Course detail

# 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 will be discussed which have applications in computer science.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Výsledky učení předmětu

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).

Prerekvizity

The knowledge acquired in the bachelor's study course "Discrete Mathematics" is assumed.

Korekvizity

Not applicable.

Plánované vzdělávací činnosti a výukové metody

Not applicable.

Způsob a kritéria hodnocení

A mid-term test.

Osnovy výuky

Not applicable.

Pracovní stáže

Not applicable.

Učební cíle

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.

Vymezení kontrolované výuky a způsob jejího provádění a formy nahrazování zameškané výuky

Not applicable.

Doporučené volitelné složky programu

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
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

Classification of course in study plans

• Programme IT-MGR-2 Master's

branch MBI , any year of study, summer semester, elective
branch MPV , any year of study, summer semester, elective
branch MGM , any year of study, summer semester, elective
branch MIS , any year of study, summer semester, elective
branch MBS , any year of study, summer semester, elective
branch MIN , any year of study, summer semester, elective
branch MMM , any year of study, summer semester, compulsory

• Programme MITAI Master's

specialization NADE , any year of study, summer semester, elective
specialization NBIO , any year of study, summer semester, elective
specialization NGRI , any year of study, summer semester, elective
specialization NNET , any year of study, summer semester, elective
specialization NVIZ , any year of study, summer semester, elective
specialization NCPS , any year of study, summer semester, elective
specialization NSEC , any year of study, summer semester, elective
specialization NEMB , any year of study, summer semester, elective
specialization NHPC , any year of study, summer semester, elective
specialization NISD , any year of study, summer semester, elective
specialization NIDE , any year of study, summer semester, elective
specialization NISY do 2020/21 , any year of study, summer semester, elective
specialization NISY , any year of study, summer semester, elective
specialization NMAL , any year of study, summer semester, elective
specialization NMAT , any year of study, summer semester, elective
specialization NSEN , any year of study, summer semester, elective
specialization NVER , any year of study, summer semester, elective
specialization NSPE , any year of study, summer semester, elective

• Programme IT-MGR-2 Master's

branch MSK , 1. 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)

Fundamentals seminar

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Relational systems and universal algebras
2. Sets, cardinal numbers and cardinal arithmetic
3. Sentences, propositional connectives, truth tables,tautologies and contradictions
4. Independence of propositional connectives, axioms of propositional logic
5. Deduction theorem and proving formulas of propositional logic
6. Terms and formulas of predicate logics
7. Interpretation, satisfiability and truth
8. Axioms and rules of inference of predicate logic
9. Deduction theorem and proving formulas of predicate logic
10. Transforming formulas into prenex normal forms
11. First-order theories and some of their models
12. Monadic logics SkS and WSkS
13. Intuitionistic, modal and temporal logics, Presburger arithmetics