = 0 with only one molecular type,namely 1. The molecules are given in hexadecimal notation.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

conc

entr

atio

n

time [generation]

c(s) = 0, even

c(s) = 1, odd

external perturbation

00.20.40.60.8

1

0 5 10 15 20 25 30

inno

vativ

ity

time [generation]

Figure 1.10: Coexistence of the odd and even cluster in a typical run of AC3. Att = 15 the system is perturbed by replacing 50% randomly selected molecules of thepopulation by 2.


Molecules: The molecules are positive natural numbers bounded by m. Formally:S = {0, 1, 2, . . . , m 1}. For the simulations in this section we set m = 220.Reactions: The reaction scheme of AC3 is the same as for AC1 and AC2:

s1 + s2 + s3 s1 + s2 + s4. (1.15)The product s4 will depend only on s1 and s2. It is defined by the function

s4 = r(s1, s2) =

{(s1 + s2) mod m if odd(s1) odd(s2),(s1 + s2 + 1) mod m otherwise.

(1.16)

The expression odd(s) is true iff s is an odd number.

Dynamics: The dynamics is the same as for AC1 and AC2.

Simulation: Figure 1.8 shows a typical simulation with population size M = 1000where the population is initialized with the molecular type 1. Therefore, therelative diversity at t = 0 is 1/M . Figure 1.8 also shows an estimate of the prob-ability that a collision produces a new molecule which has not been present in thepopulation before. This probability is called innovativity. Both, diversity andinnovativity, increase rapidly and approach the maximum value of one. If the sim-ulation continuous, the innovativity will (must) decrease, because S is finite. It isalso convenient to define a measure of innovation which does not take the wholehistory into account (see Sec. 4.2.4).

The macroscopic behavior (Fig. 1.8) indicates a high degree of disorder in the pop-ulation which is illustrated in Fig. 1.9 by listing the most common molecules fort = 1, t = 8, and t = 14. For t = 1 and t = 8 the influence of the initial conditioncan be clearly observed. But for t = 14 the population looks nearly like a randomlyinitialized population. The reason for this can be found in the reaction mechanism.A common class of deterministic random number generators are based on the sameunderlying mechanism - a linear congruence generator (Knuth 1972).

But there is a hidden order in the population. To reveal this order the moleculeswill be assigned to classes. To do this in systematic way the concept of a classifieris introduced. A classifier C is a function C : S [0, 1] which maps a molecule toa value between 0 and 1. A classifier is an indicator for a certain property, e.g., sizeor stability of a molecule. In general a large value C(s) indicates that the molecules has the property represented by classifier C. The restriction of the return value tothe interval [0, 1] eases comparison between different classifier as it is, for example,required in a cluster analysis. The concept of classifiers is in detail introduced alongthe mesoscopic analysis in Sec. 4.3. Here, as an example, we simply define a discreteclassifier which detects whether a molecule is odd:

C(s) =

{1 if s is odd,0 otherwise.

(1.17)

The population can now be clustered according to the classifier into two clusters.The time evolution of their size is depicted in Fig. 1.10. The figure shows that both


clusters coexist. Their concentrations level off. The stability of the coexistence isdemonstrated by injecting M/2 molecules of type 2 so that the number of evennumbers suddenly increases drastically. After a few generations the perturbation iscompensated.

To understand this phenomena we take a look at the reaction table on cluster level:

s1 / s2 C(s2) = 0 C(s2) = 1C(s1) = 0 C(s4) = 1 C(s4) = 0C(s1) = 1 C(s4) = 0 C(s4) = 0

Loosely speaking, the table shows how clusters interact. For example, the productof a reaction between molecules from cluster 0 will always belong to cluster 1. Thetable can be easily inferred from Eq. (1.16). Looking at the table we can intuitivelysee that the coexistence should be stable.

To support that intuition an ordinary differential equation (ODE) can be derivedby relying on conventional mass action kinetics (p. 1.1.5). This ODE allows (1) toderive the steady state concentration of the clusters, and (2) to prove its stability.Let x0 and x1 be the normalized concentration of cluster C(s) = 0 and C(s) = 1,respectively such that 0 xi 1 and 1 =

xi. The ODE reads:

x1 = x20 x1 = f(x0, x1), (1.18)

x0 = 1 x1. (1.19)

There is only one fixed point x(0)1 = (3

5)/2 0.382. The fixed point is asymp-

totically stable because dfdx

(x(0)1 ) < 0.

