Table Of ContentBoolean Systems
Boolean Systems
Topics in Asynchronicity
Serban E. Vlad
OradeaCityHalland
TheSocietyofMathematicalSciencesofRomania
Oradea,Romania
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Contents
Preface xix
1. Boolean functions 1
1.1. ThebinaryBoolealgebra 1
1.2. Affinespacesdefinedbytwopoints 4
1.3. Booleanfunctions 6
1.4. Duality 7
1.5. Iterates 8
1.6. Cartesianproductoffunctions 10
1.7. Successorsandpredecessors 11
1.8. FunctionsthatarecompatiblewiththeaffinestructureofBn 15
1.9. TheHammingdistance.Lipschitzfunctions 18
2. Morphisms of generator functions 21
2.1. Definition 21
2.2. Examplesofmorphisms 22
2.3. Composition 23
2.4. Isomorphisms 23
2.5. Synonymousfunctions 24
2.6. Symmetryrelativetotranslations 26
2.7. Morphismsvs.duality 26
2.8. Morphismsvs.iterates 28
2.9. Morphismsvs.Cartesianproductoffunctions 28
2.10. Morphismsvs.successorsandpredecessors 29
2.11. Morphismsvs.fixedpoints 30
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viii Contents
3. State portraits 31
3.1. Preliminaries 31
3.2. Stateportraits 32
3.3. Stateportraitsvs.generatorfunctions 33
3.4. Examples 35
3.5. Statesubportrait 36
3.6. Isomorphisms.Duality 37
3.7. Indegree,outdegree,balancedstateportraits 38
3.8. Path,path-connectedness 39
3.9. Hamiltonianpath,Eulerianpath 40
4. Signals 41
4.1. Definition 41
4.2. Initialvalueandfinalvalue,initialtimeandfinaltime 42
4.3. Duality 42
4.4. Monotonicity 43
4.5. Orbit,orbitalequivalence 43
4.6. Omega-limitset,omega-limitequivalence 44
4.7. Theforgetfulfunction 47
4.8. Theimageofasignalviaafunction 48
4.9. Periodicity 50
5. Computation functions. Progressiveness 53
5.1. Maindefinitions 53
5.2. Morphismsofprogressivecomputationfunctions 55
5.3. Specialcasesofprogressivecomputationfunctions 57
6. Flows and equations of evolution 59
6.1. Flows 59
6.2. Reachability 62
6.3. Examples 62
Contents ix
6.4. Consistency,causalityandcomposition 64
6.5. Equationsofevolution 67
6.6. Flowswithconstantgeneratorfunctions 68
7. Systems 71
7.1. Severalequivalentperspectives 71
7.2. Definition 72
7.3. Subsystem 74
7.4. Cartesianproduct 76
8. Morphisms of flows 79
8.1. Definition 79
8.2. Inducedmorphisms 81
8.3. Morphismsofgeneratorfunctionsvs.morphismsofflows 82
8.4. Composition 83
8.5. Isomorphisms 84
8.6. Symmetryrelativetotranslations 86
8.7. Morphismscompatiblewiththesubsystems 88
8.8. Morphismsvs.duality 91
8.9. Morphismsvs.orbitsandomega-limitsets 91
8.10. Morphismsvs.Cartesianproducts 92
8.11. Morphismsvs.successorsandpredecessors 92
8.12. Morphismsvs.limits 93
8.13. Morphismsvs.orbitalandomega-limitequivalence 94
8.14. Pseudo-morphisms 95
9. Nullclines 97
9.1. Definition 97
9.2. Examples 97
9.3. Properties 99
9.4. Specialcase:NC =Bn 100
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10. Fixed points 103
10.1. Definition 103
10.2. Fixedpointsvs.finalvalues.Restposition 104
10.3. Morphismsvs.fixedpoints 105
11. Sources, isolated fixed points, transient points, sinks 107
11.1. Definition 107
11.2. Morphisms 108
11.3. Otherproperties 109
12. Sets of reachable states 111
12.1. Convergentsequencesofsets 111
12.2. Setsofreachablestates 112
12.3. Example 115
12.4. Isomorphisms 116
13. Dependence on the initial conditions 119
13.1. Definition 119
13.2. Examples 122
13.3. Subsystem 124
13.4. Cartesianproduct 124
13.5. Isomorphisms 125
13.6. Versionsofdependenceontheinitialconditions 125
14. Periodicity 127
14.1. Eventualperiodicityanddoubleeventualperiodicity 127
14.2. Maintheorems 128
14.3. Morphismsvs.periodicity 132
14.4. Otherdefinitionsofperiodicity 133
15. Path-connectedness and topological transitivity 135
15.1. Path-connectedness 135
