BOOLEAN ARITHMETIC and its Applications
Address of SITE:
Presentation of the book "BOOLEAN ARITHMETIC and its Applications"
This book is the handout of one PostGraduate Discipline, named PEA - 5737 "Boolean Equations Applied to System Engineering" (see: http://www.poli.usp.br/pea5737) of Polytechnic School of São Paulo University (EPUSP), offered since 1973. 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.
The previous paper relatively to the edition of this first draft of this book were the following:
1st 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, 1997. (*)
2nd 1997 ‘Boolean Arithmetic: Implications in Technical Linguistics’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on November 17, 1997. (*)
3rd 1997 ‘Boolean Arithmetic: Logic & Technology’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on December 5, 1997. (*)
4th 1998 ‘Boolean Arithmetic: Esperangol, Direct & Reverse Computational Language’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on January 6, 1998. (*)
5th 1998 ‘Boolean Arithmetic: Text Deduction’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on March 2, 1998. (*)
6th 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, 1998. (*)
7th 1998 ‘Boolean Arithmetic: Applications to one Non-Von Neumann Computing’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, in July, 1998. (*)
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 the following letter:
Thus, Professor D. M. Gabbay, Editor-in- Chief of the Journal of Logic and Computation, after receiving those various papers on Boolean Arithmetic, has suggested that I put it all 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 (1st Part and 2nd Part)", which marked the beginning of these researches about Boolean Arithmetic and Technical Linguistics (also called Boolean Linguistics).
Since 1973 until nowadays, I have been teaching at EPUSP the discipline at the PostGraduate 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.
Its CONTENTS has the following fourteen Chapters:
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. 51-98) 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 obtention of different aspects of this mathematical presentation and its technological applications.
The Chapter 4 (pp. 99-122) 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.
The Chapter 5 (pp. 123-152) 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. 152-178) 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. 179-266) 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 Generalisation 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. 267-290) 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. 291-312) 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.
The Chapter 10 (pp. 313-343) 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. 344-374) 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).
The 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. 375-415) 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. 416-429) 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. 430-457) 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.
THE RESULTS OF A LIFETIME RESEARCH CARRIED OUT AT UNIVERSITY OF SÃO PAULO - POLYTECHNIC SCHOOL - DEPARTMENT OF ELECTRICAL ENERGY AND AUTOMATION (ABSTRACT).
"Esperangol" replaces the present chaotic Machine Language since it is based on Mathematical Machine Language and therefore, it is Universal as well as being UNIQUE.
I believe that people would be very interested in "Esperangol", a direct and reversible computer language, due to the following:
1st.: This new language makes the introduction of any kind of "virus" in the computer process, impossible;
2nd.: It allows the automatic production of software mathematically free of failures caused by human factors;
3rd.: It allows complete safety in computer real time operation, where there will be no more need of the "Ctrl+Alt+Del" keys.
I think that even many circumstances might have been changed in these last years, until now, there was no interruption in the research carry out at that University. Although there was an incredible development of Information Science during this past time, solutions of some problems have not been found yet. These main problems are:
.The chronic crisis in real time software;
.The present viral proliferation in the sector;
.The lack of common nucleus in the different applications of Artificial Intelligence;
.The unobtainable software automatically free,
.The fifth generation computer system.
For these reasons, the research relative to this Pure Mathematical Field, led me to find a new area of studies, called Boolean Arithmetic, mathematically isomorphic to the Boolean Algebra, where problems mainly aroused by current computer language into the operations system, can be studied and solved. Then, a draft of my new book on "Boolean Arithmetic and its Applications" now is ready, whose CD-Rom's copy by download (~600 pages), is in the SITE: http://www.poli.usp.br/pea5737. Its CONTENTS presents some results of the researches, referring to the author’s Curriculum Vitæ inside annexed. This present enlarged work shows that Boolean Arithmetic solves problems using only bits. But, and as it is a mathematical logical system, it may prove to be a valuable instrument in remodeling computer programming in order to avoid the current computer languages, which may cause problems to the operational system.
I think that my approach via Boolean Arithmetic can be transformed in a valuable instrument in remodeling computer programming. Nowadays, Software Engineering programs are empirically developed. I may say that Mathematics is not yet used in Programming, whereas, Hardware Engineering is strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. This Boolean Arithmetic could be useful in searching solution for current problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks the computer operation.
For this purpose in Chapter 5 of the referred CD-Rom copy of "Boolean Arithmetic and its Applications", I presented the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represent graphically a part of a general computing program. These simple examples show us that incompatibility, may cause logical "bug", and are not due to HUMAN FACTORS as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6 (pp. 162-172), using numerical method as Partition Method, Doubling Operators, etc. The idea is to maintain the Nodal Decisions when the "forbidden eras", could be technologically eliminated. In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", these examples are again repeated as a Problem on pp. 355-369, with more details, using the BAF(BAF) properties of Chapter 7 (pp.179-266). It was shown how these "bugs" can be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter of the "Boolean Arithmetic and its applications", refers to my book, "ESÇÃO (n~m~p): A Non-Von Neumann Computer", published in 1985. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which was seen in the previous Sections of the referred book "Boolean Arithmetic and its applications", can be found in some publications referred in p.457, based on the last book. Other publications as referred on p.457, show the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles, which enable us to find the solution for the problem.
I think, in conclusion, that the most important part of this book is that it probably allows:
1st To establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd To establish a true Mathematical Model to the current Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd To establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, it will remain only the memory unit and the respective input/output devices. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any resource to the present situation;
4th To establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th To re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th To establish promising researches in the field of the Reverse Mathematics, as it was seen as an application in Chapter 11 (p. 344) in the sense of Hilbert’s Program proposal;
7th To establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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.
Thus, paradoxically, the absence of mathematics in the Information Science, which revealed the 4th Great Crisis in Mathematics and allowed the extraordinary technological development, began to demand in its growing and wide employment of the logical artificial thought, more and more fundamentals of the utilized scientific true.
THE DELIVERY OF THE DRAFT OF THE BOOK, "BOOLEAN ARITHMETIC and its Applications"
When the draft of the book was ready, in May 21 1998, I sent it to the Professor D. M. Gabbay as an answer to that amazing and stimulant invitation of his 16/7/98 letter, relative to put the entire subject about Boolean Arithmetic, together, as a book:
In 28th September 1999, I received from Professor D. M. Gabbay the following letter:
Then, a brief analysis from the exposed situation is the following:
1. Time to write the draft of the book since letter invitation (16/7/1998) to its delivery (21/5/1999) to Professor Gabbay: It was little more than 9 months;
2. Time that Professor Gabbay have had an opportunity to take a brief look through the received book: It was little more than 4 months until its receipt (28/9/1999);
After these times, Professor Gabbay, reluctantly, tell me the following:
a. That he does not believe that the material received will fit with reorganization with the book series that he edit;
b. That he does not think of an alternative to suggest to me.
At the same time that I have written to Professor Gabbay (21/5/1999), I have sent to others Professors, the following three letters
1st letter:
May 21, 1999
To
Professor Chris L. Hankin
Dept. of Computing
Imperial College
180, Queen's Gate, LONDON, SW7 2BZ, UK.
e-mail: clh@doc.ic.ac.uk
Ref.: Interview with Mr. Ricardo Wagner Colombini Martins, held in the beginning of June, 1994, at Imperial College, London, about ‘Boolean Arithmetic".
Dear Professor Hankin,
Due to Professor Gabbay’s suggestion, made in July 1998, about putting as a book some submission papers to Journal of Logic and Computation, a Partial Draft of this book entitled "Boolean Arithmetic and its Applications" is ready. I hope that, with successive revisions with your possible assistance as Editorial Board’s Member of the Journal, this text will develop into a book which may prove useful after the understanding of Boolean Arithmetic.
These researches on Boolean Arithmetic were carried out at the Department of Electrical Energy and Automation from Polytechnic School of Sao Paulo University, Brazil.
My son Ricardo, will give you a copy of this boolk and I hope to have your comments on it.
Thank you for your attention, as well as for your kindness in receiving him once more.
Yours sincerely,
(a) Prof. Wagner Waneck Martins
2nd letter:
São Paulo, 21 de maio de 1999.
À
Professora Doutora Myriam Gondim Déchamps,
Université Paris Sud – Centre d’Orsay
Mathématique – Bat. 425
91405 – ORSAY – Cedex France.
Myriam.Dechamps@math.u-psud.fr
Referência: Remessa de cópia do manuscrito do livro "Boolean Arithmetic and its applications".
Prezada Professora,
Conforme entendimentos havidos através de seu irmão e meu colega de turma, Engº Pedro Gondim, o portador da presente é meu filho Ricardo Wagner Colombini Martins, que lhe entregará para seu conhecimento, uma cópia do manuscrito do livro em referência, acompanhado de cópias das cartas aos Professores D. M. Gabbay, Chris L. Hankin e Dana S. Scott, relativas a entregas de outras cópias.
Tendo já há algum tempo tido notícias pelo seu irmão, sobre o sucesso de suas atividades no campo da Matemática, tomei a liberdade de entrar no site http://www.math.u-psud.fr/~dechamps e fiquei deslumbrado com a descrição do seu detalhamento e dos resultados alcançados. Creio não exagerar, que esses trabalhos e pesquisas realizadas pela prezada Professora constituem honra e glória para nós brasileiros, razão pela qual peço que aceite as minhas mais sinceras congratulações.
Como dentro de suas atividades de interesse geral ressaltam as referentes ao ensino da Matemática, tomo liberdade de anexar à presente, também para seu conhecimento, a cópia de um trabalho correlato com o mesmo assunto do livro em referência, apresentado em setembro de 1976 durante o "The 3rd International Congress of Mathematical Education", realizado em Karlsruhe, Alemanha e intitulado "BOOLEAN ARITHMETIC: Implications in Mathematical Teaching", acompanhado de sua versão para o idioma germânico.
Ao seu esposo, estimado Professor Dechamps, ao qual também apresento meus cordiais cumprimentos, tomo a liberdade também de apresentar uma cópia do trabalho intitulado "Could ‘Magnetic Super Pulse’ possibily be the Macroscopic Manifestation of the ‘Quantum Leap’?", que talves possa interessá-lo no campo da Física.
Para terminar, antecipadamente agradeço qualquer atenção de sua parte relativamente ao assunto, no sentido da divulgar essas ideias junto ao pessoal especializado dessa Universidade e eventualmente interessado em contatar-me a respeito, sou,
Atenciosamente
(a) Wagner Waneck Martins
3rd letter:
May 21, 1999
To
Professor Dana S. Scott,
Sch. of Computer Science,
Carnegie Mellon University,
Pittsburgh – PA 15213 – USA.
e-mail: dana_scott@proof.ergo.cmu.edu
Subject: 1st Partial draft of the book ‘Boolean
Arithmetic and its applications’
Dear Professor Scott,
In the late 1976, during the realisation of "LOGIC COLLOQUIUM’76 – OXFORD – 19 to 30 July, I had the pleasure to contact you personally. After our meeting, besides my admittance as member of A.S.L., the abstract entitled "Boolean equations systems: A numerical method of solution", was published in "THE JOURNAL OF SYMBOLIC LOGIC" – September, 1977, pp. 460-461 (Annexes 1 and 2).
