Course detail
Logic
FIT-LOGAcad. year: 2020/2021
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.
Supervisor
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" is assumed.
Co-requisites
Not applicable.
Recommended optional programme components
Not applicable.
Recommended or required reading
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
Planned learning activities and teaching methods
Not applicable.
Assesment methods and criteria linked to learning outcomes
A mid-term test.
Exam prerequisites:
Regular attendance at exercises and passing both check tests.
Language of instruction
Czech
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.
Classification of course in study plans
- Programme IT-MGR-2 Master's
branch MBI , any year of study, summer semester, 5 credits, elective
branch MPV , any year of study, summer semester, 5 credits, elective
branch MGM , any year of study, summer semester, 5 credits, elective
branch MIS , any year of study, summer semester, 5 credits, elective
branch MBS , any year of study, summer semester, 5 credits, elective
branch MIN , any year of study, summer semester, 5 credits, elective
branch MMI , any year of study, summer semester, 5 credits, elective
branch MMM , any year of study, summer semester, 5 credits, compulsory - Programme MITAI Master's
specialization NADE , any year of study, summer semester, 5 credits, elective
specialization NBIO , any year of study, summer semester, 5 credits, elective
specialization NGRI , any year of study, summer semester, 5 credits, elective
specialization NNET , any year of study, summer semester, 5 credits, elective
specialization NVIZ , any year of study, summer semester, 5 credits, elective
specialization NCPS , any year of study, summer semester, 5 credits, elective
specialization NSEC , any year of study, summer semester, 5 credits, elective
specialization NEMB , any year of study, summer semester, 5 credits, elective
specialization NHPC , any year of study, summer semester, 5 credits, elective
specialization NISD , any year of study, summer semester, 5 credits, elective
specialization NIDE , any year of study, summer semester, 5 credits, elective
specialization NISY , any year of study, summer semester, 5 credits, elective
specialization NMAL , any year of study, summer semester, 5 credits, elective
specialization NMAT , any year of study, summer semester, 5 credits, elective
specialization NSEN , any year of study, summer semester, 5 credits, elective
specialization NVER , any year of study, summer semester, 5 credits, elective
specialization NSPE , any year of study, summer semester, 5 credits, elective - Programme IT-MGR-2 Master's
branch MSK , 1. year of study, summer semester, 5 credits, compulsory-optional
Type of course unit
Lecture
26 hours, optionally
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)
Fundamentals seminar
26 hours, compulsory
Teacher / Lecturer
Syllabus
- 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