Course detail

# Mathematical Logic

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 like the modal and temporal ones will be discussed which have applications in computer science.

Language of instruction

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Learning outcomes of the course unit

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.
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).

Prerequisites

The knowledge acquired in the bachelor's study course "Discrete Mathematics" and the master's study course "Mathematical Structures in Informatics" is assumed.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

The subject is evaluated according to the result of the final exam, the minimum for passing the exam is 50/100 points.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

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
G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996
Melvin Fitting, First order logic and automated theorem proving, Springer, 1996

Classification of course in study plans

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme DIT-EN Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT-EN Doctoral, any year of study, summer semester, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Basics of set theory and cardinal arithmetics
2. Language, formulas and semantics of propositional calculus
3. Formal theory of the propositional logic
4. Provability in propositional logic, completeness theorem
5. Language of the (first-order) predicate logic, terms and formulas
6. Semantic of predicate logics
7. Axiomatic theory of the first-order predicate logic
8. Provability in predicate logic
9. Theorems on compactness and completeness, prenex normal forms
10. First-order theories and their models
11. Undecidabilitry of first-order theories, Gödel's incompleteness theorems
12. Second-order theories (monadic logic, SkS and WSkS)
13. Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)

Guided consultation in combined form of studies

26 hours, optionally

Teacher / Lecturer