In these past 23 years there was no interruption of the researches carried out on Boolean Arithmetic and Technical Linguistics (or, Boolean Linguistics) whose beginning was at the Polytechnic School of University of São Paulo, Brazil.
Recently, after submitting various papers on Boolean Arithmetic to Journal of Logic and Computation, I had the honor to receive in July, 1998, a suggestion from Professor D. M. Gabbay, to put it all together as a book. Because of this suggestion, the 1st Partial draft of the book "Boolean Arithmetic and its applications", is ready.
Although there was an incredible development of Information Science, solutions of some problems have not been found yet. These main problems are the chronic crisis in real time software; the present viral proliferation in the sector; the lack of common nucleus in the different applications of Artificial Intelligence; the unobtainable software automatically free.
We believe that this new area of studies, Boolean Arithmetic, may present opportune mathematical contributions to find the solutions for these problems.
As you were responsible for the publication of this abstract I believe you may be interested in this book, which will how besides Boolean Arithmetic can be derived from Boolean Algebra, as well as some practical applications.
This present enlarged work shows that Boolean Arithmetic solves problems using bits. But, and as it is a mathematical logical system, it may prove to be a valuable instrument in remodeling computer programming in order to avoid the current computer languages which cause sometimes problems to the system of operation.
I hope that, with successive revisions and your possible assistance as advisor to OUP of Journal of Logic and Computation, this text will develop into a book where the usefulness of Boolean Arithmetic can be achieved.
Looking forward to hearing from you, I remain
Yours Sincerely,
a. Professor Wagner Waneck Martins
Only this third letter, addressed to the to Professor Dana S. Scott of Carnegie Mellon University, in 8th June 1999, I have had the honor to receive an answer, witch carry in Boolean Algebra language, suggestion about the example on p. 118/158 of the book.
This letter was the following:
Thus, I have had an excellent opportunity with my answer to Professor Scott, to translate its suggestion into Boolean Arithmetic language, using the isomorphic numerical properties with Boolean Algebra, as it is referred in the book.
As this suggestion stands for me a very relevant question proposed by a remarkable Professor, I spend in 16/8/1999 a long answer that, I think, it is better to release:
August 16, 1999
To
Professor Dana S. Scott,
Computer Science Department
Carnegie Mellon University,
5000 Forbes Avenues
Pittsburgh, Pennsylvania PA 15213–3891, USA.
E-mail: dana.scott@cs.cmu.edu
Subject: 1st Partial draft of the book:
‘Boolean Arithmetic and its applications’
Your letter ~8th June 1999.
Dear Professor Scott:
Thank you for your letter referred above and your suggestion about the example on p. 118/158 of the book.
I would like to apply this suggestion in general, with your permission, using the isomorphic numerical language of Boolean Arithmetic referred in the book. As the example on p. 118 it refer to two Boolean Functions, "Y1 = P" and "Y2 = R", the application after, will be made in three part. : Part a): Only with the Boolean Functions, "Y1 = P". Part b): Only with the Boolean Functions, "Y2 = R". Part c): The entire example, with simultaneous Boolean Functions, "Y1" and "Y2" (or, "P and R").
Then, some preliminary observations are given:
1. On p. 119 of the first solution of this example after Fig. 5, is written that it is a "historical solution", because of the late origin of these researches which led to Boolean Arithmetic and Technical Linguistics. At that time, the properties used in the second solution presented on p.139 and in the third solution presented on page 158 were unknown. These properties refer to the employed ANALYTICAL METHOD OF SOLUTION of Chapter 5 (p. 123-151) and the "DOUBLING OPERATOR" (p. 66) and PARTITION METHOD (p. 153). This "historical solution" was, as a façon de parler, the beginning of these researches, when the paths through these numerical mathematical fields were discovered;
2. For all applications of Boolean Arithmetic it will be adopted the concepts detailed from Boolean Arithmetical Variables (BAV), Boolean Arithmetical Function (BAF) through its Numerical Transform (NT) on pages (51-55)/(102-104) and the items a) to e) on pages (89-90). The item a) refers to the use of "Hindrance" of Negative Logic as logical unit instead of the dual concept of "Transmission" of Positive Logic (see Fig. 1 – p. 88), which allows that the classical representation of Equations in Mathematics with the second member null can be maintained;
3. A formal Boolean Equation is a logical interrelation between two Boolean Functions of any given variables. This logical interrelation could be a "logical implication" which is called on page 105, "Symbol 4": X ® Y (or vice versa, X Y), or both, that is, "logical equality" which is called on pages 105-106, "Symbol 5": X = Y. As Boolean Arithmetical Functions (BAF) and Boolean Functions are isomorphic, the properties of the isomorphism guarantee that these interrelations are maintained in Boolean Arithmetic (p. 104);
4. The Boolean variables used in the example on p. 118 and repeated on p. 158 are replaced by the Boolean variables used in your suggestion, according to the following table:
Y1 Y2 X1 X2 X3 X3 (X2,X1)
P R q p r r =Q(p,q)
Thus, in the general Boolean Arithmetical Equation language, your suggestion could have the following translation, adopting one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}":
"If the Boolean Arithmetical Equation (BAE) is given, , we want to determine the following Boolean Arithmetical Function incognito: , such that we have satisfied the numerical expression:
Thus, applying the Property (20a) - p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE, we have:
However, we can suppose that the incognito function is:
Then, making the substitution of expression (2) in (1), we have:
Therefore, in order to determine all the possible solutions, we must have:
In expression (4) we know the Boolean Arithmetical Function (BAF), , that is, the binary components of the abscissa of the Numerical Transform (NT). Then, through that expression (4) we can determine all the possible numerical values for the four incognito components { } of the abscissa of the Numerical Transform (NT) of the Boolean Arithmetical Function (BAF) incognito, expressed in (2).
In order to apply that useful suggestion, adequately translated to the numerical language of Boolean Arithmetical Equation, the example on p. 118 will be separate in two preliminaries parts:
Part a): "Given the Boolean Arithmetical Equation (BAE), we want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical expression: ."
Note: In the solution of this part, as an illustration, all details of the employment of Boolean Arithmetic are given.
Initially, we consider the Boolean Arithmetical Function (BAF) of the first member of the given BAE. Then, applying the definition of Numerical Transform (NT) of a Boolean Arithmetical Function (BAF) - p. 54-55 (Truth Table and Boolean Arithmetical Variables procedures), or, using the properties of the "Doubling Operators" - p. 66, we have:
1st Mode: Through the Truth Table procedure (p. 91):
S1Þ ß T1 r p q
t0 0 0 0 P(0,0,0) = a 0 = 0
t1 0 0 1 P(0,0,1) = a 1 = 0
t2 0 1 0 P(0,1,0) = a 2 = 0 Therefore, we obtain the
t3 0 1 1 P(0,1,1) = a 3 = 0 following expression (5):
t4 1 0 0 P(1,0,0) = a 4 = 1
t5 1 0 1 P(1,0,1) = a 5 = 0
t6 1 1 0 P(1,1,0) = a 6 = 1
t7 1 1 1 P(1,1,1) = a 7 = 1
2nd Mode: Through the Boolean Arithmetical Variables (BAVs) procedures (p. 92):
Choosing one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", we have:
Applying the distributive property of NTs to the given Boolean Arithmetical Function, we have the following expression:
Substituting the numerical values (6) of the NTs of the BAVs in the expression (7), we have:
Then, we have the same final Boolean Arithmetical Equation:
(5)
3rd Mode: Through the "Doubling Operators" (p. 66):
Applying the distributive property of NTs to the given Boolean Arithmetical Function, we have the following expression:
But, applying the "Doubling Operators" properties (p. 66) we have:
Substituting the numerical values given by expression (9) in expression (8), we have:
Then, we have the same final Boolean Arithmetical Equation:
(5)
However, we can suppose that the incognito function is:
such that
(10)
Thus, applying the Property (20a) on p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE as we have seen, we have:
or,
, or,
Therefore, in order to determine all the possible solutions of BAE (11), we must have:
The first solution, as null solution, is rejected:
NT{r = Q(p,q)} = [0000]2.{2;p q} = 0
The second solution is accepted as a unique full solution:
In Boolean arithmetical language:
13. NT{r = Q(p,q)} = [0010]2.{2;p q}
In Boolean algebraic language, the Anti-Transform is:
r = Q(p,q) = NT-1{ [0010]2 .{2;p,q} } = P1 =p’ v q \ r = p’v q
Proof:
, where r = p’v q
Then: P = (p’v q) v (p & q’) = (p’v q v p) & (p’ v q v q’) = 0 (QED)
Part b): "Given the Boolean Arithmetical Equation (BAE), we want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical expression: ."
Choosing one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", and applying the "Doubling Operators" properties (p. 66) to the expression (14):
14.
we have:
Substituting the numerical values given by expression (15) in expression (14), we have:
Then, we have the final Boolean Arithmetical Equation:
(16)
However, we can suppose that the incognito function is:
such that
(17)
Thus, applying the Property (20a) on p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE as we have seen, we have:
or,
or,
Therefore, in order to determine all the possible solutions of BAE (11), we must have:
.
Then, we have the following two solutions restricted in times "t2" and " t3":
In Boolean arithmetical language, these solutions are:
1st solution: NT{ r = Q(p,q)} = [ii11] 2 {2;p q} (rejected)
2nd solution: NT{ r = Q(p,q)} = [ii10] 2 {2;p q} (accepted), but restricted in "t2" and " t3".
In Boolean algebraic language, the correspondent Anti-Transform of this 2nd solution is:
19. r = Q(p,q) = NT-1{ [ii10]2 .{2;p,q} }, that is (see example on p. 352):
a. To obtain the semantic interpretation of their restriction, we make the replacement of this symbolic bit "i", by the bit "0" and the other bit elements by bit "1" – than the NODAL RESTRICTION is obtained. This NODAL RESTRICTION corresponds in the expression (19), to times "t2" and " t3" (forbidden eras – p.355).
b. Then, the NODAL RESTRICTION is given through Boolean algebraic language, or the following Anti-Transform:
19. r = N(p,q) = NT-1{ [0011]2 .{2;p,q} } = S3 v S2 =(p’ & q’) v (p’ & q) = p’
c) If we make the replacement of the symbolic bit "i" by the bit "1" and remaining others without any alteration, the NODAL CONCLUSION is obtained.
d) Then, the unique NODAL CONCLUSION is:
r = Q(p,q) = NT-1 { [1110]2 .{2;p q} } =S0 = p & q, but restricted in times "t2" and " t3" (forbidden eras – p.355).
Note: This semantic interpretation of the NODAL RESTRICTION was possible because it was a consequence of the employment of Boolean Equation concepts.
Now, I’m applying the general Boolean Arithmetical Equation language, as it was referred in the beginning of this letter, on the entire example in discussion, which is the following:
Part c): "Given the simultaneous System of Boolean Arithmetical Functions (SBAFs):
We want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical logical equality expression:
."
Note: In the solution of this entire example, as a last application of your useful suggestion, it is different from the other three presented in the book. The first is referred on p. 118, and it is a historical solution. The second is repeated on p. 139 and it is an application of an Analytical Method of solution for System of Boolean Equations. The third is repeated on page 158 and it is an application of the "DOUBLING OPERATOR" and PARTITION METHOD properties.
