Boolean Functions

SERBAN E. VLAD is an Analyst Programmer at Oradea City Hall, Romania. He is a member of the Romanian and the German societies of industrial and applied mathematics, ROMAI and GAMM. He is the author of many papers and several books and book chapters.

• Preface xi1 Boolean Functions 11.1 The Binary Boole Algebra 21.2 Definition of the Boolean Functions. Examples. Duality 41.3 Iterates 61.4 State Portraits. Stable and Unstable Coordinates 111.5 Modeling the Asynchronous Circuits 141.6 Sequences of Sets 141.7 Predecessors and Successors 151.8 Source, Isolated Fixed Point, Transient Point, Sink 181.9 Translations 192 Affine Spaces Defined by Two Points 212.1 Definition 212.2 Properties 232.3 Functions that Are Compatible with the Affine Structure of Bn 252.4 The Hamming Distance. Lipschitz Functions 282.5 Affine Spaces of Successors 313 Morphisms 353.1 Definition 353.2 Examples 363.3 The Composition 383.4 A Fixed Point Property 393.5 Symmetrical Functions Relative to Translations. Examples 393.6 The Dual Functions Revisited 413.7 Morphisms vs. Predecessors and Successors 424 Antimorphisms 454.1 Definition 454.2 Examples 464.3 The Composition 484.4 A Fixed Point Property 514.5 Antisymmetrical Functions Relative to Translations. Examples 514.6 Antimorphisms vs Predecessors and Successors 525 Invariant Sets 555.1 Definition 555.2 Examples 575.3 Properties 585.4 Homomorphic Functions vs Invariant Sets 605.5 Special Case of Homomorphic Functions vs Invariant Sets 625.6 Symmetry Relative to Translations vs Invariant Sets 635.7 Antihomomorphic Functions vs Invariant Sets 645.8 Special Case of Antihomomorphic Functions vs Invariant Sets 655.9 Antisymmetry Relative to Translations vs Invariant Sets 665.10 Relatively Isolated Sets, Isolated Set 675.11 Isomorphic Functions vs Relatively Isolated Sets 685.12 Antiisomorphic Functions vs Relatively Isolated Sets 696 Invariant Subsets 716.1 Definition 716.2 Examples 726.3 Maximal Invariant Subset 726.4 Minimal Invariant Subset 746.5 Connected Components 766.6 Disconnected Set 777 Path Connected Set 817.1 Definition 817.2 Examples 827.3 Properties 847.4 Path Connected Components 847.5 Morphisms vs Path Connectedness 857.6 Antimorphisms vs Path Connectedness 858 Attractors 878.1 Preliminaries 888.2 Definition 898.3 Properties 908.4 Morphisms vs Attractors 948.5 Antimorphisms vs Attractors 959 The Technical Condition of Proper Operation 979.1 Definition 979.2 Examples 1009.3 Iterates 1019.4 The Sets of Predecessors and Successors 1019.5 Source, Isolated Fixed Point, Transient Point, Sink 1039.6 Isomorphisms vs tcpo 1049.7 Antiisomorphisms vs tcpo 10510 The Strong Technical Condition of Proper Operation 10710.1 Definition 10710.2 Examples 10910.3 Iterates 11010.4 The Sets of Predecessors and Successors 11010.5 Source, Isolated Fixed Point, Transient Point, Sink 11110.6 Isomorphisms vs Strong tcpo 11110.7 Antiisomorphisms vs Strong tcpo 11211 The Generalized Technical Condition of Proper Operation 11511.1 Definition 11511.2 Examples 11911.3 Iterates 12011.4 The Sets of Predecessors and Successors 12111.5 Source, Isolated Fixed Point, Transient Point, Sink 12511.6 Isomorphisms vs the Generalized tcpo 12611.7 Antiisomorphisms vs the Generalized tcpo 12811.8 Other Properties 12912 The Strong Generalized Technical Condition of Proper Operation 13112.1 Definition 13112.2 Examples 13512.3 Iterates 13612.4 Source, Isolated Fixed Point, Transient Point, Sink 13712.5 Asynchronous and Synchronous Transient Points 14112.6 The Sets of Predecessors and Successors 14112.7 Isomorphisms vs the Strong Generalized tcpo 14412.8 Antiisomorphisms vs the Strong Generalized tcpo 14613 Time-Reversal Symmetry 14713.1 Definition 14813.2 Examples 15013.3 The Uniqueness of the Symmetrical Function 15113.4 Isomorphisms and Antiisomorphisms vs Time-Reversal Symmetry 15113.5 Other Properties 15214 Time-Reversal Symmetry vs tcpo 15514.1 Time-Reversal Symmetry vs tcpo 15514.2 Time-Reversal Symmetry vs the Strong tcpo 15614.3 Examples 15915 Time-Reversal Symmetry vs the Generalized tcpo 16315.1 Time-Reversal Symmetry vs the Generalized tcpo 16315.2 Examples 168Appendix A The Category As 171Appendix B Notations 175Bibliography 177Index 181