Screen LibraryComputational analysis of one-dimensional cellular automata /
Cellular automata provide one of the most interesting avenues into the study of complex systems in general, as well as having an intrinsic interest of their own. Because of their mathematical simplicity and representational robustness they have been used to model economic, political, biological, eco...
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| Main Authors: | |
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| Corporate Authors: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore : |
| Publication Dates: | 1996. |
| Literature type: | eBook |
| Language: | English |
| Series: |
World scientific series on nonlinear science. Series A ;
v. 15 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/2712#t=toc |
| Summary: |
Cellular automata provide one of the most interesting avenues into the study of complex systems in general, as well as having an intrinsic interest of their own. Because of their mathematical simplicity and representational robustness they have been used to model economic, political, biological, ecological, chemical, and physical systems. Almost any system which can be treated in terms of a discrete representation space in which the dynamics is based on local interaction rules can be modelled by a cellular automata. The aim of this book is to give an introduction to the analysis of cellular automata (CA) in terms of an approach in which CA rules are viewed as elements of a nonlinear operator algebra, which can be expressed in component form much as ordinary vectors are in vector algebra. Although a variety of different topics are covered, this viewpoint provides the underlying theme. The actual mathematics used is not hard, and the material should be accessible to anyone with a junior level university background, and a certain degree of mathematical maturity. |
| Carrier Form: | 1 online resource (x,275pages) : illustrations (some color). |
| Bibliography: | Includes bibliographical references. |
| ISBN: | 9789812798671 |
| CLC: | TP301.1 |
| Contents: | ch. 1. The operator algebra of cellular automata. 1. Basic definitions and notation. 2. Boolean forms of CA rules. 3. The canonical basis operators. 4. Symmetry transformations on CA rules. 5. CA rules as maps of the interval. 6. Exercises for chapter 1 -- ch. 2. Cellular automata arithmetic. 1. Products of CA rules. 2. The division algorithm. 3. Residue arithmetic of cellular automata. 4. Exercises for chapter 2 -- ch. 3. Fixed points and cycles. 1. Fixed points and shift cycles via rule decomposition. 2. Jen's invariance matrix method. 3. Exercises for chapter 3 -- ch. 4. Commutation of CA rules. 1. Computation of commutator sets. 2. Idempotence. 3. Ito relations. 4. Some interesting sub-groups. 5. Exercises for chapter 4 -- ch. 5. Additive rules: I. Basic analysis. 1. Conditions for additivity. 2. Cyclotomic analysis. 3. Injectivity. 4. Another view of injectivity. 5. Exercises for chapter 5 -- ch. 6. Additive rules: II. Cycle structures and entropy. 1. State transition diagrams. 2. Cycle periods. 3. Reachability. 4. Conditions for states on cycles. 5. Entropy reduction. 6. Exercises for chapter 6 -- ch. 7. Additive rules: III. Computation of predecessors. 1. Predecessors of 3-site rules. 2. k-site rules. 3. Examples. 4. The operator B. 5. Exercises for chapter 7 -- ch. 8. The binary difference rule. 1. Basic properties of D. 2. The graph of D:[0,1]-->[0,1]. 3. Some numerical relations. 4. A density result. 5. Exercises for chapter 6 -- ch. 9. Computation of pre-images. 1. Pre-images and predecessors. 2. The rule graph and basic matrix. 3. Computation of pre-images from the basic matrix. 4. Pre-images and the Jen recurrence relations. 5. exercises for chapter 9 -- ch. 10. The garden of Eden. 1. GE(X) and GE*(X). 2. Computation of GE*(X). 3. GE*(X) for 3-site rules. 4. Classification with GE*. 5. Exercises for chapter 10 -- ch. 11. Time series simulation. 1. Cellular automata generating time series. 2. Statistics of time series. 3. Exercises for chapter 11 -- ch. 12. Surjectivity of cellular automata rules. 1. A kitchen sink theorem. 2. The de Bruijn diagram. 3. The subset diagram. 4. The semi-group [symbol](X). 5. The subset matrix and some replacement diagrams. 6. Exercises for chapter 12. |