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Introduction.
Purpose.
New sections of the discrete mathematics course, although implemented in the form curricula and series of lectures, do not yet exist in the form of monographs, at least in Russian, since the course of discrete mathematics for technical universities is focused on old applied problems that engineers had to solve. In particular, in mathematical logic it was the minimization of logical circuits, which has lost its relevance today.
It is interesting to note that the theory of logic circuit synthesis, having gone through almost a complete “biological cycle” before the eyes of one generation of researchers, is a very instructive example of how industries are highly susceptible to obsolescence technical sciences, weakly related to fundamental science. 10 years ago everything technical magazines were filled with articles on minimization and synthesis of logic circuits. Most of the minimization methods developed by scientists are now forgotten and are not in demand in practice. And those ideas that were considered purely theoretical at that time found practical application in modern technology. For example, fuzzy logic, Petri nets and algorithm theory have stood the test of time and are widely used in various fields of cybernetics and programming, such as system Programming, computational complexity and artificial intelligence.
And the theory of algorithms became the central section of discrete mathematics. However, unlike most monographs in Russian, in the course of lectures these issues are presented as a means of solving practical, engineering problems.
As is known, after each decade, the component base of computers, OS, the means of access and the programs themselves are changing radically. However, the structures and algorithms underlying them remain unchanged for much longer. These foundations began to be laid thousands of years ago, when formal logic was developed and the first algorithms were developed.
Mathematical logic and the theory of algorithms traditionally belong to fundamental science and are considered to be of little practical relevance and difficult to understand. Indeed, when J. Bull created mathematical apparatus Boolean algebra, it took him a long time to find practical application, however, in the 20th century it was precisely this mathematical apparatus that made it possible to design all computer components. Consequently, the first of these prejudices is successfully refuted by the development of computer technology.
As for the prejudice about the difficulty of understanding this discipline, it largely stems from the fact that books on mathematical logic and the theory of algorithms were written by mathematicians for mathematicians.
Now, when the capabilities of computing technology have increased many times over, and there are much more personal computers themselves than there are people who know how to use them effectively, understanding what can and cannot be done with the help of modern computing technology is of exceptional importance.
Exactly general theory algorithms have shown that there are problems that are unsolvable no matter how powerful the computing power is, and its rapidly developing branch, the theory of computational complexity, gradually leads to the understanding that there are problems that can be solved, but are objectively complex, and their complexity may turn out to be in some sense absolute, those. practically inaccessible to modern computers.
This course set the following objectives:
1. Present all the issues under consideration as simply as possible, but not simpler than is required for a highly qualified specialist.
2. Practical problems of design and analysis of information systems are the starting point, and the formal apparatus is a means of systematically solving these problems. It is our deep conviction that a student is not a vessel that needs to be filled, but a torch that needs to be lit.
3. Each section of the course contains self-test questions. To complete this course, the student must answer all these questions.
As a result of mastering this course, the student, based on a clear understanding of the relevant theoretical sections, should be able to:
Implement the simplest type of logical transformation of information in an arbitrary basis of logical functions;
Highlight in evidentiary reasoning natural language logical structure, build formal proof schemes and check their correctness.
1.2 Logical representations
Logical representations - description of the system, process, phenomenon under study in the form of a set complex statements made up of simple (elementary) statements And logical connectives between them. Logical representations and their components are characterized by certain properties and a set of permissible transformations over them (operations, inference rules, etc.), implementing those developed in formal (mathematical) logic correct methods reasoning - the laws of logic.
Methods (rules) of formal presentation of statements, construction of new statements from existing ones using logically correct transformations, as well as methods (methods) of establishing the truth or falsity of statements are studied in mathematical logic. Modern mathematical logic includes two main sections: logic of statements and covering it predicate logic(Fig. 1.1), for the construction of which there are two approaches (languages), forming two variants of formal logic: algebra of logic And logical calculus. There is a one-to-one correspondence between the basic concepts of these languages of formal logic. Their isomorphism is ultimately ensured by the unity of the underlying admissible transformations.
Rice. 1.1
The main objects of traditional branches of logic are statements.
Statement - declarative sentence (statement, judgment), o which it makes sense to say that it true or false. All scientific knowledge(laws and phenomena of physics, chemistry, biology, etc., mathematical theorems, etc.), events Everyday life, situations arising in economics and management processes are formulated in the form of statements. Imperative and interrogative sentences are not statements.
Examples of statements: “Twice two is four”, “We live in the 21st century”, “The ruble is the Russian currency”, “Alyosha is Oleg’s brother”, “The operations of union, intersection and addition are Boolean operations on sets”, “Man is mortal” , “Rearranging the places of the terms does not change the sum,” “Today is Monday,” “If it rains, you should take an umbrella.”
In order to further operate with these sentences as statements, we must know for each of them whether it is true or false, i.e. know them truth value (truth). Note that in some cases the truth or falsity of a statement depends on what specific reality (system, process, phenomenon) we are trying to describe with its help. In this case, the given statement is said to be true (or false) in a given interpretation (context). We further assume that the context is given and the statement has a certain truth value.