The macroscopic equilibrium apparent in Fig. 1.8 is an example for a constructiveequilibrium. The term constructive equilibrium is introduced here to describea phenomenon where the system is stable with respect to macroscopic quantities(e.g., diversity or innovativity) but permanently produces on a microscopic level newcomponents. This phenomenon has especially observed in a matrix-multiplicationchemistry and will be discussed in detail in Chapter 5

Chapter 2

Motivation for Artificial Chemistry Re-search

2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.1 Artificial Chemistries as Models of Chemical Systems . . 40

2.1.2 Artificial Chemistries as Models of Non-Chemical Systems 41

2.1.3 Artifical Chemistries as Sub-Systems of Complex Models 42

2.1.4 Explaining General Phenomena and Mechanisms . . . . . 45

2.2 Information Processing . . . . . . . . . . . . . . . . . . . . 46

2.2.1 Real Chemical Computing . . . . . . . . . . . . . . . . . . 47

2.2.2 Artificial Chemical Computing . . . . . . . . . . . . . . . 48

2.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Discussion of a Potential Theory for Constructive Dy-namical Systems . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.1 Current State . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.2 Components of a Theory . . . . . . . . . . . . . . . . . . 49

2.5 Artificial Chemistry and Evolution Theory . . . . . . . . 50

2.6 Strange Systems . . . . . . . . . . . . . . . . . . . . . . . . 51

2.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . 53

The motivations for artificial chemistry research are as divers as the different ap-proaches that can be found in the field. The motivation provides another way toclassify AC research. This way of classification is quite independent from technicalcharacteristics described in the previous chapter. For this reason a separate chapter

39

40 CHAPTER 2. MOTIVATION FOR ARTIFICIAL CHEMISTRY RESEARCH

Modelling

Information processingControlProving

OptimizationNetworksTravelling Salesman

Parallel ProcessingSociologyBiology

Figure 2.1: Illustration of the motivations underlying artificial chemistry research.

is devoted to the discussion of the various motivations and aims underlying artificialchemistries.

Thinking in terms of artificial chemistries can be useful in certain domains whichshould be organized in three main groups (Fig. 2.1): (1) modeling, (2) informa-tion processing, and (3) optimization. They differ according to their motivations,whether the aim is: (1) to model a system, (2) to build an information processingsystem, or (3) to find a good solution for an optimization problem. These threefundamental directions of the motivation will now be discussed in detail.

2.1 Modeling

The application of artificial chemistries as models is surely the most common use ofartificial chemistries. In that application domain, artificial chemistries are mostlyapplied to model biological or bio-chemical systems. Exceptions are for example theChemical Abstract Machine which models concurrent computer processes (Berry andBoudol 1992) (Sec. 3.1.2) or the Random Prolog Processor applied to model socialsystems (Szuba 1997; Szuba 1998; Szuba and Stras 1997).

2.1.1 Artificial Chemistries as Models of Chemical Systems

As it has been shown in many publications (s. Chapter 3) complex chemical pro-cesses can be modeled by quite abstract artificial chemistries where detailed reactionmechanisms are replaced by mathematical or algorithmic operations. These oper-ations should not accurately emulate real chemical reactions but should capturefundamental principles, (Laing 1972; Varela, Maturana, and Uribe 1974; Hofstadter1979; McCaskill 1988; Kauffman 1993; Fontana 1992; Banzhaf 1993b; Lancet, Ke-dem, and Pilpel 1994; Dittrich 1995; Segre, Pilpel, Glusman, and Lancet 1997). In

2.1. MODELING 41

general, artificial chemistries modeling (bio-)chemical systems can be subdivided ac-cording to their level of abstraction (Sec. 1.1.4). ACs, such as the polymer automata(Mayer and Rasmussen 1998a) or the Belusov-Zhabotinsky reaction formulated inARMS (Suzuki and Tanaka 1997; Suzuki and Tanaka 1998), model concrete chemi-cal systems in an analogous way. That means that a molecule and a reaction in themodel correspond to a molecule and reaction in reality, respectively.

Approaches, such as the chemistries AC1-AC3 in Sec. 1.2, AlChemy, tape-rewritingsystems, assembler automata, number chemistries or random chemistries, modelthe dynamics of chemical-like organizations in an abstract way. These are the ap-proaches to which the term artificial refers to. Therefore, strictly speaking weshould have these abstract models of chemistry in mind when using the term artifi-cial chemistry in a modeling context. If we encompass all kinds of chemical modelsthe term artificial chemistry would lose some substance. Of course one shouldalso be careful not to draw a black-white picture.

