Presentation of the book BOOLEAN ARITHMETIC and its Applications

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1 Presentation of the book BOOLEAN ARITHMETIC and its Applications This book is the handout of one Post Graduate Discipline, offered since 1973, named PEA Boolean Equations Applied to System Engineering, available as an e-book with past 600 pages by download, in the site of Polytechnic School of São Paulo University (EPUSP): It is also the handout of another subject, named EA 029 The New Information Science and the Boolean Arithmetic, a University Extension Course (PECE) of the same School (EPUSP), offered since 1999 (see The previous paper relatively to the edition of this first draft of this book were the following: 1 st 1997 Boolean Arithmetic: A numerical method of solution for System of Boolean Equations, paper submitted to the Journal of Logic and Computation, Oxford, UK, on October 27, (*) 2 nd 1997 Boolean Arithmetic: Implications in Technical Linguistics, paper submitted to the Journal of Logic and Computation, Oxford, UK, on November 17, (*) 3 rd 1997 Boolean Arithmetic: Logic & Technology, paper submitted to the Journal of Logic and Computation, Oxford, UK, on December 5, (*) 4 th 1998 Boolean Arithmetic: Esperangol, Direct & Reverse Computational Language, paper submitted to the Journal of Logic and Computation, Oxford, UK, on January 6, (*) 5 th 1998 Boolean Arithmetic: Text Deduction, paper submitted to the Journal of Logic and Computation, Oxford, UK, on March 2, (*) 6 th 1998 Boolean Arithmetic: Applications of Boolean Matrix in a Logic-Based System, paper submitted to the Journal of Logic and Computation, Oxford, UK, on April 7, (*) 7 th 1998 Boolean Arithmetic: Applications to one Non-Von Neumann Computing, paper submitted to the Journal of Logic and Computation, Oxford, UK, in July, (*) In consequence of the length of these seven papers, marked with (*), Professor D. M. Gabbay of Department of Computer Science, King s College, Strand, London, as Editor-in-Chief of Journal of Logic and Computation edited by Oxford University Press, Oxford, UK, send to me a letter where he suggest that I put all, material about Boolean Arithmetic, together as a book, because it seemed to him that we had a lot to say about the topic. Indeed it is truth, as these papers refer to a lifetime research at the Polytechnic School of Sao Paulo University (EPUSP) about Boolean Mathematics and its technological applications. Our hitherto unpublished theses were presented at EPUSP in 1965 for the degree of Ph.D. and in 1967, as requirement for Professorship Promotion at Electrical Engineering Department. These theses entitled respectively, Contributions to the study of switching circuits (1 st Part and 2 nd Part), which marked the beginning of these researches about Boolean Arithmetic and Technical Linguistics (also called Boolean Linguistics).

2 NOTE: There is a extra-chapter, called CORRESPONDENCE, where in one of its Sections, it is shown an opportune Professor Dana S. Scott's suggestion given in a letter, that was utilized in terms of the Boolean Arithmetic's language. Since 1973 until nowadays, I have been teaching at EPUSP the discipline at the Post Graduate Course, PEA 5737: Introduction of the Boolean Equations in Engineering Systems, where a series of these handouts were distributed as part of the content. This book tries to call attention of the reader interested in mathematical fundamentals of Boolean Arithmetic and Technical Linguistics sectors. The CONTENTS of the book BOOLEAN ARITHMETIC and its Applications", has the following fourteen Chapters, what are available Section by Section, separately by download, later on: The Chapter 1 (pp.12-22) Introduction, gives a general overview of the role of Mathematical Logic in the Information Science and its technological development in all aspects of the subject to applications in hardware, software and firmware engineering. The Chapter 2 (pp.23-50) The absence of mathematics in the Information Science, has the following Sections: Section 2.1 The three great mathematical crises (p. 23); Section 2.2 The concept of time in Mathematics (p. 27); Section 2.3 The present great crisis in Mathematics: The fourth (p. 38); Section 2.4 Conclusions (p. 45). This Chapter provides a general overview of the Logic in Mathematics and its consequent absence in the Information Science, showing the last (the fourth) great crisis in Mathematics. The Chapter 3 (pp ) Boolean Mathematics: General Theory, has the following Sections: Section 3.1 Isomorphic properties (p. 51); Section 3.2 Properties of Boolean Algebra (p. 55); Section 3.3 Properties of Boolean Arithmetic (p. 60); Section 3.4 Technological Values (p. 87); Section 3.5 Practical examples (p. 90). This Chapter begins with a general overview of a Boolean Mathematics through isomorphic properties and empirical procedures that allows the obtainment of different aspects of this mathematical presentation and its technological applications. The Chapter 4 (pp ) Boolean Arithmetic: Theoretical Background, has the following Sections: Section 4.1 Boolean Arithmetical Field (p. 99); Section 4.2 Boolean Arithmetical Variables (p. 102); Section 4.3 Boolean Arithmetical Functions (p. 103); Section 4.4 Boolean Arithmetical Equations (p. 104); Section 4.5 System of Boolean Arithmetical Functions (p. 110); Section 4.6 System of Arithmetical Boolean Equations (p. 117); Section 4.7 Practical Application ( 118). This Chapter focused on the theoretical background of the Boolean Arithmetic and its corresponding concepts to Boolean Algebra.

