ISBN: 3540655069
TITLE: Abstract Compositional Analysis of Iterated Relations
AUTHOR: Geurts, Frederic
TOC:

Foreword by Michel Sintzoff V
Preface VII
1. Prologue: Aims, Themes, and Motivations 1
1.1 Complex Relational Dynamical Systems 2
1.1.1 The Context: A First Contact with Dynamical Systems 2
1.1.2 Mutual Exclusion 4
1.1.3 Social Pressure 7
1.1.4 On the Chaotic Demography of Rabbits 9
1.2 Tools and Motivations 14
1.3 Overview of the Monograph 16
Part I. Mathematical Framework: Iterated Relations and Composition
2. Dynamics of Relations 21
2.1 Functional Discrete-Time Dynamical Systems 22
2.2 Relational Dynamical Systems 24
2.2.1 Point-Level Nondeterministic Dynamics 25
2.2.2 Set-Level Deterministic Dynamics 26
2.2.3 Comparison 26
2.3 Preliminary Definitions and Properties 28
2.3.1 Basic Definitions About Relations 28
2.3.2 Notions from Topology 31
2.3.3 Monotonicity and General Junctivity Properties 33
2.3.4 Fixpoint Theorems 37
2.3.5 Elementary Properties 39
2.3.6 Metric Properties 40
2.4 Transfinite Iterations 44
2.4.1 Motivation 44
2.4.2 Transfinite Fixpoint Theorem 45
2.4.3 Transfinite Limits of Iterations 47
2.5 Discussion 48
2.5.1 Relations vs Functions 48
2.5.2 Set-Level Dynamics and Predicate-Transformers 49
2.5.3 Point-Level Dynamics and Trace Semantics 50
2.5.4 Nondeterminism and Probabilistic Choices 50
2.5.5 Transfinite Iterations 51
2.5.6 Time Structure 51
3. Dynamics of Composed Relations 53
3.1 Structural Composition 53
3.2 Composition of Relations 54
3.2.1 Unary Operators 55
3.2.2 N-Ary Operators 56
3.2.3 Composed Dynamical Systems 59
3.3 Dynamics of Composed Relations 62
3.3.1 One-Step Set-Level Evolution of Composed Relations 62
3.3.2 Point-Level Dynamics of Composed Systems 67
3.4 Algebraic Properties of Composition Operators 71
3.4.1 Composition of Unary Operators 72
3.4.2 Composition of Unary and N-Ary Operators 72
3.4.3 Composition of N-Ary Operators 73
3.4.4 Fixpoint Theory for the Composition 75
3.5 Discussion 77
3.5.1 Composition Operators 77
3.5.2 Nondeterminism and Probabilities Revisited 78
3.5.3 Fixpoint Operator and Composition 79
Part II. Abstract Complexity: Abstraction, Invariance, Attraction
4. Abstract Observation of Dynamics 83
4.1 Observation of Systems 83
4.2 Trace-Based Dynamics 85
4.3 Symbolic Observation 86
4.4 Abstraction of Systems 88
4.5 Qualitative Abstract Verification 89
4.6 Observation as Abstraction 91
4.7 Discussion 91
4.7.1 Observation and Abstraction: Related Work 92
4.7.2 Symbolic Dynamics vs Astract Observation 92
4.7.3 Qualitative Abstract Verification 93
5. Invariance, Attraction, Complexity 95
5.1 Invariance 96
5.1.1 Forward and Backward Invariance 96
5.1.2 Global Invariance 100
5.1.3 Strong Invariance 100
5.2 Structure of Invariants 102
5.2.1 Trace-Parametrized Invariants 103
5.2.2 Fullness and Atomicity 104
5.2.3 Chaos 106
5.2.4 Fullness Implies Trace Chaos 108
5.2.5 Fullness and Atomicity Imply Knudsen Chaos 108
5.2.6 Devaney vs Trace vs Knudsen Chaos 109
5.3 Fullness and Atomicity Criteria 110
5.3.1 Criteria 110
5.3.2 Case Studies: Dyadic Map, Cantor Relation, Logistic Map 113
5.4 Attraction 119
5.4.1 Intuition: From Reachability to Attraction 120
5.4.2 From Weak to Full Attraction 121
5.4.3 A Taxonomy of Attraction 123
5.5 Attraction Criteria 125
5.6 Attraction by Invariants 126
5.7 Discussion 128
5.7.1 Invariance and Attraction: Related Notions 128
5.7.2 Energy-Like Functions 129
5.7.3 Dynamical Complexity 130
Part III. Abstract Compositional Analysis of Systems: Dynamics and Computations
6. Compositional Analysis of Dynamical Properties 135
6.1 Aims and Informal Results 135
6.2 Inversion 138
6.3 Restrictions 140
6.3.1 Domain Restriction 140
6.3.2 Range Restriction 141
6.4 Negation 143
6.5 Sequential Composition 144
