Courses in academic year 2019/2020
Ac.Y. 2018/19
Summer Schedule 2018/19
Bachelor Study
Master Study
Ph.D. Study
Ac.Year:
Sem:
Curriculums:
Field/
IT-MSC-2 MBI - Elective
IT-MSC-2 MBS - Elective
IT-MSC-2 MGM - Elective
IT-MSC-2 MIN - Elective
IT-MSC-2 MIS - Elective
IT-MSC-2 MMI - Elective
IT-MSC-2 MMM 2nd Compulsory
IT-MSC-2 MPV - Elective
IT-MSC-2 MSK 1st Compulsory-Elective - group M
credit+exam
Type of
instruction:
Hour/sem
Guarantor:
Šlapal Josef, prof. RNDr., CSc. (DADM)
Faculty of Mechanical Engineering BUT
Department of Algebra and Discrete Mathematics FME BUT
The aim of the course is to acquaint students with the basic methods of reasoning in mathematics. The students should learn about general principles of predicate logic and, consequently, acquire the ability of exact mathematical reasoning and expression. They should also get familiar with some other important formal theories utilizied in informatics too.
In the course, the basics of propositional and predicate logics will be taught. First, the students will get acquainted with the syntax and semantics of the logics, then the logics will be studied as formal theories with an emphasis on formula proving. The classical theorems on correctness, completeness and compactness will also be dealt with. After discussing the prenex forms of formulas, some properties and models of first-order theories will be studied. We will also deal with the undecidability of first-order theories resulting from the well-known Gödel incompleteness theorems. Finally, some further important logics will be discussed which have applications in computer science.
Knowledge and skills required for the course:
The knowledge acquired in the bachelor's study course "Discrete Mathematics" is assumed.
Subject specific learning outcomes and competencies:
The students will acquire the ability of understanding the principles of axiomatic mathematical theories and the ability of exact (formal) mathematical expression. They will also learn how to deduct, in a formal way, new formulas and to prove given ones. They will realize the efficiency of formal reasonong and also its limits.
Generic learning outcomes and competencies:
The students will learn exact formal reasoning to be able to devise correct and efficient algorithms solving given problems. They will also acquire an ability to verify the correctness of given algorithms (program verification).
Syllabus of lectures:
Basics of set theory and cardinal arithmetics
Language, formulas and semantics of propositional calculus
Formal theory of the propositional logic
Provability in propositional logic, completeness theorem
Language of the (first-order) predicate logic, terms and formulas
Semantic of predicate logics
Axiomatic theory of the first-order predicate logic
Provability in predicate logic
Theorems on compactness and completeness, prenex normal forms
First-order theories and their models
Undecidabilitry of first-order theories, Gödel's incompleteness theorems
Second-order theories (monadic logic, SkS and WSkS)
Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)
Syllabus of numerical exercises:
Relational systems and universal algebras
Sets, cardinal numbers and cardinal arithmetic
Sentences, propositional connectives, truth tables,tautologies and contradictions
Independence of propositional connectives, axioms of propositional logic
Deduction theorem and proving formulas of propositional logic
Terms and formulas of predicate logics
Interpretation, satisfiability and truth
Axioms and rules of inference of predicate logic
Deduction theorem and proving formulas of predicate logic
Transforming formulas into prenex normal forms
First-order theories and some of their models
Monadic logics SkS and WSkS
Intuitionistic, modal and temporal logics, Presburger arithmetics
Fundamental literature:
E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993
D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intelligence and Logic Programming, Oxford Univ. Press 1993
G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996
Melvin Fitting, First order logic and automated theorem proving, Springer, 1996
Sally Popkorn, First steps in modal logic, Cambridge Univ. Press, 1994
Study literature:
D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993
A. Sochor, Klasická matematická logika, Karolinum, 2001
V. Švejnar, Logika, neúplnost a složitost, Academia, 2002
Progress assessment:
A mid-term text.