The Boolean functions may be iterated either asynchronously, when their coordinates are computed independently of each other, or synchronously, when their coordinates are computed at the same time. In Boolean Systems: Topics in Asynchronicity, a book addressed to mathematicians and computer scientists interested in Boolean systems and their use in modelling, author Serban E. Vlad presents a consistent and original mathematical theory of the discrete-time Boolean asynchronous systems. The purpose of the book is to set forth the concepts of such a theory, resulting from the synchronous Boolean system theory and mostly from the synchronous real system theory, by analogy, and to indicate the way in which known synchronous deterministic concepts generate new asynchronous nondeterministic concepts. The reader will be introduced to the dependence on the initial conditions, periodicity, path-connectedness, topological transitivity, and chaos. A property of major importance is invariance, which is present in five versions. In relation to it, the reader will study the maximal invariant subsets, the minimal invariant supersets, the minimal invariant subsets, connectedness, separation, the basins of attraction, and attractors. The stability of the systems and their time-reversal symmetry end the topics that refer to the systems without input. The rest of the book is concerned with input systems. The most consistent chapters of this part of the book refer to the fundamental operating mode and to the combinational systems (systems without feedback). The chapter Wires, Gates, and Flip-Flops presents a variety of applications. The first appendix addresses the issue of continuous time, and the second one sketches the important theory of Daizhan Cheng, which is put in relation to asynchronicity. The third appendix is a bridge between asynchronicity and the symbolic dynamics of Douglas Lind and Brian Marcus.
1. Boolean Systems - p. Page i 2. Cover - p. Page i 3. Front Matter - p. Page i 4. Table of Contents - p. Page iii 5. Copyright - p. Page iii 6. Dedication - p. Page iv 7. Contents - p. Page vi 8. Preface - p. Page vi 9. List of Illustrations - p. Page xi 10. List of Tables - p. Page xii 11. 1 : Boolean functions - p. Page xii 12. 2 : Morphisms of generator functions - p. Page 20 13. 3 : State portraits - p. Page 30 14. 4 : Signals - p. Page 40 15. 5 : Computation functions. Progressiveness - p. Page 52 16. 6 : Flows and equations of evolution - p. Page 58 17. 7 : Systems - p. Page 69 18. 8 : Morphisms of flows - p. Page 77 19. 9 : Nullclines - p. Page 95 20. 10 : Fixed points - p. Page 101 21. 11 : Sources, isolated fixed points, transient points, sinks - p. Page 106 22. 12 : Sets of reachable states - p. Page 110 23. 13 : Dependence on the initial conditions - p. Page 117 24. 14 : Periodicity - p. Page 126 25. 15 : Path-connectedness and topological transitivity - p. Page 134 26. 16 : Chaos - p. Page 144 27. 17 : Nonwandering points and Poisson stability - p. Page 149 28. 18 : Invariance - p. Page 154 29. 19 : Relatively isolated sets, isolated set - p. Page 175 30. 20 : Maximal invariant subset - p. Page 182 31. 21 : Minimal invariant superset - p. Page 193 32. 22 : Minimal invariant subset - p. Page 202 33. 23 : Connectedness and separation - p. Page 209 34. 24 : Basins of attraction - p. Page 220 35. 25 : Basins of attraction of the states - p. Page 236 36. 26 : Local basins of attraction - p. Page 244 37. 27 : Local basins of attraction of the states - p. Page 252 38. 28 : Attractors - p. Page 258 39. 29 : Stability - p. Page 274 40. 30 : Time-reversal symmetry - p. Page 284 41. 31 : Generator functions with one parameter - p. Page 293 42. 32 : Input flows and equations of evolution - p. Page 304 43. 33 : Input systems - p. Page 311 44. 34 : The fundamental (operating) mode - p. Page 320 45. 35 : Combinational systems with one level - p. Page 326 46. 36 : Combinational systems - p. Page 334 47. 37 : Wires, gates, and flip flops - p. Page 359 48. A : Continuous time - p. Page 385 49. B : Theory of Cheng - p. Page 391 50. C : Symbolic dynamics - p. Page 403 51. Notations - p. Page 423 52. Bibliography - p. Page 427 53. Index - p. Page 427