6.6 Intersection 146
6.7 Union 147
6.8 Products 154
6.8.1 Free Product 154
6.8.2 Connected Product 155
6.9 Combining Union with Free Product 156
6.10 Discussion 156
6.10.1 Compositionality: Summary 157
6.10.2 Limitations and Open Problems 157
6.10.3 Related Work 159
6.10.4 Emergence of Complexity by Structural Composition 160
7. Case Studies: Compositional Analysis of Dynamics 163
7.1 A Collection of Complex Behaviors 163
7.2 Smale Horseshoe Map 164
7.3 Cantor Relation 168
7.4 From Cantor Relation to Truncated Logistic Map 169
7.5 Paperfoldings 172
7.5.1 Introduction 172
7.5.2 Paperfolding Sequences 173
7.5.3 Dynamical Complexity of Paperfoldings 177
7.5.4 Partial Conclusions 180
7.6 Discussion: Compositional Dynamical Complexity 180
8. Experimental Compositional Analysis of Cellular Automata 183
8.1 Aims and Motivations: Attraction-Based Classification and Composition 184
8.2 Preliminary Notions 186
8.2.1 Cellular Automata 186
8.2.2 Transfinite Attraction 188
8.2.3 Shifted Hamming Distance 188
8.3 Experimental Classification 189
8.4 Formal Attraction-Based Classification 191
8.4.1 Introduction 192
8.4.2 Type-N Cellular Automata 193
8.4.3 Type-F Cellular Automata 193
8.4.4 Type-P Cellular Automata 194
8.4.5 Type-S Cellular Automata 194
8.4.6 Type-A Cellular Automata 195
8.4.7 Discussion 196
8.5 Structural Organizations of CA Classes 196
8.5.1 Motivation: Simulation vs Theoretical Results 196
8.5.2 Linear Periodicity Hierarchy 198
8.5.3 Periodicity Clustering 199
8.5.4 Organization w.r.t. Shifted Hamming Distance 199
8.5.5 Dynamical Complexity in CA 201
8.6 Conjectures in CA Composition 201
8.7 Complexity by Composition of Shifts 203
8.7.1 Rules 2 and 16 203
8.7.2 Cantor Relation 204
8.7.3 Comparison 206
8.7.4 A More Precise Conjecture 206
8.8 Qualitative Analysis and Complexity Measures 206
8.9 Compositional Analysis of Complex CA 208
8.9.1 Local Disjunction, Local Union, and Global Union 208
8.9.2 Comparison and Summary of Results 210
8.10 Discussion 211
8.10.1 Summary and Partial Conclusion 211
8.10.2 Open Questions 212
8.10.3 Classification: State-of-the-Art 213
8.10.4 Aperiodicity in Cellular Automata 215
8.10.5 Related Work in Composition 216
9. Compositional Analysis of Computational Properties 217
9.1 Automata as Dynamical Systems 217
9.2 Comparing Dynamical Systems 220
9.2.1 Extrinsic Method 220
9.2.2 Intrinsic Method 221
9.2.3 Our Comparison 221
9.3 From Locality to Globality 221
9.3.1 Turing Machines 222
9.3.2 Cellular Automata223
9.3.3 Continuous Functions 224
9.3.4 General Model 224
9.4 Comparison Through Simulation 227
9.4.1 Simulation 227
9.4.2 Choice of Coding 228
9.4.3 From TM to CA 228
9.4.4 From CA to CF 231
9.4.5 Weak Hierarchy 232
9.5 Topological and Metric Properties 232
9.5.1 Continuity 233
9.5.2 Shift-Invariance 233
9.5.3 Lipschitz Property 234
9.5.4 Shift-Vanishing Effect 235
9.5.5 Nondeterminism 235
9.5.6 Summary 237
9.6 Computability of Initial Conditions 238
9.7 Hierarchy of Systems 239
9.8 Discussion 240
9.8.1 Composition and Computation 240
9.8.2 Further Work 240
9.8.3 Related Work 241
10. Epilogue: Conclusions and Directions for Future Work 243
10.1 Contributions and Related Work 244
10.1.1 Mathematical Framework 245
10.1.2 Compositional Analysis 246
10.2 Directions for Future Research 247
10.2.1 A Patchwork of Open Technical Issues 248
10.2.2 Fractal Image Compression 248
10.2.3 Distributed Dynamical Optimization 249
10.2.4 Distributed Systems and Self-Stabilization 250
10.2.5 Probabilistic Systems and Measures 250
10.2.6 Higher-Order Systems, Control, and Learning 251
10.2.7 Design of Attraction-Based Systems 252
10.3 The Garden of Structural Similarities 253
10.4 Coda: Compositional Complexity Revisited 255
Bibliography 257
Glossary of Symbols 273
Index 277
END
