Mathematical Logic

Vztahy:

Předmět BIE-LOG.21 nesmí být zapsán, je-li v témže semestru zapsán anebo již dříve absolvován předmět BIE-MLO (vztah je symetrický)

Předmět BIE-LOG.21 nesmí být zapsán, je-li v témže semestru zapsán anebo již dříve absolvován předmět BIE-MLO (vztah je symetrický)

Garant předmětu:

Kateřina Trlifajová

Přednášející:

Kateřina Trlifajová

Cvičící:

Kateřina Trlifajová

Předmět zajišťuje:

katedra aplikované matematiky

Anotace:

The course focuses on the basics of propositional and predicate logic. It starts from the semantic point of view. Based on the notion of truth, satisfiability, logical equivalence, and the logical consequence of formulas are defined. Methods for determining the satisfiability of formulas, some of which are used for automated proving, are explained. This relates to the P vs. NP problem and Boolean functions in propositional logic. In predicate logic, the course further deals with formal theories, such as arithmetics, and their models. The syntactic approach to mathematical logic is demonstrated on the axiomatic system of propositional logic and its properties. Gödel's incompleteness theorems is explained.

Požadavky:

Knowledge of basic mathematical structures from algebra and analysis

Osnova přednášek:

1.Historical introduction. Syntax and semantics of propositional logic. Proof by induction.

2.Logical equivalence. Full and minimal conjunctive and disjunctive normal forms.

3.Logical consequence. Tableau method for propositional logic.

4.Resolution method. SAT problem. P vs. NP problem.

5.Boole algebra. Boolean functions.

6.Predicate logic. Syntax. Interpretation.

7.Logical truth, satisfiability, contradictions. Logical equivalence.

8.Logical consequence. Tableau method for predicate logic.

9.Prenex normal forms. Resolution method for predicate logic.

10.First-order theories and its models. Ordering, equivalence, arithmetic.

11.Axiomatic system of propositional logic.

12.Consistency, correctness, completeness.

13.Gödel incompleteness theorems.

Osnova cvičení:

1.Propositional formulas. Truth tables. Formalization.

2.Basic logical laws. Universal system of connectives.

3.Disjunctive and conjunctive normal forms. Logical consequence.

4.Tableau method. Resolution method.

5.Boole algebra: properties, counting, ordering, atoms.

6.Predicate logic. Language, terms, formulas. Formalization.

7.Three levels of truth. Logical equivalence.

8.Interpretation. Satisfiable formulas.

9.Logical consequence. Tableau method.

10.Prenex form. Resolution method.

11.Theories and their models. Isomorphism and elementary equivalence.

12.Hilbert axiomatic system.

13.Repetition.

Cíle studia:

The goal is to learn to work in formal mathematical logic, to understand its syntax and semantics. Work with theories as axiomatic systems and derive their consequences. Know what the correctness, completeness, consistency, and decidability of theories mean, and which problems are relied to. Understand Boolean algebra as a generalization of propositional logic.

Studijní materiály:

1.Trlifajová K., Vašata D.,: Matematická logika. CVUT, 2017. ISBN 978-80-01-05342-3.

2.Mendelson E.: Introduction to Mathematical Logic (6th Edition). Chapman and Hall, 2015. ISBN 978-1482237726.

3.Bergmann M., Moor J., Nelson J.: The Logic Book (6th Edition). McGraw-Hill, 2013. ISBN 978-0078038419.

Poznámka:

Rozvrh na zimní semestr 2024/2025:

Rozvrh není připraven

Rozvrh na letní semestr 2024/2025:

Rozvrh není připraven

Předmět je součástí následujících studijních plánů:

• Bachelor specialization, Computer Engineering, 2021 (volitelný předmět)
• Bachelor specialization, Information Security, 2021 (volitelný předmět)
• Bachelor specialization, Software Engineering, 2021 (volitelný předmět)
• Bachelor specialization, Computer Science, 2021 (PS)
• Bachelor specialization, Computer Networks and Internet, 2021 (VO)
• Bachelor specialization Computer Systems and Virtualization, 2021 (volitelný předmět)