3 The Chapter 5 (pp ) An Analytical Method of Solution for Systems of Boolean Equations, has the following Sections: Section 5.1 Applications of De Morgan s Laws (p. 123); Section 5.2 Analytical determination of the general solution (p. 131); Section 5.3 Practical Applications (p. 139). This Chapter 5 gives an analytical method of solution for a System of Boolean Equation, whose deduction was obtained after the equivalent numerical method of solution was known. The Chapter 6 (pp ) A Numerical Method of Solution for Systems of Boolean Equations, has the following Sections: Section 6.1 Partition Method (p. 153); Section 6.2 Practical Applications (p. 158). This Chapter 6 gives a numerical method of solution for a System of Arithmetical Boolean Equation, which is termed as Partition Method. The Chapter 7 (pp ) Boolean Arithmetical Function of Boolean Arithmetical Function and its Inverse Operations, has the following Sections: Section 7.1 Logical Implication, as a Direct Operation (p. 179); Section 7.2 Generalization of the Direct BAF(BAF) Operation (p. 191); Section 7.3 First Inverse Antecedent BAF(BAF) Operation (p. 205); Section 7.4 Second Inverse Consequent BAF(BAF) Operation (p. 233); Section 7.5 Practical examples The Svoboda Problem (p. 258). This Chapter 7 shows how it is possible to establish a numerical work to make a job with Boolean Arithmetical Function of Boolean Arithmetical Function, BAF(BAF), as one Direct Arithmetical Operation and its corresponding Inverse Operations. The Chapter 8 (pp ) Implications in Technical Linguistics, has the following Sections: Section 8.1 Areas of Thought (p. 267); Section 8.2 Laws of Minimum Effort of Technical Linguistics (p. 272); Section 8.3 Practical examples (p. 280). This Chapter 8 focused on the implications of the Boolean Arithmetic in the Technical Linguistics and in its respective Laws of Minimum Effort. The Chapter 9 (pp ) Esperangol, Direct & Reverse Computational Language, has the following Sections: Section 9.1 The General Problem of System Engineering (p. 292); Section 9.2 The Automatic Rational Structured Programming (p. 295); Section 9.3 Esperangol, Direct and Reverse Computational Language (p. 298); Section 9.4 Practical examples (p. 300). This Chapter 9, besides introducing a concept of a General Problem of System Engineering (GPSE) and a general definition of an Automatic Rational Structured Programming (ARSP), gives the fundamental concepts of the so called Esperangol, Direct & Reverse Computational Language.

