Boolean Functions - Vlad, Serban E

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Résumé :

Preface xi

1 Boolean Functions 1

1.1 The Binary Boole Algebra 2

1.2 Definition of the Boolean Functions. Examples. Duality 4

1.3 Iterates 6

1.4 State Portraits. Stable and Unstable Coordinates 11

1.5 Modeling the Asynchronous Circuits 14

1.6 Sequences of Sets 14

1.7 Predecessors and Successors 15

1.8 Source, Isolated Fixed Point, Transient Point, Sink 18

1.9 Translations 19

2 Affine Spaces Defined by Two Points 21

2.1 Definition 21

2.2 Properties 23

2.3 Functions that Are Compatible with the Affine Structure of Bn 25

2.4 The Hamming Distance. Lipschitz Functions 28

2.5 Affine Spaces of Successors 31

3 Morphisms 35

3.1 Definition 35

3.2 Examples 36

3.3 The Composition 38

3.4 A Fixed Point Property 39

3.5 Symmetrical Functions Relative to Translations. Examples 39

3.6 The Dual Functions Revisited 41

3.7 Morphisms vs. Predecessors and Successors 42

4 Antimorphisms 45

4.1 Definition 45

4.2 Examples 46

4.3 The Composition 48

4.4 A Fixed Point Property 51

4.5 Antisymmetrical Functions Relative to Translations. Examples 51

4.6 Antimorphisms vs Predecessors and Successors 52

5 Invariant Sets 55

5.1 Definition 55

5.2 Examples 57

5.3 Properties 58

5.4 Homomorphic Functions vs Invariant Sets 60

5.5 Special Case of Homomorphic Functions vs Invariant Sets 62

5.6 Symmetry Relative to Translations vs Invariant Sets 63

5.7 Antihomomorphic Functions vs Invariant Sets 64

5.8 Special Case of Antihomomorphic Functions vs Invariant Sets 65

5.9 Antisymmetry Relative to Translations vs Invariant Sets 66

5.10 Relatively Isolated Sets, Isolated Set 67

5.11 Isomorphic Functions vs Relatively Isolated Sets 68

5.12 Antiisomorphic Functions vs Relatively Isolated Sets 69

6 Invariant Subsets 71

6.1 Definition 71

6.2 Examples 72

6.3 Maximal Invariant Subset 72

6.4 Minimal Invariant Subset 74

6.5 Connected Components 76

6.6 Disconnected Set 77

7 Path Connected Set 81

7.1 Definition 81

7.2 Examples 82

7.3 Properties 84

7.4 Path Connected Components 84

7.5 Morphisms vs Path Connectedness 85

7.6 Antimorphisms vs Path Connectedness 85

8 Attractors 87

8.1 Preliminaries 88

8.2 Definition 89

8.3 Properties 90

8.4 Morphisms vs Attractors 94

8.5 Antimorphisms vs Attractors 95

9 The Technical Condition of Proper Operation 97

9.1 Definition 97

9.2 Examples 100

9.3 Iterates 101

9.4 The Sets of Predecessors and Successors 101

9.5 Source, Isolated Fixed Point, Transient Point, Sink 103

9.6 Isomorphisms vs tcpo 104

9.7 Antiisomorphisms vs tcpo 105

10 The Strong Technical Condition of Proper Operation 107

10.1 Definition 107

10.2 Examples 109

10.3 Iterates 110

10.4 The Sets of Predecessors and Successors 110

10.5 Source, Isolated Fixed Point, Transient Point, Sink 111

10.6 Isomorphisms vs Strong tcpo 111

10.7 Antiisomorphisms vs Strong tcpo 112

11 The Generalized Technical Condition of Proper Operation 115

11.1 Definition 115

11.2 Examples 119

11.3 Iterates 120

11.4 The Sets of Predecessors and Successors 121

11.5 Source, Isolated Fixed Point, Transient Point, Sink 125

11.6 Isomorphisms vs the Generalized tcpo 126

11.7 Antiisomorphisms vs the Generalized tcpo 128

11.8 Other Properties 129...

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