2.1.2 Artificial Chemistries as Models of Non-Chemical Systems

The chemical metaphor assists also to build models of non-chemical systems, suchas evolving populations1, ecological systems, social systems, or economic markets.The similarity of these systems and their relation to the chemical metaphor becomesparticularly clear in the replicator equation (Taylor and Jonker 1978; Schusterand Sigmund 1983)

xi = xi(fi(x1, . . . , xn) (x1, . . . , xn)), i = 1, . . . , n (2.1)

which describes the dynamics of n replicating types. The function fi can be inter-preted as the catalytic influence of all other types on the replication rate of xi or asthe fitness of type i which may depend on the concentration of all other types (ingeneral). The dilution flux is given by

(x1, . . . , xn) =n

i=1

xifi(x1, . . . , xn) (2.2)

and can also be interpreted as the average fitness ((Hofbauer and Sigmund 1998),p. 67). The dilution flux ensures that the state of the system stays inside theconcentration simplex Sn = {(x1, . . . , xn) :

xi = 1, xi 0}. On molecular level

the replicator equation is applied to model ensembles of replicating molecules (e.g.,polymers). In this category falls the hypercycle (Eigen and Schuster 1977) whichdynamics can be described by a special form of the replicator equation. On anecological scale the replicator equation describes the dynamics of interacting popu-lations. In this case replication rate constants can be associated with fitness values ofcertain groups or species. The replicator equation is equivalent to the Lotka-Volteraequation (Hofbauer 1981) which demonstrates the close relation between chemical

1Note that this is the biological population terminus.


and ecological dynamics. Even for social and economic systems the replicator equa-tion can serve as a model (Hofbauer and Sigmund 1988).

We will now give examples where artificial chemistries are used as models of ecolog-ical, social, and technical systems.

Ecological Modeling with Artifical Chemistries

Suzuki, Takabayashi, and Tanaka (1999) applied ARMS (abstract chemical re-writing system) to model an ecological system in which plants respond to herbivorefeeding activity by producing volatiles that in turn attract carnivorous natural en-emies of the herbivores. Such defense mechanisms have been reported in severaltritrophic systems. The volatiles are not the mere result of mechanical damage,but are produced by the plant as a specific response to herbivore damage. Suzuki,Takabayashi, and Tanaka (1999) compared the case where plants produce herbivore-induced volatiles vs the case where they do not. They where able to reproduce theseemingly surprising phenomenon where herbivore-induced volatiles, which attractcarnivores, result in the population increase of herbivores.

Social Modeling with Artificial Chemistries

We can also model high-level systems, such as language, fashion, or societies, bychemical-like systems. For example Szuba and Stras (1997) introduced an artificialchemistry based on logic, called the Random Prolog Processor (RPP), and appliedit to model a society of actors solving cooperatively an inference problem. Withreference to this model Szuba and Stras (1997) derived methods to evaluate theinference power of human social structures such as cities or nations. See Sec. 3.1.5,p. 61 for more details.

Modeling Technical Systems

But also technical systems can be modeled by artificial chemistries: The Chemi-cal Abstract Machine developed by Berry and Boudol (1992) is a tool to describeconcurrent computational processes. It is based on the chemical metaphor of the language by Benatre and Le Metayer (1993). See Sec. 3.1.2, p. 58 for more details.

2.1.3 Artifical Chemistries as Sub-Systems of Complex Models

Artificial chemistries can be used as sub-systems in complex models, e.g., modelsof growing cells or ecological simulations. Reasons for using an artificial chemistryinstead of a realistic analogous model of the chemical processes are: (1) lack ofknowledge about the molecules involved and their chemical properties, or (2) searchfor general principles which are independent from a concrete instance of a chemicalsystem.

2.1. MODELING 43

AC as a Sub-System of a Cellular Growth Model

An artificial chemistry as a sub-system can mostly be found in models of interactingand growing cells. These models consists of a collection of cells where each cellcontains a chemical reaction system - sometimes called metabolism. In additioncells are spatially related to one another inside some kind of media and interact bydiffusion of molecules. The dynamics of the chemical reaction system inside the cellsis called intra-cellular dynamics. Cell-cell interaction by diffusion is governed bythe inter-cellular dynamics. Dependent on the application additional propertiesare assigned to substances, e.g., whether they are able to penetrate a membrane,whether they may form a membrane, or whether they initiate a cell division.

