Theoretical Computer Science

• 39 hrs lectures
• 13 hrs 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 are capable of applying the acquainted knowledge when solving complex theoretical as well as practical problems in the areas of system modelling, programming, formal specification and verification, and artificial intelligence.

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.

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

1. An overview of the applications of the formal language theory, the modelling and decision power of formalisms, operations over languages.
2. Regular languages and their properties, Kleene's theorem, Nerod's theorem, Pumping lemma.
3. Minimalization of finite-state automata, the relation of indistinguishability of automata states, construction of a reduced finite-state automaton.
4. Closure properties of regular languages, regular languages as a Boolean algebra, decidable problems of regular languages.
5. Context-free languages and their properties, normal forms of context-free grammars, unambiguous and deterministic context-free languages, Pumping lemma for context-free languages.
6. Closure properties of context-free languages, closedness wrt. substitution and its consequences, decidable problems of context-free languages.
7. Turing machines (TMs), the language accepted by a TM, recursively enumerable and recursive languages and problems, TMs and functions, methods of  constructing TMs.
8. 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.
9. TMs and type-0 languages, diagonalisation, properties of recursively enumerable and recursive languages, linearly bounded automata and type-1 languages.
10. Computable functions, initial functions, primitive recursive functions, mu-recursive functions, the relation of TMs and computable functions.
11. The Church-Turing thesis, universal TMs, undecidability, the halting problem, reductions, the Post's correspondence problem.
12. Undecidable problems of the formal language theory.
13. An introduction to the computational complexity, Turing complexity, the P and NP classes and beyond.

The minimal total score 20 points (the projects and the mid-term exam).

A written mid-term exam, a regular evaluation of homeworks.