Boolean Systems
Topics in Asynchronicity
- 1st Edition - January 6, 2023
- Latest edition
- Author: Serban E. Vlad
- Language: English
The Boolean functions may be iterated either asynchronously, when their coordinates are computed independently of each other, or synchronously, when their coordinates are co… Read more
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Description
Description
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.
Key features
Key features
- Presents a consistent and original theory of the discrete-time Boolean asynchronous systems, which are useful for mathematicians and computer scientists interested in Boolean Networks, dynamical systems, and modeling.
- Studies the flows and equations of evolution, nullclines, dependence on initial conditions, periodicity, path-connectedness, topological transitivity, chaos, nonwandering points, invariance, connectedness, and separation, as well as the basins of attraction, attractors, stability, and time-reversal symmetry.
- Explains the fundamental operating mode of the input systems and the combinational systems (systems without feedback).
- Includes a chapter of applications of the Boolean systems and their modeling techniques.
- Makes use of the unbounded delay model of computation of the Boolean functions.
Readership
Readership
Table of contents
Table of contents
Preface
Chapter 1. Boolean Functions
1. The binary Boole algebra
2. Affine spaces defined by two points
3. Boolean functions
4. Duality
5. Iterates
6. Cartesian product of functions
7. Successors and predecessors
8. Functions that are compatible with the affine structure of Bn
9. The Hamming distance. Lipschitz functions
Chapter 2. Morphisms of Generator Functions
1. Definition
2. Examples of morphisms
3. Composition
4. Isomorphisms
5. Synonymous functions
6. Symmetry relative to translations
7. Morphisms vs. duality
8. Morphisms vs. iterates
9. Morphisms vs. Cartesian product of functions
10. Morphisms vs. successors and predecessors
11. Morphisms vs. fixed points
Chapter 3. State Portraits
1. Preliminaries
2. State portraits
3. State portraits vs. generator functions
4. Examples
5. State subportrait
6. Isomorphisms. Duality
7. Indegree, outdegree, balanced state portraits
8. Path, path connectedness
9. Hamiltonian path, Eulerian path
Chapter 4. Signals
1. Definition
2. Initial value and final value, initial time and final time
3. Duality
4. Monotonicity
5. Orbit, orbital equivalence
6. Omega-limit set, omega-limit equivalence
7. The forgetful function
8. The image of a signal via a function
9. Periodicity
Chapter 5. Computation Functions and Progressiveness
1. Main definitions
2. Morphisms of progressive computation functions
3. Special cases of progressive computation functions
Chapter 6. Flows and Equations of Evolution
1. Flows
2. Reachability
3. Examples
4. Consistency, causality and composition
5. Equations of evolution
6. Flows with constant generator functions
Chapter 7. Systems
1. Several equivalent perspectives
2. Definition
3. Subsystem
4. Cartesian product
Chapter 8. Morphisms of Flows
1. Definition
2. Induced morphisms
3. Morphisms of generator functions vs. morphisms of flows
4. Composition
5. Isomorphisms
6. Symmetry relative to translations
7. Morphisms compatible with the subsystems
8. Morphisms vs. duality
9. Morphisms vs. orbits and omega limit sets
10. Morphisms vs. Cartesian products
11. Morphisms vs. successors and predecessors
12. Morphisms vs. limits
13. Morphisms vs. orbital and omega-limit equivalence
14. Pseudo-morphisms
Chapter 9. Nullclines
1. Definition
2. Examples
3. Properties
4. Special case: NCi = Bn
Chapter 10. Fixed points
1. Definition
2. Fixed points vs. final values. Rest position
3. Morphisms vs. fixed points
Chapter 11. Sources, Isolated Fixed Points, Transient Points, Sinks
1. Definition
2. Morphisms
3. Other properties
Chapter 12. Sets of Reachable States
1. Convergent sequences of sets
2. Sets of reachable states
3. Example
4. Isomorphisms
Chapter 13. Dependence on the Initial Conditions
1. Definition
2. Examples
3. Subsystem
4. Cartesian product
5. Isomorphisms
6. Versions of dependence on the initial conditions
Chapter 14. Periodicity
1. Eventual periodicity and double eventual periodicity
2. Main theorems
3. Morphisms vs. periodicity
4. Other definitions of periodicity
hapter 15. Path Connectedness and Topological Transitivity
1. Path connectedness
2. Topological transitivity
3. Examples
4. Some properties
5. Morphisms
