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Boolean Functions: Topics in Asynchronicity

Boolean Functions: Topics in Asynchronicity

Serban E. Vlad

ISBN: 978-1-119-51751-1

Jan 2019

208 pages

$108.99

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Description

The aim of this books is twofold: the first is to provide a formal mathematical description of problems arising in the practical realization of switching circuits; the second is the identification of basic, necessary formal conditions for the avoidance of obvious known problems in the realization of practical switching circuits. This book contains 14 chapters, beginning with the definition of Boolean functions and examples. Next, the author covers the Hamming distance and Lipschitz functions. Morphisms are covered next, followed by anti-morphisms. The author then discusses invariant sets and invariant subsets, followed by coverage of path connected sets. Proper operation is then introduced, followed by the generalized technical condition of proper operation. The strong generalized technical condition of proper operation is then examined. Next, time-reversal symmetry is covered, followed by time-reversal symmetry vs. TCPO. Finally, the book concludes with a chapter on time-reversal symmetry vs. the generalized TCPO. Definitions and examples are provided in each chapter. 

Contents

Preface

1 Boolean functions

1.1 The binary Boole algebra

1.2 Definition of the Boolean functions. Examples. Duality

1.3 Iterates

1.4 State portraits. Stable and unstable coordinates

1.5 Modeling the asynchronous circuits

1.6 Sequences of sets

1.7 Predecessors and successors

1.8 Source, isolated fixed point, transient point, sink

1.9 Translations

2 Affine spaces defined by two points

2.1 Definition

2.2 Properties

2.3 Functions that are compatible with the affine structure of Bn

2.4 The Hamming distance. Lipschitz functions

2.5 Affine spaces of successors

3 Morphisms

3.1 Definition

3.2 Examples

3.3 The composition

3.4 A fixed point property

3.5 Symmetrical functions relative to translations. Examples

3.6 The dual functions revisited

3.7 Morphisms vs predecessors and successors

4 Antimorphisms

4.1 Definition

4.2 Examples

4.3 The composition

4.4 A  fixed point property

4.5 Antisymmetrical functions relative to translations. Examples

4.6 Antimorphisms vs predecessors and successors

5 Invariant sets

5.1 Definition

5.2 Examples

5.3 Properties

5.4  Homomorphic functions vs invariant sets

5.5  Special case of homomorphic functions vs invariant sets

5.6  Symmetry relative to translations vs invariant sets

5.7  Antihomomorphic functions vs invariant sets

5.8  Special case of antihomomorphic functions vs invariant sets

5.9  Antisymmetry relative to translations vs invariant sets

5.10 Relatively isolated sets, isolated set

5.11 Isomorphic functions vs relatively isolated sets

5.12 Antiisomorphic functions vs relatively isolated sets

6 Invariant subsets

6.1 Definition

6.2 Examples

6.3 Maximal invariant subset

6.4 Minimal invariant subset

6.5 Connected components

6.6 Disconnected set

7 Path connected set

7.1 Definition

7.2 Examples

7.3 Properties

7.4 Path connected components

7.5 Morphisms vs path connectedness

7.6 Antimorphisms vs path connectedness

8 Attractors

8.1 Preliminaries

8.2 Definition

8.3 Properties

8.4 Morphisms vs attractors

8.5 Antimorphisms vs attractors

9 The technical condition of proper operation

9.1 Definition

9.2 Examples

9.3 Iterates

9.4 The sets of predecessors and successors

9.5 Source, isolated  fixed point, transient point, sink

9.6 Isomorphisms vs tcpo

9.7 Antiisomorphisms vs tcpo

10 The strong technical condition of proper operation

10.1 Definition

10.2 Examples

10.3 Iterates

10.4 The sets of predecessors and successors

10.5 Source, isolated  fixed point, transient point, sink

10.6 Isomorphisms vs strong tcpo

10.7Antiisomorphisms vs strong tcpo

11 The generalized technical condition of proper operation

11.1 Definition

11.2 Examples

11.3 Iterates

11.4 The sets of predecessors and successors

11.5 Source, isolated  fixed point, transient point, sink

11.6 Isomorphisms vs the generalized tcpo

11.7 Antiisomorphisms vs the generalized tcpo

11.8 Other properties

12 The strong generalized technical condition of proper operation

12.1 Definition

12.2 Examples

12.3 Iterates

12.4 Source, isolated  fixed point, transient point, sink

12.5 Asynchronous and synchronous transient points

12.6 The sets of predecessors and successors

12.7 Isomorphisms vs the strong generalized tcpo

12.8 Antiisomorphisms vs the strong generalized tcpo

13 Time-reversal symmetry

13.1 Definition

13.2 Examples

13.3 The uniqueness of the symmetrical function

13.4 Isomorphisms and antiisomorphisms vs time-reversal symmetry

13.5 Other properties

14 Time-reversal symmetry vs tcpo

14.1 Time-reversal symmetry vs tcpo

14.2 Time-reversal symmetry vs the strong tcpo

14.3 Examples

15 Time-reversal symmetry vs the generalized tcpo

15.1 Time-reversal symmetry vs the generalized tcpo

15.2 Examples

Bibliography

Appendix A The category As

Appendix B Notations

Appendix C Index