Course detail
Mathematical Logic
FIT-MLDAcad. year: 2021/2022
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
Mode of study
Guarantor
Learning outcomes of the course unit
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
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Course curriculum
Work placements
Aims
Specification of controlled education, way of implementation and compensation for absences
Recommended optional programme components
Prerequisites and corequisites
Basic literature
Recommended reading
A. Sochor, Klasická matematická logika, Karolinum, 2001
D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intelligence and Logic Programming, Oxford Univ. Press 1993
D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993
E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
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
V. Švejnar, Logika, neúplnost a složitost, Academia, 2002
Classification of course in study plans
- Programme DIT Doctoral 0 year of study, summer semester, compulsory-optional
- Programme DIT Doctoral 0 year of study, summer semester, compulsory-optional
- Programme CSE-PHD-4 Doctoral
branch DVI4 , 0 year of study, summer semester, elective
- Programme CSE-PHD-4 Doctoral
branch DVI4 , 0 year of study, summer semester, elective
- Programme DIT-EN Doctoral 0 year of study, summer semester, compulsory-optional
- Programme DIT-EN Doctoral 0 year of study, summer semester, compulsory-optional
- Programme CSE-PHD-4 Doctoral
branch DVI4 , 0 year of study, summer semester, elective
- Programme CSE-PHD-4 Doctoral
branch DVI4 , 0 year of study, summer semester, elective
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
- 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)
Guided consultation in combined form of studies
Teacher / Lecturer