Choosing a Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", we have:
Then, by expression (20) we must have:
The fundamental logical "Symbol 5" (pages 105-106), between the abscissa of the "NTs", brings about the following equality,
[1101 0000]2 = [1100 1110]2 , that is:
[1101 0000]2 v ([1100 1110]2)’ & ([1101 0000]2)’ [1100 1110]2 = 0
But,
([1100 1110]2)’ = [0011 0001]2 and ([1101 0000]2)’ = [0010 1111]2
Then, we have:
[1101 0000]2 v [0011 0001]2 & [0010 1111]2 v [1100 1110]2 = 0
[0001 0000]2 & [0000 1110]2 = 0, or
The desired solution of this SESABE (Solving Equation of System of Arithmetical Boolean Equation) or the searched solution is, as we have supposed:
Then, applying the Property (20a) – p. 69, to the first member of the above (SESABE), we have:
Replacing expression (2) in expression (23), we have:
or,
Therefore, in order to determine all the possible solutions of Boolean Arithmetical Equation (BAE) expressed by (25), we must have:
That is, we have the unique numerical solution in Boolean arithmetical language:
(27) NT{r = Q(p,q)} = [1110]2.{2;p q}
In Boolean algebraic language, the Anti-Transform is:
r = Q(p,q) = NT-1{ [1110]2 .{2;p,q} } = S0 =p & q \ r = p & q
Proof: , where r = p & q. Then: P = (p & q) v (p & q’) = p
R = p & r’ v q, where r = p & q. Then: R = p & (p & q)’ v q = p & p’ v q’ v q = p
That is, we have P = R, when r = p & q (QED).
In your letter, you referred to my approach via Boolean Arithmetic that it can be "a valuable instrument in remodelling computer programming."
With your permission, some of the problems provoked by current computer language into the system of operation, can be presented as follows:
Nowadays, Software Engineering build its programs empirically, that is, Mathematics is not yet used in Programming, whereas, Hardware Engineering built its chips strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. This Boolean Arithmetic could be useful to the searching of solution of problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks its running.
For this purpose in Chapter 5, I present the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represents graphically a part of a general computing program. These simple examples show us that incompatibilities, which cause logical "bug", are not due to HUMAN FACTORS as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6, respectively on pp. 162-170 and 170-172, using numerical method as Partition Method, Doubling Operators, etc, to maintain the Nodal Decisions when the "forbidden eras", technologically could be eliminated. In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", these examples are again repeated as a Problem on pp. 355-369, with more details, using the BAF(BAF) properties of Chapter 7 (pp.179-266). It was shown how these "bugs" could be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter refers to the book published in 1985 as is referred on p.457 [05]. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which we seen in the previous Sections, can be found in some publications referred in this p.457 [13], based on that book. Other publications as referred on the same p.457, [02], [03] and [14], shows the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles an attempt to have the solution for the problem.
However, I think, that the most important part of this book is that it probably allows:
1st To establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd To establish a true Mathematical Model to the current Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd To establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, it will remain only the memory unit and the respective input/output devices. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any resource to the present situation;
4th To establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th To re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th To establish promising researches in the field of the Reverse Mathematics, as it was seen as an application in Chapter 11 (p. 344) in the sense of Hilbert’s Program proposal;
7th To establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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 the computing of empirical models and isolated facts.
Thus, paradoxically, the absence of mathematics in the Information Science, which revealed the 4th Great Crisis in Mathematics and allowed the extraordinary technological development, began to demand in its crescent and wide employment of the logical artificial thought, more and more fundamentals of the utilised scientific true.
Looking forward to hearing from you, I remain
Yours Sincerely,
(a) Professor Wagner Waneck Martins
4th letter:
In 1/6/1999 I sent as an individual member of "The Association of Symbolic Logic", a fourth letter addressed to Professor Andreas R. Blass of Department of Mathematics of University of Michigan, one of the Editor of "The Bulletin of Symbolic Logic", published by that Association.
This letter was the following:
1st June, 1999
Professor Andreas R. Blass
Department of Mathematics,
University of Michigan
Ann Arbor, Michigan, MI 48109, USA.
e-mail:
Subject: 1st draft of the book:
"Boolean Arithmetic and its applications".
Your letter 3rd March 1995.
Dear Professor Blass,
In the late 1976, after I received a letter on 14 July from the Chairman of "LOGIC COLLOQUIUM’76 – OXFORD – UK – 19 to 30 July (see annexed copy), I had the pleasure to contact personally Professor Dana S. Scott, to present the abstract of the paper entitled, "Boolean equations systems: A numerical Method of solution". This abstract was published in "THE JOURNAL OF SYMBOLIC LOGIC" – September 1977, pp. 460-461. Since then I became an individual member of THE ASSOCIATION FOR SYMBOLIC LOGIC.
After your letter in reference, re-examining the matter, I sent the following papers to Professor D. M. Gabbay, Editor-in-Chief of "JOURNAL OF LOGIC AND COMPUTATION", edited by OXFORD UNIVERSITY PRESS:
[1] "Boolean Arithmetic: A numerical method of solution for systems of Boolean equations", paper submitted to Journal of Logic and Computation in October, 1997;
[2] "Boolean Arithmetic: Implications in Technical Linguistics", paper submitted to Journal of Logic and Computation in November, 1997;
[3] "Boolean Arithmetic: Logic and Technology", paper submitted to Journal of Logic and Computation in December, 1997;
[4] "Boolean Arithmetic: Esperangol, Direct and Reverse Computational Language", paper submitted to Journal of Logic and Computation in January, 1998
[5] "Boolean Arithmetic: Text Deduction" paper submitted to Journal of Logic and Computation in March, 1998;
[6] "Boolean Arithmetic: Applications of Boolean Matrix in a Logic-Based System", paper submitted to Journal of Logic and Computation in April, 1998;
[7] "Boolean Arithmetic: Applications to one Non-Von Neumann Computing", paper submitted to Journal of Logic and Computation in July, 1998.
Then, I had the honor to receive the letter 16 of July, 1998 from Professor Gabbay, where he suggested to put all these topics together as a book, whose entire manuscript reviewed, he would be for consideration in one of their many book series (see annex copy).
Now, the 1st draft of the book is ready. In 21st May, 1999 I wrote a letter to Professors D. M. Gabbay, Chris L. Hankin and Dana S. Scott (copies annexed), where I say that I believe with their possible assistance, this draft will develop into a book, where the usefulness of Boolean Arithmetic could be presented to a bigger audience. For this reason, as a 23-year-old member of the Association for Symbolic Logic, I am now sending you one copy of the above book.
I would any comments/suggestions from you about this book.
Yours Sincerely,
(a) Prof. Wagner Waneck Martins
5th letter:
In 16/2/2000, seven months and half later, I sent the following second letter to Professor Andreas R. Blass:
February 16, 2000.
Professor Andreas R. Blass
Department of Mathematics,
University of Michigan
Ann Arbor, Michigan, MI 48109, USA.
e-mail:
Subject: 1st draft of the book :
"Boolean Arithmetic and its applications".
My letter June 8, 1999 (Copy annexed)
Dear Professor Blass,
In June 8, 1999, I send to you a letter, in which the 1st draft of the book entitled "Boolean Arithmetic and its applications" was annexed. This subject was related to the development of my researches about Boolean Arithmetic at Polytechnic School of University of Saint Paul, Brazil.
Have you ever received the letter? I’m afraid it could have not reached you. I would be glad if you could send me any information about it.
Yours Sincerely,
(a) Professor Wagner Waneck Martins
Now, in 16/3/2000, after eight months and half later, I received the following answer from Professor Blass:
6th letter:
In 6/9/1999 I sent to Mrs. Susan V. Berresford, President of Ford Foundation the following letter:
September 6, 1999
To
Mrs. Susan V. Berresford,
President
Ford Foundation
(Headquarters)
320 East 43rd Street
NY 10017, New York, USA.
Subject: 1st Partial draft of the book ‘Boolean
Arithmetic and its applications’
Dear Mrs. Berresford,
In 1974, motivated by the following introductory words of The Ford Foundation (TFF)’ Report 1972,
"The Foundation works principally by granting funds to Institutions and Organisations for experimental, demonstration and developmental efforts that give promise of producing significant advances in various fields...",
Mr K. N. Rao, of TFF’s Program Officer – Latin American and Caribbean Sector, received me on February 8, 1974.
At that time, the subject was about the possibility of applying the results of our researches in the field of mathematical fundamentals to practical professional activities. A Study Group of the Technologists Mathematical Movement (SGTMM) was created in the Polytechnic School of São Paulo University (EPUSP) with the purpose to consider the possibility of practical professional applications for these researches.
Then, to my surprise, after some appointments we talked about the incompleteness’ problem, Mr Rao took the initiative to arrange an interview with Professor Kurt Gödel. This unforgettable interview was held on February 15, 1974, in his office at the Institute for Advanced Study, Princeton University.
Thus, in the beginning of the Foreword of that book (referred on pages 6 to 8) we found the complete interview provided by The Ford Foundation, 25 years ago. This interview made me feel enthusiastic about my researches, which led to the production of the referred book.
In the past 25 years there was no interruption of the researches carried out on Boolean Arithmetic and Technical Linguistics (or, Boolean Linguistics) that began at the Polytechnic School of University of São Paulo, Brazil.
After submitting various papers on Boolean Arithmetic to Journal of Logic and Computation, I had the honour to receive in July 1998, a suggestion from Editor-in-Chief, Professor D. M. Gabbay, to put it all together as a book. Because of this suggestion, the 1st Partial draft of the book "Boolean Arithmetic and its applications", is ready.
Although there was an amazing development in the field of Information Science, solutions to some problems have not yet been found. Some of these main problems are:
1. The chronic crisis in real time software; the present viral proliferation in the sector;
2. The lack of common nucleus in the different applications of Artificial Intelligence;
3. The current impossibility of obtaining software developed automatically.
We believe that Boolean Arithmetic is a new area of research that may present relevant mathematical contributions to find the solutions for these problems.
Now, through your site at the Internet, I found in the President’s Message, the following comments from the Foundation’s 1998 annual report:
"Values-judgements about what is right and important in life-help steer our lives and institutions. Without explicit attention to values, we risk relying on traditions and ideas that may have lost their power to attract and motivate people."
With your permission, besides the several and important examples, I would like to include one more about the state-of-the-art of Information Science:
Nowadays, Software Engineering build its programs empirically, that is to say, Mathematics is not yet used in Programming. On the other hand Hardware Engineering build its chips strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. Boolean Arithmetic could be useful in the searching for solution of the current problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks its running.
This present enlarged work represented by the annexed book shows that Boolean Arithmetic can solve important problems in Information Science, using bits directly. As a mathematical logical system, it may prove to be a valuable instrument in remodelling computer programming in order to avoid the current computer languages which cause sometimes problems to the system of operation. As the optical instruments expand the range of the eyesight, this Boolean Arithmetic could represent the development of computing instruments into logical apparatus that might allow mankind to have more safety in the decision-making processes.
For this purpose in Chapter 5, I present the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represent graphically part of a general computing program.
These simple examples show us that incompatibilities, which cause logical "bug", are not due to HUMAN FACTORS, as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6, respectively on pp. 162-170 and 170-172, using numerical method as Partition Method, Doubling Operators, etc. These solutions, maintain the Nodal Decisions whereas the "forbidden eras", technologically are eliminated.
