Course detail

Logic

FIT-LOGAcad. year: 2017/2018

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.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

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.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Regular attendance at exercises and passing both check tests.

Course curriculum

    Syllabus of lectures:
    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)

    Syllabus of numerical exercises:
    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

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

A mid-term test.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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

Recommended reading

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 Intellogence 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 A. Sochor, Klasická matematická logika, Karolinum, 2001 V. Švejnar, Logika, neúplnost a složitost, Academia, 2002

Classification of course in study plans

  • Programme IT-MSC-2 Master's

    branch MMI , 0 year of study, summer semester, elective
    branch MBI , 0 year of study, summer semester, elective
    branch MSK , 1 year of study, summer semester, compulsory-optional
    branch MMM , 0 year of study, summer semester, compulsory
    branch MBS , 0 year of study, summer semester, elective
    branch MPV , 0 year of study, summer semester, elective
    branch MIS , 0 year of study, summer semester, elective
    branch MIN , 0 year of study, summer semester, elective
    branch MGM , 0 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

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 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

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