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# Abstract Compositional Analysis of Iterated Relations : A Structural Approach to Complex State Transition Systems

### Lecture Notes in Computer Science,

Paperback ISBN: 9783540655060
Number Of Pages: 280

### Paperback

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This self-contained monograph is an integrated study of generic systems defined by iterated relations using the two paradigms of abstraction and composition. This accommodates the complexity of some state-transition systems and improves understanding of complex or chaotic phenomena emerging in some dynamical systems. The main insights and results of this work concern a structural form of complexity obtained by composition of simple interacting systems representing opposed attracting behaviors. This complexity is expressed in the evolution of composed systems (their dynamics) and in the relations between their initial and final states (the computation they realize). The theoretical results are validated by analyzing dynamical and computational properties of low-dimensional prototypes of chaotic systems, high-dimensional spatiotemporally complex systems, and formal systems.

 Foreword p. V Preface p. VII Prologue: Aims, Themes, and Motivations p. 1 Complex Relational Dynamical Systems p. 2 The Context: A First Contact with Dynamical Systems p. 2 Mutual Exclusion p. 4 Social Pressure p. 7 On the Chaotic Demography of Rabbits p. 9 Tools and Motivations p. 14 Overview of the Monograph p. 16 Mathematical Framework: Iterated Relations and Composition Dynamics of Relations p. 21 Functional Discrete-Time Dynamical Systems p. 22 Relational Dynamical Systems p. 24 Point-Level Nondeterministic Dynamics p. 25 Set-Level Deterministic Dynamics p. 26 Comparison p. 26 Preliminary Definitions and Properties p. 28 Basic Definitions About Relations p. 28 Notions from Topology p. 31 Monotonicity and General Junctivity Properties p. 33 Fixpoint Theorems p. 37 Elementary Properties p. 39 Metric Properties p. 40 Transfinite Iterations p. 44 Motivation p. 44 Transfinite Fixpoint Theorem p. 45 Transfinite Limits of Iterations p. 47 Discussion p. 48 Relations vs Functions p. 48 Set-Level Dynamics and Predicate-Transformers p. 49 Point-Level Dynamics and Trace Semantics p. 50 Nondeterminism and Probabilistic Choices p. 50 Transfinite Iterations p. 51 Time Structure p. 51 Dynamics of Composed Relations p. 53 Structural Composition p. 53 Composition of Relations p. 54 Unary Operators p. 55 N-Ary Operators p. 56 Composed Dynamical Systems p. 59 Dynamics of Composed Relations p. 62 One-Step Set-Level Evolution of Composed Relations p. 62 Point-Level Dynamics of Composed Systems p. 67 Algebraic Properties of Composition Operators p. 71 Composition of Unary Operators p. 72 Composition of Unary and N-Ary Operators p. 72 Composition of N-Ary Operators p. 73 Fixpoint Theory for the Composition p. 75 Discussion p. 77 Composition Operators p. 77 Nondeterminism and Probabilities Revisited p. 78 Fixpoint Operator and Composition p. 79 Abstract Complexity: Abstraction, Invariance, Attraction Abstract Observation of Dynamics p. 83 Observation of Systems p. 83 Trace-Based Dynamics p. 85 Symbolic Observation p. 86 Abstraction of Systems p. 88 Qualitative Abstract Verification p. 89 Observation as Abstraction p. 91 Discussion p. 91 Observation and Abstraction: Related Work p. 92 Symbolic Dynamics vs Astract Observation p. 92 Qualitative Abstract Verification p. 93 Invariance, Attraction, Complexity p. 95 Invariance p. 96 Forward and Backward Invariance p. 96 Global Invariance p. 100 Strong Invariance p. 100 Structure of Invariants p. 102 Trace-Parametrized Invariants p. 103 Fullness and Atomicity p. 104 Chaos p. 106 Fullness Implies Trace Chaos p. 108 Fullness and Atomicity Imply Knudsen Chaos p. 108 Devaney vs Trace vs Knudsen Chaos p. 109 Fullness and Atomicity Criteria p. 110 Criteria p. 110 Case Studies: Dyadic Map, Cantor Relation, Logistic Map p. 113 Attraction p. 119 Intuition: From Reachability to Attraction p. 120 From Weak to Full Attraction p. 121 A Taxonomy of Attraction p. 123 Attraction Criteria p. 125 Attraction by Invariants p. 126 Discussion p. 128 Invariance and Attraction: Related Notions