In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", on pp. 355-369, these examples are again repeated as a Problem, with more details, using the BAF(BAF) properties of Chapter 7 (pp. 179-266). It was shown how these "bugs" could be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter refers to the book published in 1985 as referred on p.457 [05]. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which we have seen in the previous Sections, can be found in some publications referred on this p.457 [13], based in that book. Other publications referred on the same p.457, [02], [03] and [14] show the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles as an attempt to have the solution for the problem.
However, as I have told in my answer to Professor Dana S. Scott (copies annexed), I think the most important aim of this book is:
Thus paradoxically, this absence of mathematics in the Information Science (Chapter 2 of the book), besides revealing the 4th Great Crisis in Mathematics (Section 2.3), paradoxically, allows its extraordinary technological development. But now, it begins to demand, in its crescent and wide employment of the logical artificial thought, more and more fundamentals of the mathematics’ scientific truth.
1st to establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd to establish a true Mathematical Model to the present Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd to establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, only the memory unit and the respective input/output devices will remain. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any microprocessor, in contrast to the present situation, when a number of microprocessors are used;
4th to establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th to re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th to establish promising researches in the field of the Reverse Mathematics, as it was seen in Chapter 11 (p. 344), an application of Hilbert’s Program proposal;
7th to establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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 Chapter 11, for instance, could be used instead of the computing on empirical models and isolated facts, to reach the same objective.
For these reasons, this present letter aim at finding out the possibility of The Ford Foundation sponsoring the publication of that book, considering its Contents and Purposes related.
In case of positive answer, I hope that with some revisions and your possible assistance, this text will be developed into a book where the usefulness of Boolean Arithmetic can be spread.
Looking forward to hearing from you, I remain
Yours Sincerely,
(a) Professor Wagner Waneck Martins
Postal Address: Rua Barão de Itaúna, 155 05078-080 – São Paulo – SP – Brazil
Tel.: 0055 011 261-4641 Fax: 0055 011 261-4703 Tel. Cell: 0055 011 9262-0847
e-mail:
(Annexes: a)1st draft of the book referred in the letter; b) mail with Prof. Dana S. Scott; c) Curriculum Vitæ; d) a copy of our project NR. 2000-1227 submitted for the "2000 Rolex Awards for Enterprise", entitled: "A Macroscopic Quantum Leap Effect and its application to the Automobile Motors" – last item referred in that Curriculum).
In 16/3/2000 I received the following answer from Ford Foundation, signed by Senior Director, Mr. L. Steven Zwerling:
7th letter:
In 6/9/1999 I sent to Ecc.ma. Sigra. Dottssa Professoressa Mirella Manaresi, Direttricce dell Dipartimento Di Matematica. UNIVERSITÀ DEGLI STUDI DI BOLOGNA, the following letter:
San Paolo, 10 marzo 2000.
Ecc.ma Sig.ra Dott.ssa
Professoressa Mirella Manaresi
Insigne Direttricce dell Dipartimento Di Matematica.
UNIVERSITÀ DEGLI STUDI DI BOLOGNA
Piazza di Porta S. Donato, 5 - 40127 - BOLOGNA (ITALIA)
RIFERIMENTO: proposta di "PROGRAMMA PROFESSORI VISITATORI":
Corso "ARITMETICA BOOLEANA: Un'introduzione delle Equazioni Booleane nell’Ingegneria di Sistemi". Lettera del Professore Galligani Ilio del 07.07.1984.
Chiarissima Professoressa,
Com'è palese nell’insieme dei documenti che costituiscono l’ALLEGATO 1 di questa lettera, il 17 Settembre 1983, come Professore presso la "Escola Politécnica da Universidade de São Paulo", ho spedito al Professor Galligani Ilio una lettera facendo proposta di un corso di "Matematica Booleane", riempiendo lo stampato avente il titolo "Programma Professori Visitatori - Modello B", portante la data dello 11 Gennaio 1983 ed indirizzato, conforme alle istruzioni, AL CONSIGLIO NAZIONALE DELLE RICERCHE, corredato della bibliografia a disposizione a quell’epoca.
Inizialmente, vorrei ricordare il brano della mia succitata lettera, in cui ho detto:
"Ricordando due anni fa, all’occasione della realizzazione presso codest'Università, del "Simposio Internazionale di Macchine Elettriche" (Settembre 1981), conservo ancora con tanto calore, i ricordi del nostro colloquio nel suo ufficio, offerto dal Manifico Rettore. In quell’occasione ho avuto l’opportunità di riferirle alcune idee sui Fondamenti della Matematica, longinqua conseguenza del grande influsso ricevuto nella mia gioventù, dall’umanesimo della Matematica Italiana, tramite le indimenticabili aule, amministrate dal Professor Emerito di cara memoria Giacomo Albanese, nel passato 1946, presso questa Scuola".
Codeste ricerche matematiche sono continuate e adesso, trascorsi oltre 16 anni fin da quel primo contatto, le sue conclusioni sono state consolidate nel libro "Boolean Arithmetic and its Applications" (Vide ALLEGATO 2 - copia in CD-ROM), il cui testo è basico al Corso che prendo la libertà di proporre tramite la presente lettera.
Lo scopo di questo Corso è di riempire con la cognizione dell’Aritmetica Booleana e della Linguistica Tecnica (ossia, Linguistica Booleana), la lacuna che oggi esiste dovuta alla mancanza della Matematica nell’Ingegneria del Software e nell’Ingegneria del Firmware (Microprogrammazione), di modo ad offrire a codesti settori la medesima sicurezza che oggi esiste nell’Ingegneria dell’Hardware. Codesta sicurezza, che oggi esiste appena nell’Ingegneria dell’Hardware, è stata resa presente dall’Algebra Booleana e dalla Geometria Booleana che hanno permesso alla Tecnologia dare affidabilità logica nella fabbricazione dei "chips"della Microelettronica.
A principio, il Programma del Corso è questo:
01 La presenza dell’Algebra Booleana nell’Informatica Moderna.
02 L’assenza della Matematica Booleana nell’Informatica Moderna.
03 La creazione dell’ Aritmetica Booleana e della Linguistica Tecnica nella Nuova Informatica.
04 La soluzione generale, analitica e numerica, di un Sistema Simultaneo d'Equazioni Booleane.
05 Funzioni Aritmetiche Booleane di Funzioni Aritmetiche Booleane e sue Operazioni Inverse.
06 Esperangol, Diritto e Riverso: il linguaggio matematico computazionale della Nuova Informatica.
07 Un’analisi Aritmetica Booleana dei Sillogismi Aristotelici.
08 Deduzione di Testi e un’Analisi Aritmetiche Booleane della Matematica Riversa di Hilbert.
09 La Computazione "Non-Von Neumann".
10 Il Modello Matematico della Computazione "Von Neumann".
Come conseguenza di queste ricerche, abbiamo ottenuto la creazione dell’ Aritmetica Booleana, matematicamente isomorfa all’Algebra Booleana, il cui linguaggio letterale è totalmente sostituito dal corrispettivo simbolismo numerico del linguaggio aritmetico delle cifre "0/1". Questo ha reso possibile l’ottenimento di una completa e generale soluzione di qualsiasi Sistema d'Equazioni Booleane, non ancora ampiamente diffuso, essendo un argomento di gran rilievo per la Logica Matematica Computazionale.
D’altra parte, l’ Aritmetica Booleana ha reso possibile la creazione della Linguistica Tecnica (ossia, Linguistica Booleana), causando l’arrivo di un unico linguaggio computazionale matematico che ho denominato "Esperangol" (Vide Capitolo 9 pagg. 291-312 - ALLEGATO 2).
I risultati ottenuti in queste ricerche si trovano riassunti nell’ALLEGATO 3.
Penso che le Aree d’Interesse per il Corso potrebbero essere le seguenti: Matematica Applicata, Ingegneria d’Energia ed Automazione Elettrica, Ingegneria Elettronica, Ingegneria Meccanica, Ingegneria Meccatronica, Ingegneria di Hardware, di Software e di Firmware e Interesse Generale.
Nella fiduciosa attesa che la promissiva evoluzione riferita di queste ricerche, frutto longinquo delle aule ripiene d’umanesimo della Matematica Italiana profferite presso la Escola Politécnica da Universidade de São Paulo dall’indimenticabile e di grata memoria Dottor Giacomo Albanese che nel passato 1946 ha fatto destare in me l’attenzione ai problemi concettuali dei Fondamenti della Matematica, renda possibile e pertanto si possa realizzare, la presente offerta del Corso Proposto come Professor Visitatore, colgo l’opportunità d’offrire all’Eccellenza Vostra, la mia dichiarazione di grande stima e straordinaria considerazione, sono
Rispettosamente
(a) Professor Wagner Waneck Martins
Indirizzo Postale: Rua Barão de Itaúna, 156
05078-060 - São Paulo - SP - BRASIL.
Telefono: 0055011 261-4641 Fax: 0055 011 261-4703 Telefono Cellulare: 0055 011 9262-0847
e-mail:
ALLEGATI:
ALLEGATO 1: a) Copia della mia lettera del 17.09.1983; b) "Programma Professori Visitatori - Modello B", avente la data dello 11 Gennaio 1983 e indirizzato, conforme alle istruzioni AL CONSIGLIO NAZIONALE DELLE RICERCHE, corredato dalla bibliografia a disposizione a quell’epoca; c) Programma dell’Attività scientifica che il Prof. Wagner Waneck Martins intende svolgere durante la permanenza in Italia; d) Lettera di risposta del Professor Galligani Ilio del 07.07.1984; e) Copia della mia lettera del 26.03.1985 sull’argomento, per occasione dell’invio del libro del quale sono autore, col titolo, "ESÇÃO (n;m;p): Un Computer Non-Von Neumann".
ALLEGATO 2: Copia in CD-ROM del Libro "Boolean Arithmetic and its Applications", nella cui parte finale c’è il "Curriculum Vitae" dell’autore.
ALLEGATO 3: The results of this lifetime research carried out at University of São Paulo - Polytechnic School - Department of Electrical Energy and Automation (Abstract).
ALLEGATO 4: Extract from Curriculum Vitæ: - Recherches in the Boolean Mathematical Field.
PS. This letter was sent also by mail
Copy to E-mail: galligan@dm.unibo.it
So far, I don’t have received any answer from this letter.
Presentation of the book "BOOLEAN ARITHMETIC and its Applications"
This book is the handout of one PostGraduate Discipline, named PEA - 5737 "Boolean Equations Applied to System Engineering" (see: http://www.poli.usp.br/pea5737) of Polytechnic School of São Paulo University (EPUSP), offered since 1973. 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.
The previous paper relatively to the edition of this first draft of this book were the following:
1st 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, 1997. (*)
2nd 1997 ‘Boolean Arithmetic: Implications in Technical Linguistics’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on November 17, 1997. (*)
3rd 1997 ‘Boolean Arithmetic: Logic & Technology’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on December 5, 1997. (*)
4th 1998 ‘Boolean Arithmetic: Esperangol, Direct & Reverse Computational Language’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on January 6, 1998. (*)
5th 1998 ‘Boolean Arithmetic: Text Deduction’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, on March 2, 1998. (*)
6th 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, 1998. (*)
7th 1998 ‘Boolean Arithmetic: Applications to one Non-Von Neumann Computing’, paper submitted to the ‘Journal of Logic and Computation’, Oxford, UK, in July, 1998. (*)
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 the following letter:
Thus, Professor D. M. Gabbay, Editor-in- Chief of the Journal of Logic and Computation, after receiving those various papers on Boolean Arithmetic, has suggested that I put it all 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 (1st Part and 2nd Part)", which marked the beginning of these researches about Boolean Arithmetic and Technical Linguistics (also called Boolean Linguistics).