4 The Chapter 10 (pp ) An Arithmetical Analysis of the Aristotelian Categorical Syllogisms, has the following Sections: Section 10.1 The Aristotelian Syllogisms (p. 313); Section 10.2 Boolean Arithmetical Equations of Aristotelian Data Problem (p. 316); Section 10.3 Boolean Arithmetical Determination on the Aristotelian Conclusion (p. 334); Section 10.4 The Boolean Arithmetical Results (p. 337). This Chapter 10 shows as an application of Boolean Arithmetic and how it is possible to establish a numerical work that can be introduced into Mathematics, the millenarian Aristotelian Categorical Syllogisms. The Chapter 11 (pp ) Text Deduction, has the following Sections: Section 11.1 The complete deduction of a set of Linguistic Postulates (p. 345); Section 11.2 The Nodal Automatic Transition Analysis (NATA) (p. 354); Section 11.3 An application of the Reverse Mathematics (p. 370). This Chapter 11 shows how it is possible the application of the Partition Method to obtain the Complete Solution of a given set of Linguistics Text Postulates and also establish a numerical work to make another application to solve the problem Hilbert s Reverse Mathematics. The Chapter 12 (pp ) Applications of Boolean Matrix in a Logic- Based System, has the following Sections: Section 12.1 Network of Logical Arithmetical Functions (p. 376); Section 12.2 Properties of Logical Arithmetical Functions (p. 382); Section 12.3 A Cartesian Model of Logic-Based Database System (p. 395); Section 12.4 Practical examples (p. 412). This Chapter 12 shows how it is possible to establish an application of a Cartesian Model to Logic-Based Database System and how it is possible works numerically with Network of Logical Arithmetical Functions (LAF s). The Chapter 13 (pp ) Logic & Technology, has the following Sections: Section 13.1 Technological background (p. 416); Section 13.2 The Mathematical Von Neumann Deviation (p. 421); Section 13.3 A Mathematical Model to the Von Neumann s Machine (p. 425). This Chapter 13 shows, after a historical and technological background, the present situation of the Logic Safety in Computing and how it can be extended. The Chapter 14 (pp ) Applications to one Non-Von Neumann Computing, has the following Sections: Section 14.1 The Design Matrix Method (DMM) (p. 430); Section 14.2 A Non-Von Neumann Computing Generation (p. 434); Section 14.3 A Non-Von Neumann Programming (p. 441); Section 14.4 Database Programs: Gray Code Routes for Robotics (p. 444); Section 14.5 An Example of Logical Sequential Design (p. 449). This Chapter 14 shows how it is possible to establish an application of one Non- Von Neumann Computing Technology and their mathematical computing model.

5 The Chapter 15 (pp ) Genetic Engineering and Boolean Arithmetic, has the following Sections: Section 15.1 An Historical Contribution (p.459) Section 15.2 Introduction (p. 463) Section 15.3 The Genetic Code (p. 465) Section 15.4 Boolean Arithmetic applied to Genetic Code (p. 468) Section 15.5 Determination of NT of DNA Amino Acids (p. 490) Section 15.6 Conclusions (p. 510) Section 15.7 Glossary / Bibliography (p. 511) This Chapter 15, written IN MEMORIAN a C. E. Shannon ( ), beginning referring to hitherto unpublished Ph. D. thesis at M. I. T., April 15, 1940, on the mathematics of genes and heredity entitled, An Algebra for Theoretical Genetics, that was therefore, the first work which was applied Boolean Algebra to Biology. The numerical expressions of Boolean Arithmetical Functions (BAFs) of the twenty Amino acids and its respective Initiator and Terminator Codons, which are functions of the four basics nucleotides involved in the Genetic Code of RNA ( Uracil, Guanine. Cytosine and Adenine ) and the four basics nucleotides involved in the Genetic Code of DNA ( Thymine, Guanine. Cytosine and Adenine ) are presented, as a contribution to the introduction of the Boolean Arithmetic into the Genetic Engineering, with the purpose of its possible utilization by the genetic researches to put in terms of Boolean Equation, its genome and proteinome sequences data of the living being. Enough, the two Genome- Sequencing Strategies utilized in Bioinformatics ( Nested Shotgun Approach of the Human Genome Project and Whole Shotgun Approach of the Celera Genomics) are in reality, empirical methods based on practical experience and experimental rather than theory, that is, trial and error method are used to achieve higher production rates. This method means that we have an absence of mathematics in that Genetic Engineer which is responsible for expenses in time and resources, and do not still offer the singleness of its results guarantee. This comment reinforces that was seen in Iten 8 th relative to THE RESULTS OF A LIFETIME RESEARCH CARRIED OUT AT UNIVERSITY OF SÃO PAULO - POLYTECHNIC SCHOOL - DEPARTMENT OF ELECTRICAL ENERGY AND AUTOMATION: 8 th To establish new promising researches in the field of Genetics through Boolean Mathematical Models in computing applications of the Theoretical Biology. The Text Deduction of the Chapter 11, for instance, could be suggested its use instead of the computing of empirical models and isolated facts.