Theoretical Computer Science

• 39 hrs lectures
• 10 hrs seminar
• 16 hrs exercises
• 13 hrs projects

• 60 pts final exam (written part)
• 25 pts written tests (written part)
• 15 pts projects

To acquaint students with more advanced parts of the formal language theory, with basics of the theory of computability, and with basic terms of the complexity theory.

The students are acquainted with basic as well as more advanced terms, approaches, and results of the theory of automata and formal languages and with basics of the theory of computability and complexity allowing them to better understand the nature of the various ways of describing and implementing computer-aided systems. 
The students acquire basic capabilities for theoretical research activities.

The course acquaints students with fundamental principles of computer science and allows them to understand where boundaries of computability lie, what the costs of solving various problems on computers are, and hence where there are limits of what one can expect from solving problems on computing devices - at least those currently known. Further, the course acquaints students, much more deeply than in the bachelor studies, with a number of concrete concepts, such as various kinds of automata and grammars, and concrete algorithms over them, which are commonly used in many application areas (e.g., compilers, text processing, network traffic analysis, optimisation of both hardware and software, modelling and design of computer systems, static and dynamic analysis and verification, artificial intelligence, etc.). Deeper knowledge of this area will allow the students to not only apply existing algorithms but to also extend them and/or to adjust them to fit the exact needs of the concrete problem being solved as often needed in practice. Finally, the course builds the students capabilities of abstract and systematic thinking, abilities to read and understand formal texts (hence allowing them to understand and apply in practice continuously appearing new research results), as well as abilities of exact communication of their ideas.

Basic knowledge of discrete mathematics concepts including algebra, mathematical logic, graph theory and formal languages concepts, and basic concepts of algorithmic complexity.

• Češka, M. a kol.: Vyčíslitelnost a složitost, Nakl. VUT Brno, 1993. ISBN 80-214-0441-8
• Češka, M., Rábová, Z.: Gramatiky a jazyky, Nakl. VUT Brno, 1992. ISBN 80-214-0449-3
• Češka, M., Vojnar, T.: Studijní  text k předmětu Teoretická informatika (http://www.fit.vutbr.cz/study/courses/TIN/public/Texty/TIN-studijni-text.pdf), 165 str. (in Czech)
• Kozen, D.C.: Automata and Computability, Springer-Verlag, New Yourk, Inc, 1997. ISBN 0-387-94907-0
• Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2nd ed., 2000. ISBN 0-201-44124-1
• Meduna, A.: Formal Languages and Computation. New York, Taylor & Francis, 2014.
• Aho, A.V., Ullmann, J.D.: The Theory of Parsing,Translation and Compiling, Prentice-Hall, 1972. ISBN 0-139-14564-8
• Martin, J.C.: Introduction to Languages and the Theory of Computation, McGraw-Hill, Inc., 3rd ed., 2002. ISBN 0-072-32200-4
• Brookshear, J.G. : Theory of Computation: Formal Languages, Automata, and Complexity, The Benjamin/Cummings Publishing Company, Inc, Redwood City, California, 1989. ISBN 0-805-30143-7

• Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2nd ed., 2000. ISBN 0-201-44124-1
• Kozen, D.C.: Automata and Computability, Springer-Verlag, New Yourk, Inc, 1997. ISBN 0-387-94907-0
• Martin, J.C.: Introduction to Languages and the Theory of Computation, McGraw-Hill, Inc., 3rd ed., 2002. ISBN 0-072-32200-4
• Brookshear, J.G. : Theory of Computation: Formal Languages, Automata, and Complexity, The Benjamin/Cummings Publishing Company, Inc, Redwood City, California, 1989. ISBN 0-805-30143-7
• Aho, A.V., Ullmann, J.D.: The Theory of Parsing,Translation and Compiling, Prentice-Hall, 1972. ISBN 0-139-14564-8

1. An introduction to the theory of formal languages, regular languages and grammars, finite  automata, regular expressions.
2. Minimization of finite-state automata, pumping theorem, Nerod's theorem, decidable problems of regular languages.
3. Context-free languages and grammars, push-down automata, transformations and normal forms of context-free grammars.
4. Advanced properties of context-free languages, pumping theorem for context-free languages, decidable problems of context-free languages, deterministic context-free languages. 
5. Turing machines (TMs), the language accepted by a TM, recursively enumerable and recursive languages and problems.
6. TMs and type-0 languages, properties of recursively enumerable and recursive languages, linearly bounded automata and type-1 languages. 
7. The Church-Turing thesis, undecidability, the halting problem, reductions, Post's correspondence problem, undecidable problems of the formal language theory, diagonalization. 
8. Predicate logic and its axiomatization.
9. Gödel's incompleteness theorems.
10. An introduction to the computational complexity, Turing complexity, asymptotic complexity.
11. P and NP classes, polynomial reduction, completeness.
12. Amortized complexity, problems beyond NP.

1. Basics of discrete mathematics, Formal languages, and operations over them. Grammars, the Chomsky hierarchy of grammars and languages. 
2. Regular languages and finite-state automata (FSA) and their determinization.
3. Conversion of regular expressions to FSA. Minimization of FSA. Pumping lemma
4. Context-free languages and grammars. Transformations of context-free grammars. 
5. Operations on context-free languages and their closure properties. Pumping lemma for context-free languages. 
6. Push-down automata, (nondeterministic) top-down and bottom-up syntax analysis. Deterministic push-down languages. 
7. Turing machines.
8. Decidability, semi-decidability, and undecidability of problems, reductions of problems I. 
9. Decidability, semi-decidability, and undecidability of problems, reductions of problems II.
10. Predicate logic I.
11. Predicate logic II.
12. Complexity classes. Properties of space and time complexity classes. 
13. P and NP problems. Polynomial reduction. 

1. Assignment  in the area of regular and contex-free languages.
2. Assignment  in the area of context-free languages and Turing machines.
3. Assignment  in the area of undecidability and complexity.

The evaluation of the course consists of the test in the 4th week (max. 15 points) and the test in the 9th week (max. 15 points), the assignments (max 3-times 5 points), and the final exam (max 60 points).

The written test in the 4th week focuses on the regular languages and on a basic knowledge of the context-free languages. The written test in the 9th week focuses on advance topics in the context-free languages, and on the area of decidability and undecidability including constructions of Turing machines.

The requirements to obtain the accreditation that is required for the final exam: The minimal total score of 18 points achieved from the assignments and from the tests in the 4th and 9th week (i.e. out of 40 points).

The final exam has 4 parts. Students have to achieve at least 4 points from each part and at least 25 points in total, otherwise the exam is evaluated by 0 points.

The minimal total score of 18 points achieved from the first two assignments, and from the exams in the 4th and 9th week (i.e. out of 40 points).