Since 1973 until nowadays, I have been teaching at EPUSP the discipline at the PostGraduate 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.
Its CONTENTS has the following fourteen Chapters:
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. 51-98) 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 obtention of different aspects of this mathematical presentation and its technological applications.
The Chapter 4 (pp. 99-122) 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.
The Chapter 5 (pp. 123-152) 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. 152-178) 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. 179-266) 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 Generalisation 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. 267-290) 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. 291-312) 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.
The Chapter 10 (pp. 313-343) 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. 344-374) 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).
The 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. 375-415) 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. 416-429) 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. 430-457) 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.
THE RESULTS OF A LIFETIME RESEARCH CARRIED OUT AT UNIVERSITY OF SÃO PAULO - POLYTECHNIC SCHOOL - DEPARTMENT OF ELECTRICAL ENERGY AND AUTOMATION (ABSTRACT).
"Esperangol" replaces the present chaotic Machine Language since it is based on Mathematical Machine Language and therefore, it is Universal as well as being UNIQUE.
I believe that people would be very interested in "Esperangol", a direct and reversible computer language, due to the following:
1st.: This new language makes the introduction of any kind of "virus" in the computer process, impossible;
2nd.: It allows the automatic production of software mathematically free of failures caused by human factors;
3rd.: It allows complete safety in computer real time operation, where there will be no more need of the "Ctrl+Alt+Del" keys.
I think that even many circumstances might have been changed in these last years, until now, there was no interruption in the research carry out at that University. Although there was an incredible development of Information Science during this past time, solutions of some problems have not been found yet. These main problems are:
.The chronic crisis in real time software;
.The present viral proliferation in the sector;
.The lack of common nucleus in the different applications of Artificial Intelligence;
.The unobtainable software automatically free,
.The fifth generation computer system.
For these reasons, the research relative to this Pure Mathematical Field, led me to find a new area of studies, called Boolean Arithmetic, mathematically isomorphic to the Boolean Algebra, where problems mainly aroused by current computer language into the operations system, can be studied and solved. Then, a draft of my new book on "Boolean Arithmetic and its Applications" now is ready, whose CD-Rom's copy by download (~600 pages), is in the SITE: http://www.poli.usp.br/pea5737. Its CONTENTS presents some results of the researches, referring to the author’s Curriculum Vitæ inside annexed. This present enlarged work shows that Boolean Arithmetic solves problems using only bits. But, and as it is a mathematical logical system, it may prove to be a valuable instrument in remodeling computer programming in order to avoid the current computer languages, which may cause problems to the operational system.
I think that my approach via Boolean Arithmetic can be transformed in a valuable instrument in remodeling computer programming. Nowadays, Software Engineering programs are empirically developed. I may say that Mathematics is not yet used in Programming, whereas, Hardware Engineering is strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. This Boolean Arithmetic could be useful in searching solution for current problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks the computer operation.
For this purpose in Chapter 5 of the referred CD-Rom copy of "Boolean Arithmetic and its Applications", I presented the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represent graphically a part of a general computing program. These simple examples show us that incompatibility, may cause logical "bug", and are not due to HUMAN FACTORS as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6 (pp. 162-172), using numerical method as Partition Method, Doubling Operators, etc. The idea is to maintain the Nodal Decisions when the "forbidden eras", could be technologically eliminated. In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", these examples are again repeated as a Problem on pp. 355-369, with more details, using the BAF(BAF) properties of Chapter 7 (pp.179-266). It was shown how these "bugs" can be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter of the "Boolean Arithmetic and its applications", refers to my book, "ESÇÃO (n~m~p): A Non-Von Neumann Computer", published in 1985. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which was seen in the previous Sections of the referred book "Boolean Arithmetic and its applications", can be found in some publications referred in p.457, based on the last book. Other publications as referred on p.457, show the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles, which enable us to find the solution for the problem.
I think, in conclusion, that the most important part of this book is that it probably allows:
1st To establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd To establish a true Mathematical Model to the current Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd To establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, it will remain only the memory unit and the respective input/output devices. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any resource to the present situation;
4th To establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th To re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th To establish promising researches in the field of the Reverse Mathematics, as it was seen as an application in Chapter 11 (p. 344) in the sense of Hilbert’s Program proposal;
7th To establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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.
Thus, paradoxically, the absence of mathematics in the Information Science, which revealed the 4th Great Crisis in Mathematics and allowed the extraordinary technological development, began to demand in its growing and wide employment of the logical artificial thought, more and more fundamentals of the utilized scientific true.
THE DELIVERY OF THE DRAFT OF THE BOOK, "BOOLEAN ARITHMETIC and its Applications"
When the draft of the book was ready, in May 21 1998, I sent it to the Professor D. M. Gabbay as an answer to that amazing and stimulant invitation of his 16/7/98 letter, relative to put the entire subject about Boolean Arithmetic, together, as a book:
In 28th September 1999, I received from Professor D. M. Gabbay the following letter:
Then, a brief analysis from the exposed situation is the following:
1. Time to write the draft of the book since letter invitation (16/7/1998) to its delivery (21/5/1999) to Professor Gabbay: It was little more than 9 months;
2. Time that Professor Gabbay have had an opportunity to take a brief look through the received book: It was little more than 4 months until its receipt (28/9/1999);
After these times, Professor Gabbay, reluctantly, tell me the following:
a. That he does not believe that the material received will fit with reorganization with the book series that he edit;
b. That he does not think of an alternative to suggest to me.
At the same time that I have written to Professor Gabbay (21/5/1999), I have sent to others Professors, the following three letters
1st letter:
May 21, 1999
To
Professor Chris L. Hankin
Dept. of Computing
Imperial College
180, Queen's Gate, LONDON, SW7 2BZ, UK.
e-mail: clh@doc.ic.ac.uk
Ref.: Interview with Mr. Ricardo Wagner Colombini Martins, held in the beginning of June, 1994, at Imperial College, London, about ‘Boolean Arithmetic".
Dear Professor Hankin,
Due to Professor Gabbay’s suggestion, made in July 1998, about putting as a book some submission papers to Journal of Logic and Computation, a Partial Draft of this book entitled "Boolean Arithmetic and its Applications" is ready. I hope that, with successive revisions with your possible assistance as Editorial Board’s Member of the Journal, this text will develop into a book which may prove useful after the understanding of Boolean Arithmetic.
These researches on Boolean Arithmetic were carried out at the Department of Electrical Energy and Automation from Polytechnic School of Sao Paulo University, Brazil.
My son Ricardo, will give you a copy of this boolk and I hope to have your comments on it.
Thank you for your attention, as well as for your kindness in receiving him once more.
Yours sincerely,
(a) Prof. Wagner Waneck Martins
2nd letter:
São Paulo, 21 de maio de 1999.
À
Professora Doutora Myriam Gondim Déchamps,
Université Paris Sud – Centre d’Orsay
Mathématique – Bat. 425
91405 – ORSAY – Cedex France.
Myriam.Dechamps@math.u-psud.fr
Referência: Remessa de cópia do manuscrito do livro "Boolean Arithmetic and its applications".
Prezada Professora,
Conforme entendimentos havidos através de seu irmão e meu colega de turma, Engº Pedro Gondim, o portador da presente é meu filho Ricardo Wagner Colombini Martins, que lhe entregará para seu conhecimento, uma cópia do manuscrito do livro em referência, acompanhado de cópias das cartas aos Professores D. M. Gabbay, Chris L. Hankin e Dana S. Scott, relativas a entregas de outras cópias.
Tendo já há algum tempo tido notícias pelo seu irmão, sobre o sucesso de suas atividades no campo da Matemática, tomei a liberdade de entrar no site http://www.math.u-psud.fr/~dechamps e fiquei deslumbrado com a descrição do seu detalhamento e dos resultados alcançados. Creio não exagerar, que esses trabalhos e pesquisas realizadas pela prezada Professora constituem honra e glória para nós brasileiros, razão pela qual peço que aceite as minhas mais sinceras congratulações.
Como dentro de suas atividades de interesse geral ressaltam as referentes ao ensino da Matemática, tomo liberdade de anexar à presente, também para seu conhecimento, a cópia de um trabalho correlato com o mesmo assunto do livro em referência, apresentado em setembro de 1976 durante o "The 3rd International Congress of Mathematical Education", realizado em Karlsruhe, Alemanha e intitulado "BOOLEAN ARITHMETIC: Implications in Mathematical Teaching", acompanhado de sua versão para o idioma germânico.
Ao seu esposo, estimado Professor Dechamps, ao qual também apresento meus cordiais cumprimentos, tomo a liberdade também de apresentar uma cópia do trabalho intitulado "Could ‘Magnetic Super Pulse’ possibily be the Macroscopic Manifestation of the ‘Quantum Leap’?", que talves possa interessá-lo no campo da Física.
Para terminar, antecipadamente agradeço qualquer atenção de sua parte relativamente ao assunto, no sentido da divulgar essas ideias junto ao pessoal especializado dessa Universidade e eventualmente interessado em contatar-me a respeito, sou,
Atenciosamente
(a) Wagner Waneck Martins
3rd letter:
May 21, 1999
To
Professor Dana S. Scott,
Sch. of Computer Science,
Carnegie Mellon University,
Pittsburgh – PA 15213 – USA.
e-mail: dana_scott@proof.ergo.cmu.edu
Subject: 1st Partial draft of the book ‘Boolean
Arithmetic and its applications’
Dear Professor Scott,
In the late 1976, during the realisation of "LOGIC COLLOQUIUM’76 – OXFORD – 19 to 30 July, I had the pleasure to contact you personally. After our meeting, besides my admittance as member of A.S.L., the abstract entitled "Boolean equations systems: A numerical method of solution", was published in "THE JOURNAL OF SYMBOLIC LOGIC" – September, 1977, pp. 460-461 (Annexes 1 and 2).
In these past 23 years there was no interruption of the researches carried out on Boolean Arithmetic and Technical Linguistics (or, Boolean Linguistics) whose beginning was at the Polytechnic School of University of São Paulo, Brazil.
Recently, after submitting various papers on Boolean Arithmetic to Journal of Logic and Computation, I had the honor to receive in July, 1998, a suggestion from Professor D. M. Gabbay, to put it all together as a book. Because of this suggestion, the 1st Partial draft of the book "Boolean Arithmetic and its applications", is ready.
Although there was an incredible development of Information Science, solutions of some problems have not been found yet. These main problems are the chronic crisis in real time software; the present viral proliferation in the sector; the lack of common nucleus in the different applications of Artificial Intelligence; the unobtainable software automatically free.
We believe that this new area of studies, Boolean Arithmetic, may present opportune mathematical contributions to find the solutions for these problems.
As you were responsible for the publication of this abstract I believe you may be interested in this book, which will how besides Boolean Arithmetic can be derived from Boolean Algebra, as well as some practical applications.