The Boolean functions may be iterated either asynchronously, when their coordinates are computed independently of each other, or synchronously, when their coordinates are computed at the same time. In Boolean Systems: Topics in Asynchronicity, a book addressed to mathematicians and computer scientists interested in Boolean systems and their use in modelling, author Serban E. Vlad presents a consistent and original mathematical theory of the discrete-time Boolean asynchronous systems. The purpose of the book is to set forth the concepts of such a theory, resulting from the synchronous Boolean system theory and mostly from the synchronous real system theory, by analogy, and to indicate the way in which known synchronous deterministic concepts generate new asynchronous nondeterministic concepts. The reader will be introduced to the dependence on the initial conditions, periodicity, path-connectedness, topological transitivity, and chaos. A property of major importance is invariance, which is present in five versions. In relation to it, the reader will study the maximal invariant subsets, the minimal invariant supersets, the minimal invariant subsets, connectedness, separation, the basins of attraction, and attractors. The stability of the systems and their time-reversal symmetry end the topics that refer to the systems without input. The rest of the book is concerned with input systems. The most consistent chapters of this part of the book refer to the fundamental operating mode and to the combinational systems (systems without feedback). The chapter Wires, Gates, and Flip-Flops presents a variety of applications. The first appendix addresses the issue of continuous time, and the second one sketches the important theory of Daizhan Cheng, which is put in relation to asynchronicity. The third appendix is a bridge between asynchronicity and the symbolic dynamics of Douglas Lind and Brian Marcus.
1. Boolean Systems - p. Page i 2. Cover - p. Page i 3. Front Matter - p. Page i 4. Table of Contents - p. Page iii 5. Copyright - p. Page iii 6. Dedication - p. Page iv 7. Contents - p. Page vi 8. Preface - p. Page vi 9. List of Illustrations - p. Page xi 10. List of Tables - p. Page xii 11. 1 : Boolean functions - p. Page xii 12. 2 : Morphisms of generator functions - p. Page 20 13. 3 : State portraits - p. Page 30 14. 4 : Signals - p. Page 40 15. 5 : Computation functions. Progressiveness - p. Page 52 16. 6 : Flows and equations of evolution - p. Page 58 17. 7 : Systems - p. Page 69 18. 8 : Morphisms of flows - p. Page 77 19. 9 : Nullclines - p. Page 95 20. 10 : Fixed points - p. Page 101 21. 11 : Sources, isolated fixed points, transient points, sinks - p. Page 106 22. 12 : Sets of reachable states - p. Page 110 23. 13 : Dependence on the initial conditions - p. Page 117 24. 14 : Periodicity - p. Page 126 25. 15 : Path-connectedness and topological transitivity - p. Page 134 26. 16 : Chaos - p. Page 144 27. 17 : Nonwandering points and Poisson stability - p. Page 149 28. 18 : Invariance - p. Page 154 29. 19 : Relatively isolated sets, isolated set - p. Page 175 30. 20 : Maximal invariant subset - p. Page 182 31. 21 : Minimal invariant superset - p. Page 193 32. 22 : Minimal invariant subset - p. Page 202 33. 23 : Connectedness and separation - p. Page 209 34. 24 : Basins of attraction - p. Page 220 35. 25 : Basins of attraction of the states - p. Page 236 36. 26 : Local basins of attraction - p. Page 244 37. 27 : Local basins of attraction of the states - p. Page 252 38. 28 : Attractors - p. Page 258 39. 29 : Stability - p. Page 274 40. 30 : Time-reversal symmetry - p. Page 284 41. 31 : Generator functions with one parameter - p. Page 293 42. 32 : Input flows and equations of evolution - p. Page 304 43. 33 : Input systems - p. Page 311 44. 34 : The fundamental (operating) mode - p. Page 320 45. 35 : Combinational systems with one level - p. Page 326 46. 36 : Combinational systems - p. Page 334 47. 37 : Wires, gates, and flip flops - p. Page 359 48. A : Continuous time - p. Page 385 49. B : Theory of Cheng - p. Page 391 50. C : Symbolic dynamics - p. Page 403 51. Notations - p. Page 423 52. Bibliography - p. Page 427 53. Index - p. Page 427
