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Boolean Systems. Topics in Asynchronicity

  • Book
  • January 2023
  • Elsevier Science and Technology
  • ID: 5638189

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.

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Table of Contents

1. Boolean Functions
2. Morphisms of Generator Functions
3. State Portraits
4. Signals
5. Computation Functions and Progressiveness
6. Flows and Equations of Evolution
7. Systems
8. Morphisms of Flows
9. Nullclines
10. Fixed points
11. Sources, Isolated Fixed Points, Transient Points, Sinks
12. Sets of Reachable States
13. Dependence on the Initial Conditions 
14.  Periodicity
15. Path Connectedness and Topological Transitivity
16. Chaos
17. Nonwandering Points and Poisson Stability
18. Invariance
19. Relatively Isolated Sets, Isolated Set
20. Maximal Invariant Subset
21. Minimal Invariant Superset
22. Minimal Invariant Subset
23. Connectedness and Separation
24. Basins of Attraction
25. The Basins of Attraction of the States
26. Local Basins of Attraction
27. Local Basins of Attraction of the States
28. Attractors
29. Stability
30. Time Reversal Symmetry 
31. Generator functions with one parameter
32. Input Flows and Equations of Evolution
33. Input systems
34. The Fundamental (Operating) Mode
35. Combinational Systems with One Level
36. Combinational systems
37. Wires, Gates, and Flip Flops
Appendix
A. Continuous Time 
B. Theory of Cheng
C. Notations

Authors

Serban Vlad The Society of Mathematical Sciences from Romania, SSMR; The Association of Applied Mathematics and Mechanics from Germany, GAMM. Serban E. Vlad is an analyst programmer at the Oradea City Hall, Romania. He is a member of the Society of Mathematical Sciences from Romania SSMR and of the Association of Applied Mathematics and Mechanics from Germany GAMM. He is the author of many papers and several books and book chapters.