Course details

# Advanced Mathematics

IAM Acad. year 2023/2024 Summer semester 5 credits

The course is a follow-up to compulsory mathematical courses at FIT. Students learn how to use mathematics methods on several subjects closely related to computer science. These are mainly number theory and its application in cryptography, basic set theory and logic, logical systems and decision procedures with applications in e.g. databases or software engineering, probability, statistics, and their applications in the analysis of probabilistic systems and artificial intelligence.

Guarantor

Course coordinator

Češka Milan, doc. RNDr., Ph.D. (DITS)

Hliněná Dana, doc. RNDr., Ph.D. (UMAT)

Lengál Ondřej, Ing., Ph.D. (DITS)

Language of instruction

Completion

Time span

- 26 hrs lectures
- 18 hrs exercises
- 8 hrs pc labs

Assessment points

- 50 pts mid-term test (written part)
- 50 pts numeric exercises

Department

Lecturer

Češka Milan, doc. RNDr., Ph.D. (DITS)

Hliněná Dana, doc. RNDr., Ph.D. (UMAT)

Holík Lukáš, doc. Mgr., Ph.D. (DITS)

Lengál Ondřej, Ing., Ph.D. (DITS)

Instructor

Češka Milan, doc. RNDr., Ph.D. (DITS)

Hliněná Dana, doc. RNDr., Ph.D. (UMAT)

Holík Lukáš, doc. Mgr., Ph.D. (DITS)

Lengál Ondřej, Ing., Ph.D. (DITS)

Learning objectives

- Practice mathematical writing and thinking, formulation of problems and solving them,
- obtain deeper insight into several areas of mathematics with applications in computer science,
- learn on examples that complicated mathematics can lead to useful algorithms and tools.

The ability to exactly and formally specify and solve problems, formally prove claims; also better understanding of the basic mathematical concepts, an overview of several areas of mathematics important in computer science. Improving the abilities of exact thinking, expressing ideas, and using a mathematical apparatus.

Why is the course taught

The main goal is not to cover a given area of mathematics but to infect a student with an enthusiasm for the subject. The student will be given an opportunity to walk the way from abstract theory to its actual use in the real world, which might possibly convince him/her that thinking and expressing ideas with a help of formal mathematics makes sense. These skills will be exercised by solving problem assignments that should be interesting and sometimes demanding. Obtaining a better overview of several areas of mathematics important in informatics is an added benefit.

Recommended prerequisites

- Discrete Mathematics (IDM)
- Formal Languages and Compilers (IFJ)
- Mathematical Analysis 1 (IMA1)
- Mathematical Analysis 2 (IMA2)
- Probability and Statistics (IPT)

Prerequisite knowledge and skills

Basic knowledge of sets, relations, propositional and predicate logic, algebra, and finite automata.

Study literature

- R. Smullyan. First-Order Logic. Dover, 1995.
- B. Balcar, P. Štěpánek. Teorie množin. Academia, 2005.
- C. M. Grinstead, J. L. Snell. Introduction to probability. American Mathematical Soc., 2012.
- G. Chartrand, A. D. Polimeni, P. Zhang. Mathematical Proofs: A Transition to Advanced Mathematics, 2013
- Steven Roman. Lattices and Ordered Sets, Springer-Verlag New York, 2008.
- A. Doxiadis, C. Papadimitriou. Logicomix: An Epic Search for Truth. Bloomsbury, 2009.

Fundamental literature

- A.R. Bradley, Z. Manna. The Calculus of Computation. Springer, 2007.
- D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena Scientific, 2008.
- M. Huth, M. Ryan. Logic in Computer Science. Modelling and Reasoning about Systems. Cambridge University Press, 2004.

Syllabus of lectures

- Axioms of set theory, the axiom of choice. Countable and uncountable sets, cardinal numbers. (Dana Hliněná)
- Application of number theory in cryptography. (Dana Hliněná)
- Number theory: prime numbers, Fermat's little theorem, Euler's function. (Dana Hliněná)
- Propositional logic. Syntax and semantics. Proof techniques for propositional logic: syntax tables, natural deduction, resolution. (Ondřej Lengál)
- Predicate logic. Syntax and semantics. Proof techniques for predicate logic: semantic tables, natural deduction. (Ondřej Lengál)
- Predicate logic. Craig interpolation. Important theories. Undecidability. Higher order logic. (Ondřej Lengál)
- Hoare logic. Precondition, postcondition. Invariant. Deductive verification of programs. (Ondřej Lengál)
- Decision procedures in logic: Classical decision procedures for arithmetics over integers and over rationals. (Lukáš Holík)
- Automata-based decision procedures for arithmetics and for WS1S (Lukáš Holík)
- Decision procedures for combined theories. (Lukáš Holík)
- Advanced combinatorics: inclusion and exclusion, Dirichlet's principle, chosen combinatorial theorems. (Milan Češka)
- Conditional probability, statistical inference, Bayesian networks. (Milan Češka)
- Probabilistic processes: Markov and Poisson process. Applications in informatics: quantitative analysis, performance analysis.

Syllabus of numerical exercises

- Proofs in set theory, Cantor's diagonalization, matching, Hilbert's hotel.
- Prime numbers and cryptography, RSA, DSA, cyphers.
- Proofs in number theory, Chinese remainder theorem.
- Proofs in propositional logic.
- Proofs in predicate logic.
- Decision procedures.
*Computer labs 1.**Computer labs 2.*- Automata decision procedures and combination theories.
*Computer labs 3.*- Proofs in combinatorics.
- Conditional probability and statistical inference in practice.
*Computer labs 4.*

Syllabus of computer exercises

- Proving programs corrects in VCC.
- SAT and SMT solvers.
- Tools MONA and Vampire.
- Analysis of probabilistic systems, PRISM.

Progress assessment

Two tests, midterm and final (25 points per test), activity during exercises (5 points per exercise).

Exam prerequisites

Obtaining at least 50 points from the 100 possible (50 tests, 50 exercises).

Course inclusion in study plans

- Programme BIT, 2nd year of study, Elective
- Programme BIT (in English), 2nd year of study, Elective
- Programme IT-BC-3, field BIT, 2nd year of study, Elective