6. Cartesian Products
7. Path connected components
Chapter 16. Chaos
1. Definition
2. Examples
3. Morphisms
Chapter 17. Nonwandering Points and Poisson Stability
1. Nonwandering points
2. Poisson stability
3. Properties
4. Morphisms
Chapter 18. Invariance
1.Definition
2.Examples
3. Invariant subset
4. Properties
5. Morphisms
6. Symmetry relative to translations
7. Subsystems
8. Cartesian products
9. Invariance and path connectedness vs. topological transitivity
10. A Lyapunov-Lagrange type invariance theorem
11. Other possibilities of defining invariance
Chapter 19. Relatively Isolated Sets, Isolated Set
1. Definition
2. Examples
3. Properties
4. When the orbits included in invariant sets are nullclines
5. Isomorphisms
6. Subsystem
Chapter 20. Maximal Invariant Subset
1. Definition
2. Examples
3. Main properties
4. Maximality vs. nullclines
5. Isomorphisms
6. Subsystems
7. Cartesian products
Chapter 21. Minimal Invariant Superset
1. Definition
2. Examples
3. Properties
4. Minimality vs. nullclines
5. Isomorphisms
6. Subsystems
7. Cartesian products
Chapter 22. Minimal Invariant Subset
1. Definition
2. Examples
3. Properties
4. Minimality vs. nullclines
5. Isomorphisms
6. Cartesian products
Chapter 23. Connectedness and Separation
1. Connectedness
2. Separation
3. Examples
4. Properties
5. Connectedness vs. topological transitivity
6. Connectedness vs. path connectedness
7.Connected components
8. Isomorphisms
Chapter 24. Basins of Attraction
1. Definition
2. Examples
3. Properties
4. The basin of attraction of the fixed points
5. The basin of attraction of the periodic points
6. Isomorphisms
Chapter 25. The Basins of Attraction of the States
1. Definition
2. Examples
3. Properties
4. Isomorphisms
Chapter 26. Local Basins of Attraction
1. Definition
2. Properties
3. Isomorphisms
Chapter 27. Local Basins of Attraction of the States
1. Definition
2. Properties
3. Isomorphisms
Chapter 28. Attractors
1. Definition
2. Examples
3. Properties
4. Topological transitivity
5. Path connectedness
6. Isomorphisms
7. Attractors as omega-limit sets
8. Cartesian product
9. Chaos
10. Repellers
11. Weak attractors
Chapter 29. Stability
1. Definition
2. Examples
3. Stability vs. the basins of attraction of the fixed points
4. Morphisms
5. Subsystems
6. (In)dependence on the initial conditions
Chapter 30. Time Reversal Symmetry
1. Definition
2. Examples
3. The uniqueness of the symmetrical function
4. Properties
5. Isomorphisms vs. time-reversal symmetry
6. Cartesian Product
Chapter 31. Generator functions with one parameter
1. Generator functions with one parameter
2. Iterates
3. Cartesian product of functions
4. Successors and predecessors
5. State portrait families
6. Bifurcations
7. Morphisms
Chapter 32. Input Flows and Equations of Evolution
1. Input flows
2. Causality and composition
3. Equations of evolution
4. Morphisms
Chapter 33. Input systems
1. Several equivalent perspectives
2. Definition
3. Subsystem
4. State space decomposition
5. Cartesian product
6. Autonomy
Chapter 34. The Fundamental (Operating) Mode
1. An introductory remark
2. Looking for common sense requests
3. The fundamental (operating) mode
Chapter 35. Combinational Systems with One Level
1. Definition
2. Examples
3. Stability
4. Cartesian product
5. Predecessors and successors
6. Isomorphisms
7. Symmetry relative to translations
8. Invariance
9. Subsystem
Chapter 36. Combinational systems
1.Definition
2.Levels
3. Example
4. The input-output function. Stability
5. Hazards
6. Cartesian product
7.Predecessors and successors
8. Isomorphisms
9. Symmetry relative to translations
10. Invariance
11. Basins of attraction, attractors
12. Subsystem
13. The fundamental operating mode
Chapter 37. Wires, Gates, and Flip Flops
1. Circuits
2. The wire
3. The delay element
4. Gates
5. The SR latch
6. The gated SR Flip Flop
7. The D type Flip Flop
Appendix A. Continuous Time
A.1 Limits, signals and computation functions
A. 2 Systems, several perspectives
Appendix B. Theory of Cheng
1. Semi-tensor products
2. Replacement of B with D
3. Structure matrix
4. Equations of evolution
5. Example
Appendix C. Symbolic dynamics
C.1 Blocks
C.2 Shift spaces
C.3 Languages
C.4 The timeless model of computation
C.5 The unbounded delay model of computation
C.6 The bounded delay model of computation
Notations
Bibliography
Index
Review quotes
Review quotes
Product details
Product details
- Edition: 1
- Latest edition
- Published: January 11, 2023
- Language: English
About the author
About the author
SV