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# Boolean Systems

## Topics in Asynchronicity

- 1st Edition - January 6, 2023
- Author: Serban E. Vlad
- Language: English
- Paperback ISBN:9 7 8 - 0 - 3 2 3 - 9 5 4 2 2 - 8
- eBook ISBN:9 7 8 - 0 - 3 2 3 - 9 5 5 6 9 - 0

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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Request a sales quoteThe 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.

- 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.

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

- No. of pages: 456
- Language: English
- Edition: 1
- Published: January 6, 2023
- Imprint: Academic Press
- Paperback ISBN: 9780323954228
- eBook ISBN: 9780323955690

SV

### Serban E. Vlad

*Boolean Systems*on ScienceDirect