This present enlarged work shows that Boolean Arithmetic solves problems using bits. But, and as it is a mathematical logical system, it may prove to be a valuable instrument in remodeling computer programming in order to avoid the current computer languages which cause sometimes problems to the system of operation.
I hope that, with successive revisions and your possible assistance as advisor to OUP of Journal of Logic and Computation, this text will develop into a book where the usefulness of Boolean Arithmetic can be achieved.
Looking forward to hearing from you, I remain
Yours Sincerely,
a. Professor Wagner Waneck Martins
Only this third letter, addressed to the to Professor Dana S. Scott of Carnegie Mellon University, in 8th June 1999, I have had the honor to receive an answer, witch carry in Boolean Algebra language, suggestion about the example on p. 118/158 of the book.
This letter was the following:
Thus, I have had an excellent opportunity with my answer to Professor Scott, to translate its suggestion into Boolean Arithmetic language, using the isomorphic numerical properties with Boolean Algebra, as it is referred in the book.
As this suggestion stands for me a very relevant question proposed by a remarkable Professor, I spend in 16/8/1999 a long answer that, I think, it is better to release:
August 16, 1999
To
Professor Dana S. Scott,
Computer Science Department
Carnegie Mellon University,
5000 Forbes Avenues
Pittsburgh, Pennsylvania PA 15213–3891, USA.
E-mail: dana.scott@cs.cmu.edu
Subject: 1st Partial draft of the book:
‘Boolean Arithmetic and its applications’
Your letter ~8th June 1999.
Dear Professor Scott:
Thank you for your letter referred above and your suggestion about the example on p. 118/158 of the book.
I would like to apply this suggestion in general, with your permission, using the isomorphic numerical language of Boolean Arithmetic referred in the book. As the example on p. 118 it refer to two Boolean Functions, "Y1 = P" and "Y2 = R", the application after, will be made in three part. : Part a): Only with the Boolean Functions, "Y1 = P". Part b): Only with the Boolean Functions, "Y2 = R". Part c): The entire example, with simultaneous Boolean Functions, "Y1" and "Y2" (or, "P and R").
Then, some preliminary observations are given:
1. On p. 119 of the first solution of this example after Fig. 5, is written that it is a "historical solution", because of the late origin of these researches which led to Boolean Arithmetic and Technical Linguistics. At that time, the properties used in the second solution presented on p.139 and in the third solution presented on page 158 were unknown. These properties refer to the employed ANALYTICAL METHOD OF SOLUTION of Chapter 5 (p. 123-151) and the "DOUBLING OPERATOR" (p. 66) and PARTITION METHOD (p. 153). This "historical solution" was, as a façon de parler, the beginning of these researches, when the paths through these numerical mathematical fields were discovered;
2. For all applications of Boolean Arithmetic it will be adopted the concepts detailed from Boolean Arithmetical Variables (BAV), Boolean Arithmetical Function (BAF) through its Numerical Transform (NT) on pages (51-55)/(102-104) and the items a) to e) on pages (89-90). The item a) refers to the use of "Hindrance" of Negative Logic as logical unit instead of the dual concept of "Transmission" of Positive Logic (see Fig. 1 – p. 88), which allows that the classical representation of Equations in Mathematics with the second member null can be maintained;
3. A formal Boolean Equation is a logical interrelation between two Boolean Functions of any given variables. This logical interrelation could be a "logical implication" which is called on page 105, "Symbol 4": X ® Y (or vice versa, X Y), or both, that is, "logical equality" which is called on pages 105-106, "Symbol 5": X = Y. As Boolean Arithmetical Functions (BAF) and Boolean Functions are isomorphic, the properties of the isomorphism guarantee that these interrelations are maintained in Boolean Arithmetic (p. 104);
4. The Boolean variables used in the example on p. 118 and repeated on p. 158 are replaced by the Boolean variables used in your suggestion, according to the following table:
Y1 Y2 X1 X2 X3 X3 (X2,X1)
P R q p r r =Q(p,q)
Thus, in the general Boolean Arithmetical Equation language, your suggestion could have the following translation, adopting one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}":
"If the Boolean Arithmetical Equation (BAE) is given, , we want to determine the following Boolean Arithmetical Function incognito: , such that we have satisfied the numerical expression:
Thus, applying the Property (20a) - p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE, we have:
However, we can suppose that the incognito function is:
Then, making the substitution of expression (2) in (1), we have:
Therefore, in order to determine all the possible solutions, we must have:
In expression (4) we know the Boolean Arithmetical Function (BAF), , that is, the binary components of the abscissa of the Numerical Transform (NT). Then, through that expression (4) we can determine all the possible numerical values for the four incognito components { } of the abscissa of the Numerical Transform (NT) of the Boolean Arithmetical Function (BAF) incognito, expressed in (2).
In order to apply that useful suggestion, adequately translated to the numerical language of Boolean Arithmetical Equation, the example on p. 118 will be separate in two preliminaries parts:
Part a): "Given the Boolean Arithmetical Equation (BAE), we want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical expression: ."
Note: In the solution of this part, as an illustration, all details of the employment of Boolean Arithmetic are given.
Initially, we consider the Boolean Arithmetical Function (BAF) of the first member of the given BAE. Then, applying the definition of Numerical Transform (NT) of a Boolean Arithmetical Function (BAF) - p. 54-55 (Truth Table and Boolean Arithmetical Variables procedures), or, using the properties of the "Doubling Operators" - p. 66, we have:
1st Mode: Through the Truth Table procedure (p. 91):
S1Þ ß T1 r p q
t0 0 0 0 P(0,0,0) = a 0 = 0
t1 0 0 1 P(0,0,1) = a 1 = 0
t2 0 1 0 P(0,1,0) = a 2 = 0 Therefore, we obtain the
t3 0 1 1 P(0,1,1) = a 3 = 0 following expression (5):
t4 1 0 0 P(1,0,0) = a 4 = 1
t5 1 0 1 P(1,0,1) = a 5 = 0
t6 1 1 0 P(1,1,0) = a 6 = 1
t7 1 1 1 P(1,1,1) = a 7 = 1
2nd Mode: Through the Boolean Arithmetical Variables (BAVs) procedures (p. 92):
Choosing one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", we have:
Applying the distributive property of NTs to the given Boolean Arithmetical Function, we have the following expression:
Substituting the numerical values (6) of the NTs of the BAVs in the expression (7), we have:
Then, we have the same final Boolean Arithmetical Equation:
(5)
3rd Mode: Through the "Doubling Operators" (p. 66):
Applying the distributive property of NTs to the given Boolean Arithmetical Function, we have the following expression:
But, applying the "Doubling Operators" properties (p. 66) we have:
Substituting the numerical values given by expression (9) in expression (8), we have:
Then, we have the same final Boolean Arithmetical Equation:
(5)
However, we can suppose that the incognito function is:
such that
(10)
Thus, applying the Property (20a) on p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE as we have seen, we have:
or,
, or,
Therefore, in order to determine all the possible solutions of BAE (11), we must have:
The first solution, as null solution, is rejected:
NT{r = Q(p,q)} = [0000]2.{2;p q} = 0
The second solution is accepted as a unique full solution:
In Boolean arithmetical language:
13. NT{r = Q(p,q)} = [0010]2.{2;p q}
In Boolean algebraic language, the Anti-Transform is:
r = Q(p,q) = NT-1{ [0010]2 .{2;p,q} } = P1 =p’ v q \ r = p’v q
Proof:
, where r = p’v q
Then: P = (p’v q) v (p & q’) = (p’v q v p) & (p’ v q v q’) = 0 (QED)
Part b): "Given the Boolean Arithmetical Equation (BAE), we want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical expression: ."
Choosing one Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", and applying the "Doubling Operators" properties (p. 66) to the expression (14):
14.
we have:
Substituting the numerical values given by expression (15) in expression (14), we have:
Then, we have the final Boolean Arithmetical Equation:
(16)
However, we can suppose that the incognito function is:
such that
(17)
Thus, applying the Property (20a) on p. 69, to the Boolean Arithmetical Function (BAF) of the first member of the given BAE as we have seen, we have:
or,
or,
Therefore, in order to determine all the possible solutions of BAE (11), we must have:
.
Then, we have the following two solutions restricted in times "t2" and " t3":
In Boolean arithmetical language, these solutions are:
1st solution: NT{ r = Q(p,q)} = [ii11] 2 {2;p q} (rejected)
2nd solution: NT{ r = Q(p,q)} = [ii10] 2 {2;p q} (accepted), but restricted in "t2" and " t3".
In Boolean algebraic language, the correspondent Anti-Transform of this 2nd solution is:
19. r = Q(p,q) = NT-1{ [ii10]2 .{2;p,q} }, that is (see example on p. 352):
a. To obtain the semantic interpretation of their restriction, we make the replacement of this symbolic bit "i", by the bit "0" and the other bit elements by bit "1" – than the NODAL RESTRICTION is obtained. This NODAL RESTRICTION corresponds in the expression (19), to times "t2" and " t3" (forbidden eras – p.355).
b. Then, the NODAL RESTRICTION is given through Boolean algebraic language, or the following Anti-Transform:
19. r = N(p,q) = NT-1{ [0011]2 .{2;p,q} } = S3 v S2 =(p’ & q’) v (p’ & q) = p’
c) If we make the replacement of the symbolic bit "i" by the bit "1" and remaining others without any alteration, the NODAL CONCLUSION is obtained.
d) Then, the unique NODAL CONCLUSION is:
r = Q(p,q) = NT-1 { [1110]2 .{2;p q} } =S0 = p & q, but restricted in times "t2" and " t3" (forbidden eras – p.355).
Note: This semantic interpretation of the NODAL RESTRICTION was possible because it was a consequence of the employment of Boolean Equation concepts.
Now, I’m applying the general Boolean Arithmetical Equation language, as it was referred in the beginning of this letter, on the entire example in discussion, which is the following:
Part c): "Given the simultaneous System of Boolean Arithmetical Functions (SBAFs):
We want to determine the Boolean Arithmetical Function incognito, , such that we have satisfied the numerical logical equality expression:
."
Note: In the solution of this entire example, as a last application of your useful suggestion, it is different from the other three presented in the book. The first is referred on p. 118, and it is a historical solution. The second is repeated on p. 139 and it is an application of an Analytical Method of solution for System of Boolean Equations. The third is repeated on page 158 and it is an application of the "DOUBLING OPERATOR" and PARTITION METHOD properties.
Choosing a Boolean Arithmetical Field (BAFi) whose cardinality is "k = 3" and ordinality is "w 3 = {r p q}", we have:
Then, by expression (20) we must have:
The fundamental logical "Symbol 5" (pages 105-106), between the abscissa of the "NTs", brings about the following equality,
[1101 0000]2 = [1100 1110]2 , that is:
[1101 0000]2 v ([1100 1110]2)’ & ([1101 0000]2)’ [1100 1110]2 = 0
But,
([1100 1110]2)’ = [0011 0001]2 and ([1101 0000]2)’ = [0010 1111]2
Then, we have:
[1101 0000]2 v [0011 0001]2 & [0010 1111]2 v [1100 1110]2 = 0
[0001 0000]2 & [0000 1110]2 = 0, or
The desired solution of this SESABE (Solving Equation of System of Arithmetical Boolean Equation) or the searched solution is, as we have supposed:
Then, applying the Property (20a) – p. 69, to the first member of the above (SESABE), we have:
Replacing expression (2) in expression (23), we have:
or,
Therefore, in order to determine all the possible solutions of Boolean Arithmetical Equation (BAE) expressed by (25), we must have:
That is, we have the unique numerical solution in Boolean arithmetical language:
(27) NT{r = Q(p,q)} = [1110]2.{2;p q}
In Boolean algebraic language, the Anti-Transform is:
r = Q(p,q) = NT-1{ [1110]2 .{2;p,q} } = S0 =p & q \ r = p & q
Proof: , where r = p & q. Then: P = (p & q) v (p & q’) = p
R = p & r’ v q, where r = p & q. Then: R = p & (p & q)’ v q = p & p’ v q’ v q = p
That is, we have P = R, when r = p & q (QED).