These types of cellular models have been successfully applied to explain cell differ-entiation (Furusawa and Kaneko 1998), the origin and evolution of multicellulardiversity (Furusawa and Kaneko 1998; Furusawa and Kaneko 2000), sympatricspeciation2 (Kaneko and Yomo 2000), or chemical evolution of hypotheticallypre-biotic ensembles (Segre, Pilpel, and Lancet 1998). In these works random,non-constructive artificial chemistries with explicit molecules and reaction rules areused. The number of different molecules is relatively small (e.g., 20 molecular typesin (Furusawa and Kaneko 2000)) and the dynamics is modeled by coupled ordi-nary differential equations. Analysis is performed by generating a huge number(e.g., 1000 independent ACs in (Furusawa and Kaneko 2000)) of different artificialchemistries by choosing the reaction rules randomly which allows to draw generaland statistically significant conclusions.

Another domain where artificial chemistries can be found as part of cellular growthmodels is the domain of artificial neural networks (ANN). Here an artificialchemistry controls division and diversification of neurons or growth of synapses.In order to get a desired behavior (development of an ANN with a specific targetfunctionality) the artificial chemistry has so far to be designed manually. Attemptsto automatically generate an artificial chemistry (e.g., by evolutionary algorithms)have not revealed satisfying results yet (Astor and Adami 1998), (Jens Astor, 1998,personal communication).

It should also be noted that the models described above are related to a recent devel-opment of computational matter, the so-called amorphous computing (Abel-son, Allen, Coore, Hanson, Homsy, Knight, Nagpal, Rauch, Sussman, and Weiss1999). An amorphous computer consists of many simple and similar computationaldevices located in an amorphous media. They can communicate by substancewhich are emitted and which diffuse through the media. Computation is viewed asemergence. This means that the result of a computational process is an emergentphenomenon which is represented by the collective state of many devices.

2sympatric = no spatial structure


AC as a Sub-System of an Ecological Model

In an ecological model an artificial chemistry can provide a fundamental currencyif mass conservation and energy conservation is respected. In such ecological modelsorganisms may absorb chemicals from their environment and metabolize them. Theorganisms may even consist of chemicals which are released after death.

An example for this approach is the metabolically-based artificial ecosystem modelEvolve IV (Brewster and Conrad 1998; Brewster and Conrad 1999) which is anadvancement of a series of evolutionary ecosystem computer models originated in1969 (Conrad 1969; Rizki and Conrad 1986). The objective of these models is toinvestigate the conditions under which an evolutionary process comparable to thatobserved in nature occurs (Brewster and Conrad 1998). Species formation is one ofthe investigated phenomena, e.g., the development of diverse quasi-species (Conrad1969; Conrad and Pattee 1970). The aim of Evolve IV is to generate a niche struc-ture reminiscent of the niche structure of a natural ecosystem and to investigatethe conditions under which niche diversification can occur. Brewster and Conrad(1998) describe experiments where the population is initialized with hand-writtenorganisms. They consist of 22 essential genes (e.g., sugar decomposition) and onenon-essential gene (conjugation gene). There are two complementary genes (photo-synthesis and scavenging gene) so that the population may diversify by specializingon one of these genes.

AC as a Sub-System of Gaia Models

A Gaia model is a system which models the interplay between biota and non-livingenvironment. It can be seen as a special form of an ecological model as discussedpreviously. The Gaia hypothesis (Lovelock 1979) assumes that the biota can bothaffect their environment and do so in a manner that benefits life. Gaia theory claimsthat the biota plays an important role in regulating the state of its environment suchthat favorable conditions for life are stabilized. Gaia theory views the whole worldas a highly integrated organism which regulates and self-maintains its state andenvironment.

A justification for this hypothesis is that the composition of the earths atmospherewould be radically different if there were not life on the surface. Without floraand fauna the atmosphere would contain mostly CO2, with very little nitrogen oroxygen.

The Gaia hypothesis is still extremely controversial. Especially the relation of e

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Peter Dittrich - uni-jena.deusers.minet.uni-jena.de/~dittrich/p/main.pdf · On Artiﬂcial Chemistries Dissertation to receive the degree Dr. rer. nat. from the Department of Computer