1.3 History of developed mathematical logic
Logic as a science was formed in the 4th century. BC. It was created by the Greek scientist Aristotle.The word “logic” comes from the Greek “logos”, which on the one hand means “word” or “exposition”, and on the other, thinking. IN explanatory dictionary Ozhegova S.I. It is said: “Logic is the science of the laws of thinking and its forms.” In the 17th century German scientist Leibniz planned to create a new science, which would be “the art of calculating truth” . In this logic, according to Leibniz, each statement would have a corresponding symbol, and reasoning would have the form of calculations. This idea of Leibniz, having not met the understanding of his contemporaries, was not spread or developed and remained a brilliant guess.
Only in the middle of the 19th century. Irish mathematician George Boole embodied Leibniz's idea. In 1854, he wrote the work “Investigation of the laws of thought,” which laid the foundations for the algebra of logic, in which laws similar to the laws of ordinary algebra apply, but the letters do not denote numbers , but statements. In the language of Boolean algebra, one can describe reasoning and “compute” its results. However, it does not cover all reasoning, but only a certain type of it. , Therefore, Boole algebra is considered a propositional calculus.
Boole's algebra of logic was the embryo of a new science - mathematical logic. In contrast, Aristotle's logic is called traditional formal logic. The name “mathematical logic” reflects two features of this science: firstly, mathematical logic is logic that uses the language and methods of mathematics; secondly, mathematical logic is brought to life by the needs of mathematics.
At the end of the 19th century. The set theory created by Georg Cantor seemed to be a reliable foundation for all mathematics, including mathematical logic, at least for propositional calculus (Boole algebra), because It turned out that Cantor algebra (set theory) is isomorphic to Boole algebra.
Mathematical logic itself became a branch of mathematics that at first seemed highly abstract and infinitely far from practical applications. However, this area did not remain the domain of “pure” mathematicians for long. At the beginning of the 20th century. (1910) Russian scientist Ehrenfest P.S. pointed out the possibility of using the apparatus of Boolean algebra in telephone communications to describe switching circuits. In 1938-1940, almost simultaneously, the works of the Soviet scientist V.I. Shestakov, the American scientist Shannon and the Japanese scientists Nakashima and Hakazawa appeared on the application of mathematical logic in digital technology. The first monograph devoted to the use of mathematical logic in the design of digital equipment was published in the USSR by the Soviet scientist M.A. Gavrilov. in 1950. The role of mathematical logic in the development of modern microprocessor technology is extremely important: it is used in the design of computer hardware, in the development of all programming languages and in the design of discrete automation devices.
Scientists from different countries made a great contribution to the development of mathematical logic: professor of Kazan University Poretsky P.S., de-Morgan, Peirce, Turing, Kolmogorov A.N., Heidel K. and others.
1.4 Questions for self-test.
1. Formulate the objectives of the course
The book was written based on materials from lectures and seminars conducted by the authors for junior students of the Faculty of Mechanics and Mathematics of Moscow State University. It talks about the basic concepts of mathematical logic (propositional logic, first-order languages, expressibility, propositional calculus, decidable theories, the completeness theorem, principles of model theory). The presentation is intended for students mathematics schools, mathematics students and everyone interested mathematical logic. The book includes about 200 problems of varying difficulty.
Logic of statements.
Utterances and Operations
“If the number n is rational, then n - algebraic number. But it is not algebraic. This means p is not rational.” We do not have to know what the number n is, which numbers are called rational and which are algebraic, in order to recognize that this reasoning is correct in the sense that the conclusion actually follows from the two stated premises. This kind of situation - when a certain statement is true regardless of the meaning of the statements included in it - constitutes the subject of propositional logic.
This beginning (especially considering that the logic course was part of the program of the Faculty of Philosophy, where “dialectical logic” was also studied) is alarming, but in fact our considerations will have a completely precise mathematical nature, although we will start with informal motivations.
Table of contents
Preface
1. Propositional logic
1.1. Utterances and Operations
1.2. Complete systems bundles
1.3. Schemes of functional elements
2. Propositional calculus
2.1. Propositional Calculus (PC)
2.2. Second proof of the completeness theorem
2.3. Finding a counterexample and sequent calculus
2.4. Intuitionistic propositional logic
3. First order languages
3.1. Formulas and interpretations
3.2. Definition of truth
3.3. Expressible predicates
3.4. Expressibility in arithmetic
3.5. Inexpressible predicates: automorphisms
3.6. Elimination of quantifiers
3.7. Presburger Arithmetic
3.8. Tarski-Seidenberg theorem
3.9. Elementary equivalence
3.10. Ehrenfeucht's game
3.11. Power reduction
4. Predicate calculus
4.1. Generally valid formulas
4.2. Axioms and rules of inference
4.3. Correctness of predicate calculus
4.4. Conclusions in predicate calculus
4.5. Completeness of predicate calculus
4.6. Renaming Variables
4.7. Prefixed normal form
4.8. Herbrand's theorem
4.9. Skolemov functions
5. Theories and models
5.1. Axioms of equality
5.2. Power boost
5.3. Complete theories
5.4. Incomplete and undecidable theories
5.5. Diagrams and Extensions
5.6. Ultrafilters and compactness
5.7. Non-standard analysis
Literature
Subject index
Index of names.
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