In your letter, you referred to my approach via Boolean Arithmetic that it can be "a valuable instrument in remodelling computer programming."
With your permission, some of the problems provoked by current computer language into the system of operation, can be presented as follows:
Nowadays, Software Engineering build its programs empirically, that is, Mathematics is not yet used in Programming, whereas, Hardware Engineering built its chips strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. This Boolean Arithmetic could be useful to the searching of solution of problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks its running.
For this purpose in Chapter 5, I present the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represents graphically a part of a general computing program. These simple examples show us that incompatibilities, which cause logical "bug", are not due to HUMAN FACTORS as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6, respectively on pp. 162-170 and 170-172, using numerical method as Partition Method, Doubling Operators, etc, to maintain the Nodal Decisions when the "forbidden eras", technologically could be eliminated. In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", these examples are again repeated as a Problem on pp. 355-369, with more details, using the BAF(BAF) properties of Chapter 7 (pp.179-266). It was shown how these "bugs" could be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter refers to the book published in 1985 as is referred on p.457 [05]. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which we seen in the previous Sections, can be found in some publications referred in this p.457 [13], based on that book. Other publications as referred on the same p.457, [02], [03] and [14], shows the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles an attempt to have the solution for the problem.
However, I think, that the most important part of this book is that it probably allows:
1st To establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd To establish a true Mathematical Model to the current Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd To establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, it will remain only the memory unit and the respective input/output devices. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any resource to the present situation;
4th To establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th To re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th To establish promising researches in the field of the Reverse Mathematics, as it was seen as an application in Chapter 11 (p. 344) in the sense of Hilbert’s Program proposal;
7th To establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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 the computing of empirical models and isolated facts.
Thus, paradoxically, the absence of mathematics in the Information Science, which revealed the 4th Great Crisis in Mathematics and allowed the extraordinary technological development, began to demand in its crescent and wide employment of the logical artificial thought, more and more fundamentals of the utilised scientific true.
Looking forward to hearing from you, I remain
Yours Sincerely,
(a) Professor Wagner Waneck Martins
4th letter:
In 1/6/1999 I sent as an individual member of "The Association of Symbolic Logic", a fourth letter addressed to Professor Andreas R. Blass of Department of Mathematics of University of Michigan, one of the Editor of "The Bulletin of Symbolic Logic", published by that Association.
This letter was the following:
1st June, 1999
Professor Andreas R. Blass
Department of Mathematics,
University of Michigan
Ann Arbor, Michigan, MI 48109, USA.
e-mail:
Subject: 1st draft of the book:
"Boolean Arithmetic and its applications".
Your letter 3rd March 1995.
Dear Professor Blass,
In the late 1976, after I received a letter on 14 July from the Chairman of "LOGIC COLLOQUIUM’76 – OXFORD – UK – 19 to 30 July (see annexed copy), I had the pleasure to contact personally Professor Dana S. Scott, to present the abstract of the paper entitled, "Boolean equations systems: A numerical Method of solution". This abstract was published in "THE JOURNAL OF SYMBOLIC LOGIC" – September 1977, pp. 460-461. Since then I became an individual member of THE ASSOCIATION FOR SYMBOLIC LOGIC.
After your letter in reference, re-examining the matter, I sent the following papers to Professor D. M. Gabbay, Editor-in-Chief of "JOURNAL OF LOGIC AND COMPUTATION", edited by OXFORD UNIVERSITY PRESS:
[1] "Boolean Arithmetic: A numerical method of solution for systems of Boolean equations", paper submitted to Journal of Logic and Computation in October, 1997;
[2] "Boolean Arithmetic: Implications in Technical Linguistics", paper submitted to Journal of Logic and Computation in November, 1997;
[3] "Boolean Arithmetic: Logic and Technology", paper submitted to Journal of Logic and Computation in December, 1997;
[4] "Boolean Arithmetic: Esperangol, Direct and Reverse Computational Language", paper submitted to Journal of Logic and Computation in January, 1998
[5] "Boolean Arithmetic: Text Deduction" paper submitted to Journal of Logic and Computation in March, 1998;
[6] "Boolean Arithmetic: Applications of Boolean Matrix in a Logic-Based System", paper submitted to Journal of Logic and Computation in April, 1998;
[7] "Boolean Arithmetic: Applications to one Non-Von Neumann Computing", paper submitted to Journal of Logic and Computation in July, 1998.
Then, I had the honor to receive the letter 16 of July, 1998 from Professor Gabbay, where he suggested to put all these topics together as a book, whose entire manuscript reviewed, he would be for consideration in one of their many book series (see annex copy).
Now, the 1st draft of the book is ready. In 21st May, 1999 I wrote a letter to Professors D. M. Gabbay, Chris L. Hankin and Dana S. Scott (copies annexed), where I say that I believe with their possible assistance, this draft will develop into a book, where the usefulness of Boolean Arithmetic could be presented to a bigger audience. For this reason, as a 23-year-old member of the Association for Symbolic Logic, I am now sending you one copy of the above book.
I would any comments/suggestions from you about this book.
Yours Sincerely,
(a) Prof. Wagner Waneck Martins
5th letter:
In 16/2/2000, seven months and half later, I sent the following second letter to Professor Andreas R. Blass:
February 16, 2000.
Professor Andreas R. Blass
Department of Mathematics,
University of Michigan
Ann Arbor, Michigan, MI 48109, USA.
e-mail:
Subject: 1st draft of the book :
"Boolean Arithmetic and its applications".
My letter June 8, 1999 (Copy annexed)
Dear Professor Blass,
In June 8, 1999, I send to you a letter, in which the 1st draft of the book entitled "Boolean Arithmetic and its applications" was annexed. This subject was related to the development of my researches about Boolean Arithmetic at Polytechnic School of University of Saint Paul, Brazil.
Have you ever received the letter? I’m afraid it could have not reached you. I would be glad if you could send me any information about it.
Yours Sincerely,
(a) Professor Wagner Waneck Martins
Now, in 16/3/2000, after eight months and half later, I received the following answer from Professor Blass:
6th letter:
In 6/9/1999 I sent to Mrs. Susan V. Berresford, President of Ford Foundation the following letter:
September 6, 1999
To
Mrs. Susan V. Berresford,
President
Ford Foundation
(Headquarters)
320 East 43rd Street
NY 10017, New York, USA.
Subject: 1st Partial draft of the book ‘Boolean
Arithmetic and its applications’
Dear Mrs. Berresford,
In 1974, motivated by the following introductory words of The Ford Foundation (TFF)’ Report 1972,
"The Foundation works principally by granting funds to Institutions and Organisations for experimental, demonstration and developmental efforts that give promise of producing significant advances in various fields...",
Mr K. N. Rao, of TFF’s Program Officer – Latin American and Caribbean Sector, received me on February 8, 1974.
At that time, the subject was about the possibility of applying the results of our researches in the field of mathematical fundamentals to practical professional activities. A Study Group of the Technologists Mathematical Movement (SGTMM) was created in the Polytechnic School of São Paulo University (EPUSP) with the purpose to consider the possibility of practical professional applications for these researches.
Then, to my surprise, after some appointments we talked about the incompleteness’ problem, Mr Rao took the initiative to arrange an interview with Professor Kurt Gödel. This unforgettable interview was held on February 15, 1974, in his office at the Institute for Advanced Study, Princeton University.
Thus, in the beginning of the Foreword of that book (referred on pages 6 to 8) we found the complete interview provided by The Ford Foundation, 25 years ago. This interview made me feel enthusiastic about my researches, which led to the production of the referred book.
In the past 25 years there was no interruption of the researches carried out on Boolean Arithmetic and Technical Linguistics (or, Boolean Linguistics) that began at the Polytechnic School of University of São Paulo, Brazil.
After submitting various papers on Boolean Arithmetic to Journal of Logic and Computation, I had the honour to receive in July 1998, a suggestion from Editor-in-Chief, Professor D. M. Gabbay, to put it all together as a book. Because of this suggestion, the 1st Partial draft of the book "Boolean Arithmetic and its applications", is ready.
Although there was an amazing development in the field of Information Science, solutions to some problems have not yet been found. Some of these main problems are:
1. The chronic crisis in real time software; the present viral proliferation in the sector;
2. The lack of common nucleus in the different applications of Artificial Intelligence;
3. The current impossibility of obtaining software developed automatically.
We believe that Boolean Arithmetic is a new area of research that may present relevant mathematical contributions to find the solutions for these problems.
Now, through your site at the Internet, I found in the President’s Message, the following comments from the Foundation’s 1998 annual report:
"Values-judgements about what is right and important in life-help steer our lives and institutions. Without explicit attention to values, we risk relying on traditions and ideas that may have lost their power to attract and motivate people."
With your permission, besides the several and important examples, I would like to include one more about the state-of-the-art of Information Science:
Nowadays, Software Engineering build its programs empirically, that is to say, Mathematics is not yet used in Programming. On the other hand Hardware Engineering build its chips strongly based in Boolean Mathematics, through the algebraic and geometrical isomorphic aspects. Boolean Arithmetic could be useful in the searching for solution of the current problems in the computer operation of any software or firmware engineering programs. The same safety operation which we have in the hardware computing can be achieved in the operation of the software and firmware computing, without any logical "bug" which blocks its running.
This present enlarged work represented by the annexed book shows that Boolean Arithmetic can solve important problems in Information Science, using bits directly. As a mathematical logical system, it may prove to be a valuable instrument in remodelling computer programming in order to avoid the current computer languages which cause sometimes problems to the system of operation. As the optical instruments expand the range of the eyesight, this Boolean Arithmetic could represent the development of computing instruments into logical apparatus that might allow mankind to have more safety in the decision-making processes.
For this purpose in Chapter 5, I present the 2nd Example (pp. 142-146) and the 3rd Example (pp. 146-151), which represent graphically part of a general computing program.
These simple examples show us that incompatibilities, which cause logical "bug", are not due to HUMAN FACTORS, as it is ordinarily admitted. These examples solved by Analytical Method are repeated in Chapter 6, respectively on pp. 162-170 and 170-172, using numerical method as Partition Method, Doubling Operators, etc. These solutions, maintain the Nodal Decisions whereas the "forbidden eras", technologically are eliminated.
In Section "11.2 – THE NODAL AUTOMATIC TRANSITION ANALYSIS (NATA)", on pp. 355-369, these examples are again repeated as a Problem, with more details, using the BAF(BAF) properties of Chapter 7 (pp. 179-266). It was shown how these "bugs" could be mathematically eliminated without any alteration in the CONCLUSION (or, NODAL DECISION) of that part of programming.