p. 128 Energy-Like Functions p. 129 Dynamical Complexity p. 130 Abstract Compositional Analysis of Systems: Dynamics and Computations Compositional Analysis of Dynamical Properties p. 135 Aims and Informal Results p. 135 Inversion p. 138 Restrictions p. 140 Domain Restriction p. 140 Range Restriction p. 141 Negation p. 143 Sequential Composition p. 144 Intersection p. 146 Union p. 147 Products p. 154 Free Product p. 154 Connected Product p. 155 Combining Union with Free Product p. 156 Discussion p. 156 Compositionality: Summary p. 157 Limitations and Open Problems p. 157 Related Work p. 159 Emergence of Complexity by Structural Composition p. 160 Case Studies: Compositional Analysis of Dynamics p. 163 A Collection of Complex Behaviors p. 163 Smale Horseshoe Map p. 164 Cantor Relation p. 168 From Cantor Relation to Truncated Logistic Map p. 169 Paperfoldings p. 172 Introduction p. 172 Paperfolding Sequences p. 173 Dynamical Complexity of Paperfoldings p. 177 Partial Conclusions p. 180 Discussion: Compositional Dynamical Complexity p. 180 Experimental Compositional Analysis of Cellular Automata p. 183 Aims and Motivations: Attraction-Based Classification and Composition p. 184 Preliminary Notions p. 186 Cellular Automata p. 186 Transfinite Attraction p. 188 Shifted Hamming Distance p. 188 Experimental Classification p. 189 Formal Attraction-Based Classification p. 191 Introduction p. 192 Type-<$>{\cal N}<$> Cellular Automata p. 193 Type-<$>{\cal F}<$> Cellular Automata p. 193 Type-<$>{\cal P}<$> Cellular Automata p. 194 Type-<$>{\cal S}<$> Cellular Automata p. 194 Type-<$>{\cal A}<$> Cellular Automata p. 195 Discussion p. 196 Structural Organizations of CA Classes p. 196 Motivation: Simulation vs Theoretical Results p. 196 Linear Periodicity Hierarchy p. 198 Periodicity Clustering p. 199 Organization w.r.t. Shifted Hamming Distance p. 199 Dynamical Complexity in CA p. 201 Conjectures in CA Composition p. 201 Complexity by Composition of Shifts p. 203 Rules 2 and 16 p. 203 Cantor Relation p. 204 Comparison p. 206 A More Precise Conjecture p. 206 Qualitative Analysis and Complexity Measures p. 206 Compositional Analysis of Complex CA p. 208 Local Disjunction, Local Union, and Global Union p. 208 Comparison and Summary of Results p. 210 Discussion p. 211 Summary and Partial Conclusion p. 211 Open Questions p. 212 Classification: State-of-the-Art p. 213 Aperiodicity in Cellular Automata p. 215 Related Work in Composition p. 216 Compositional Analysis of Computational Properties p. 217 Automata as Dynamical Systems p. 217 Comparing Dynamical Systems p. 220 Extrinsic Method p. 220 Intrinsic Method p. 221 Our Comparison p. 221 From Locality to Globality p. 221 Turing Machines p. 222 Cellular Automata p. 223 Continuous Functions p. 224 General Model p. 224 Comparison Through Simulation p. 227 Simulation p. 227 Choice of Coding p. 228 From TM to CA p. 228 From CA to CF p. 231 Weak Hierarchy p. 232 Topological and Metric Properties p. 232 Continuity p. 233 Shift-Invariance p. 233 Lipschitz Property p. 234 Shift-Vanishing Effect p. 235 Nondeterminism p. 235 Summary p. 237 Computability of Initial Conditions p. 238 Hierarchy of Systems p. 239 Discussion p. 240 Composition and Computation p. 240 Further Work p. 240 Related Work p. 241 Epilogue: Conclusions and Directions for Future Work p. 243 Contributions and Related Work p. 244 Mathematical Framework p. 245 Compositional Analysis p. 246 Directions for Future Research p. 247 A Patchwork of Open Technical Issues p. 248 Fractal Image Compression p. 248 Distributed Dynamical Optimization p. 249 Distributed Systems and Self-Stabilization p. 250 Probabilistic Systems and Measures p. 250 Higher-Order Systems, Control, and Learning p. 251 Design of Attraction-Based Systems p. 252 The Garden of Structural Similarities p. 253 Coda: Compositional Complexity Revisited p. 255 Bibliography p. 257 Glossary of Symbols p. 273 Index p. 277 Table of Contents provided by Publisher. 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ISBN: 9783540655060
ISBN-10: 3540655069
Series: Lecture Notes in Computer Science,
Audience: General
Format: Paperback
Language: English
Number Of Pages: 280
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.55
Weight (kg): 0.41