Theoretical Computer Science

• 39 hrs lectures
• 26 hrs exercises
• 13 hrs projects

• 60 pts final exam (written part)
• 25 pts mid-term test (written part)
• 15 pts projects

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.

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 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., 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

1. An introduction to the theory of formal languages, regular languages and grammars, finite  automata, regular expressions.
2. Context-free languages and grammars, push-down automata.
3. Regular languages as a Boolean algebra, Kleene's algebra, Kleene's theorem, minimization of finite-state automata. 
4. Pumping lemma, Nerod's theorem, decidable problems of regular languages. Transformations and normal forms of context-free grammars
5. Advanced properties of context-free languages, Pumping lemma for context-free languages, decidable problems of context-free languages. Deterministic context-free languages. 
6. Turing machines (TMs), the language accepted by a TM, recursively enumerable and recursive languages and problems, TMs and functions, methods of constructing TMs. 
7. Modifications of TMs, TMs with a tape infinite on both sides, with more tapes, nondeterministic TMs, automata with two push-down stacks, automata with counters. 
8. TMs and type-0 languages, diagonalization, properties of recursively enumerable and recursive languages, linearly bounded automata and type-1 languages. 
9. The Church-Turing thesis, universal TMs, undecidability, the halting problem, reductions, Post's correspondence problem. Undecidable problems of the formal language theory. 
10. Computable functions, initial functions, primitive recursive functions, mu-recursive functions, the relation of TMs and computable functions, asymptotic complexity. 
11. An introduction to the computational complexity, Turing complexity,
12. P and NP classes, and beyond. Polynomial reduction, Completeness.
13. Introduction to Petri nets, motivations, definition of P/T Petri nets, methods of Petri net analyses, Petri net classes.

[The first two lectures summarize and formalize the body of knowledge acquired in the IFJ course at FIT VUT. Lectures 3-5 deepen the knowledge in the area of regular and context-free languages. Lectures 6-12 introduce the fundamental principles and concepts in the area of computability and complexity of formal languages and problems. The last lecture introduces the fundamental principles in the area of mathematical description, modeling, and analysis of parallel and distributed dynamical systems using Petri Nets.]

1. Sets and relations. Strings, languages, and operations over them. Grammars, the Chomsky hierarchy of grammars and languages. 
2. Regular languages and finite-state automata (FSA), determinization and minimization of FSA, conversion of regular expressions to FSA. 
3. Kleene algebra. Pumping lemma, proofs of non-regularity of languages. 
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. Recursive and recursively enumerable languages and their properties. 
9. Decidability, semi-decidability, and undecidability of problems, reductions of problems. 
10. Computable functions. Other Turing-complete computing mechanisms (automata with multiple push-down stacks, counter automata). 
11. Complexity classes. Properties of space and time complexity classes. 
12. NP problems. Polynomial reduction. 
13. Petri nets

1. Assignment  in the area of regular and context free languages.
2. Assignment  in the area of Turing machines and the theory of undecidability.
3. Assignment  in the area of computable functions, complexity, and Petri nets.

An evaluation of the exam in the 4th week (max. 15 points) and in the 9th week (max. 15 points), an evaluation of the assignments (max 3-times 5 points) and an final exam evaluation (max 60 points).
Exam prerequisites:
The minimal total score of 15 points achieved from the first two assignments, and from the exams in the 4th and 9th week (i.e. out of 35 points).

A written exam in the 4th week focusing on the fundamental knowledge in the area of regular and context-free languages, a written exam in the 9th week focusing on advance topics in the area of regular and context-free languages, and on Turing machines, regular evaluation of the assignments, a final written exam. Students have to achieve at least 25 points, otherwise the exam is assessed by 0 points.

The minimal total score of 15 points achieved from the first two assignments, and from the exams in the 4th and 9th week (i.e. out of 35 points).