The last Chapter refers to the book published in 1985 as referred on p.457 [05]. The mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles which we have seen in the previous Sections, can be found in some publications referred on this p.457 [13], based in that book. Other publications referred on the same p.457, [02], [03] and [14] show the mathematical fundamentals of that Non-Von Machines and Non-Von Programming Styles as an attempt to have the solution for the problem.
However, as I have told in my answer to Professor Dana S. Scott (copies annexed), I think the most important aim of this book is:
Thus paradoxically, this absence of mathematics in the Information Science (Chapter 2 of the book), besides revealing the 4th Great Crisis in Mathematics (Section 2.3), paradoxically, allows its extraordinary technological development. But now, it begins to demand, in its crescent and wide employment of the logical artificial thought, more and more fundamentals of the mathematics’ scientific truth.
1st to establish that any pure sequential program can be recorded into a simple EPROM chip or CD-ROM device, without use of any current computational language;
2nd to establish a true Mathematical Model to the present Von Neumann Machines in order to eliminate the current logical bugs, mainly in the real time applications without the ordinary use of empirical methods. This elimination is guaranteed with some data alterations, but without any modification in its formal logical nodal decisions;
3rd to establish a true Non-Von Mathematical Model, with the elimination the microprocessor as a subsystem of the present computational machines. Thus, only the memory unit and the respective input/output devices will remain. Then, the "Von Neumanns’s bottleneck" disappears, establishing a pure parallel processing without any microprocessor, in contrast to the present situation, when a number of microprocessors are used;
4th to establish a mathematical programming in the field of the hardware, software and firmware engineering, with the creation of a Technical Linguistics in the adequate field of the Negative Logic;
5th to re-establish new lines of logical mathematical researches in the field of Information Sciences, which has been broken since 1938 in Shannon’s time, with the Mathematical Von Neumann Deviation, as it was described in Chapter 13 (p. 416);
6th to establish promising researches in the field of the Reverse Mathematics, as it was seen in Chapter 11 (p. 344), an application of Hilbert’s Program proposal;
7th to establish new promising researches in the field of Quantum Logic Gates of Quantum-Mechanical Computers, since it is possible to establish the Mathematical Model as a Quantum-Mechanical Non- Von Computers;
8th 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 Chapter 11, for instance, could be used instead of the computing on empirical models and isolated facts, to reach the same objective.
For these reasons, this present letter aim at finding out the possibility of The Ford Foundation sponsoring the publication of that book, considering its Contents and Purposes related.
In case of positive answer, I hope that with some revisions and your possible assistance, this text will be developed into a book where the usefulness of Boolean Arithmetic can be spread.
Looking forward to hearing from you, I remain
Yours Sincerely,
(a) Professor Wagner Waneck Martins
Postal Address: Rua Barão de Itaúna, 155 05078-080 – São Paulo – SP – Brazil
Tel.: 0055 011 261-4641 Fax: 0055 011 261-4703 Tel. Cell: 0055 011 9262-0847
e-mail:
(Annexes: a)1st draft of the book referred in the letter; b) mail with Prof. Dana S. Scott; c) Curriculum Vitæ; d) a copy of our project NR. 2000-1227 submitted for the "2000 Rolex Awards for Enterprise", entitled: "A Macroscopic Quantum Leap Effect and its application to the Automobile Motors" – last item referred in that Curriculum).
In 16/3/2000 I received the following answer from Ford Foundation, signed by Senior Director, Mr. L. Steven Zwerling:
7th letter:
In 6/9/1999 I sent to Ecc.ma. Sigra. Dottssa Professoressa Mirella Manaresi, Direttricce dell Dipartimento Di Matematica. UNIVERSITÀ DEGLI STUDI DI BOLOGNA, the following letter:
San Paolo, 10 marzo 2000.
Ecc.ma Sig.ra Dott.ssa
Professoressa Mirella Manaresi
Insigne Direttricce dell Dipartimento Di Matematica.
UNIVERSITÀ DEGLI STUDI DI BOLOGNA
Piazza di Porta S. Donato, 5 - 40127 - BOLOGNA (ITALIA)
RIFERIMENTO: proposta di "PROGRAMMA PROFESSORI VISITATORI":
Corso "ARITMETICA BOOLEANA: Un'introduzione delle Equazioni Booleane nell’Ingegneria di Sistemi". Lettera del Professore Galligani Ilio del 07.07.1984.
Chiarissima Professoressa,
Com'è palese nell’insieme dei documenti che costituiscono l’ALLEGATO 1 di questa lettera, il 17 Settembre 1983, come Professore presso la "Escola Politécnica da Universidade de São Paulo", ho spedito al Professor Galligani Ilio una lettera facendo proposta di un corso di "Matematica Booleane", riempiendo lo stampato avente il titolo "Programma Professori Visitatori - Modello B", portante la data dello 11 Gennaio 1983 ed indirizzato, conforme alle istruzioni, AL CONSIGLIO NAZIONALE DELLE RICERCHE, corredato della bibliografia a disposizione a quell’epoca.
Inizialmente, vorrei ricordare il brano della mia succitata lettera, in cui ho detto:
"Ricordando due anni fa, all’occasione della realizzazione presso codest'Università, del "Simposio Internazionale di Macchine Elettriche" (Settembre 1981), conservo ancora con tanto calore, i ricordi del nostro colloquio nel suo ufficio, offerto dal Manifico Rettore. In quell’occasione ho avuto l’opportunità di riferirle alcune idee sui Fondamenti della Matematica, longinqua conseguenza del grande influsso ricevuto nella mia gioventù, dall’umanesimo della Matematica Italiana, tramite le indimenticabili aule, amministrate dal Professor Emerito di cara memoria Giacomo Albanese, nel passato 1946, presso questa Scuola".
Codeste ricerche matematiche sono continuate e adesso, trascorsi oltre 16 anni fin da quel primo contatto, le sue conclusioni sono state consolidate nel libro "Boolean Arithmetic and its Applications" (Vide ALLEGATO 2 - copia in CD-ROM), il cui testo è basico al Corso che prendo la libertà di proporre tramite la presente lettera.
Lo scopo di questo Corso è di riempire con la cognizione dell’Aritmetica Booleana e della Linguistica Tecnica (ossia, Linguistica Booleana), la lacuna che oggi esiste dovuta alla mancanza della Matematica nell’Ingegneria del Software e nell’Ingegneria del Firmware (Microprogrammazione), di modo ad offrire a codesti settori la medesima sicurezza che oggi esiste nell’Ingegneria dell’Hardware. Codesta sicurezza, che oggi esiste appena nell’Ingegneria dell’Hardware, è stata resa presente dall’Algebra Booleana e dalla Geometria Booleana che hanno permesso alla Tecnologia dare affidabilità logica nella fabbricazione dei "chips"della Microelettronica.
A principio, il Programma del Corso è questo:
01 La presenza dell’Algebra Booleana nell’Informatica Moderna.
02 L’assenza della Matematica Booleana nell’Informatica Moderna.
03 La creazione dell’ Aritmetica Booleana e della Linguistica Tecnica nella Nuova Informatica.
04 La soluzione generale, analitica e numerica, di un Sistema Simultaneo d'Equazioni Booleane.
05 Funzioni Aritmetiche Booleane di Funzioni Aritmetiche Booleane e sue Operazioni Inverse.
06 Esperangol, Diritto e Riverso: il linguaggio matematico computazionale della Nuova Informatica.
07 Un’analisi Aritmetica Booleana dei Sillogismi Aristotelici.
08 Deduzione di Testi e un’Analisi Aritmetiche Booleane della Matematica Riversa di Hilbert.
09 La Computazione "Non-Von Neumann".
10 Il Modello Matematico della Computazione "Von Neumann".
Come conseguenza di queste ricerche, abbiamo ottenuto la creazione dell’ Aritmetica Booleana, matematicamente isomorfa all’Algebra Booleana, il cui linguaggio letterale è totalmente sostituito dal corrispettivo simbolismo numerico del linguaggio aritmetico delle cifre "0/1". Questo ha reso possibile l’ottenimento di una completa e generale soluzione di qualsiasi Sistema d'Equazioni Booleane, non ancora ampiamente diffuso, essendo un argomento di gran rilievo per la Logica Matematica Computazionale.
D’altra parte, l’ Aritmetica Booleana ha reso possibile la creazione della Linguistica Tecnica (ossia, Linguistica Booleana), causando l’arrivo di un unico linguaggio computazionale matematico che ho denominato "Esperangol" (Vide Capitolo 9 pagg. 291-312 - ALLEGATO 2).
I risultati ottenuti in queste ricerche si trovano riassunti nell’ALLEGATO 3.
Penso che le Aree d’Interesse per il Corso potrebbero essere le seguenti: Matematica Applicata, Ingegneria d’Energia ed Automazione Elettrica, Ingegneria Elettronica, Ingegneria Meccanica, Ingegneria Meccatronica, Ingegneria di Hardware, di Software e di Firmware e Interesse Generale.
Nella fiduciosa attesa che la promissiva evoluzione riferita di queste ricerche, frutto longinquo delle aule ripiene d’umanesimo della Matematica Italiana profferite presso la Escola Politécnica da Universidade de São Paulo dall’indimenticabile e di grata memoria Dottor Giacomo Albanese che nel passato 1946 ha fatto destare in me l’attenzione ai problemi concettuali dei Fondamenti della Matematica, renda possibile e pertanto si possa realizzare, la presente offerta del Corso Proposto come Professor Visitatore, colgo l’opportunità d’offrire all’Eccellenza Vostra, la mia dichiarazione di grande stima e straordinaria considerazione, sono
Rispettosamente
(a) Professor Wagner Waneck Martins
Indirizzo Postale: Rua Barão de Itaúna, 156
05078-060 - São Paulo - SP - BRASIL.
Telefono: 0055011 261-4641 Fax: 0055 011 261-4703 Telefono Cellulare: 0055 011 9262-0847
e-mail:
ALLEGATI:
ALLEGATO 1: a) Copia della mia lettera del 17.09.1983; b) "Programma Professori Visitatori - Modello B", avente la data dello 11 Gennaio 1983 e indirizzato, conforme alle istruzioni AL CONSIGLIO NAZIONALE DELLE RICERCHE, corredato dalla bibliografia a disposizione a quell’epoca; c) Programma dell’Attività scientifica che il Prof. Wagner Waneck Martins intende svolgere durante la permanenza in Italia; d) Lettera di risposta del Professor Galligani Ilio del 07.07.1984; e) Copia della mia lettera del 26.03.1985 sull’argomento, per occasione dell’invio del libro del quale sono autore, col titolo, "ESÇÃO (n;m;p): Un Computer Non-Von Neumann".
ALLEGATO 2: Copia in CD-ROM del Libro "Boolean Arithmetic and its Applications", nella cui parte finale c’è il "Curriculum Vitae" dell’autore.
ALLEGATO 3: The results of this lifetime research carried out at University of São Paulo - Polytechnic School - Department of Electrical Energy and Automation (Abstract).
ALLEGATO 4: Extract from Curriculum Vitæ: - Recherches in the Boolean Mathematical Field.
PS. This letter was sent also by mail
Copy to E-mail: galligan@dm.unibo.it
So far, I don’t have received any answer from this letter.
Até agora
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