Mathematical Principles of Computer Science (FSI-VZI)

Academic year 2021/2022
Supervisor: prof. RNDr. Ing. Miloš Šeda, Ph.D.  
Supervising institute: ÚAI all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The course objective is to make students familiar with basic mathematical structures of the branch and with the methodology of their possible implementations. It is the introduction to the applicability and adequacy of their use.
Learning outcomes and competences:
Competent development and use of nontrivial object oriented implementations of basic mathematic structures of the branch.
Prerequisites:
The understanding of algorithmization principles, structured approach to problem solving and methodology knowledge of non-object program making is expected.
Course contents:
The course provides students with the introduction to mathematical computer science. Formal languages and grammars and word processing tools in these languages are discussed.
The course completes predicate calculus, methods of proving the truth of logical formulas and classification of complexity problems with the definition of classes P and NP.
C/Python is used as an implementation tool. Practical use of theorems and consequents is demonstrated on the implementation of simple technical applications.
The course completes fundaments of graph theory, they cover graph search, Eulerian trails, Hamiltonian paths, shortest paths, minimum spanning trees, network flows, graph colouring and applications of computational geometry structures.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Course-unit credit requirements: Individually elaborated software project. Project must consistently use lectured methodology. The elaboration of project is continuously checked and consulted. Exam is held in the usual manner.
Controlled participation in lessons:
The attendance at lectures is recommended while at seminars it is obligatory. Education runs according to week schedules. The form of compensation of missed seminars is fully in the competence of a tutor.
Type of course unit:
    Lecture  13 × 4 hrs. optionally                  
    Computer-assisted exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 1. Introduction.
2. The list, queue, stack structures, designs of representation and implementation.
3. Breadth-first and depth-first search of graph, combined search; the use of queue and stack. AND/OR graphs.
4. Eulerian trails, Hamiltonian paths.
5. The shortest path, minimum spanning tree.
6. Network flows, graph colouring.
7. Voronoi diagrams and Delaunay triangulation.
8. Formal languages and grammars. Chomsky’s classification.
8. Regular grammars and finite automata.
9. Finite automata without stack, representation.
10. Context-free grammars and finite automata with stack.
11. Turing machine, enumeratibility, algorithm complexity.
12. Sorting algorithms
13. Recapitulation.
    Computer-assisted exercise 1. Principles of code security improvement, separation of overhead and data classes.
2. Implementation of list.
3. Implementation of queue and stack.
4. Implementation of tree.
5. Implementation of general oriented graph, search in graph I.
6. Implementation of general oriented graph, search in graph II.
7. Approaches to implementation of graph evaluation.
8. Searching in special graph topologies; examples of use.
9. Solution designs of simple problems realized through search in oriented evaluated graph.
10. Object implementation of finite automaton without stack.
11. Object implementation of finite automaton with stack.
12. Linguistic variable implementation, if-then operation.
13. Accreditation.
Literature - fundamental:
1. Češka, M.: Teoretická informatika, učební text FIT VUT v Brně, 2002
2. Jungnickel, D: Graphs, networks and algorithms, 2008
3. Greenshaw, R. and Hoover, H.J.: Fundamentals of the Theory of Computation Principle and Practice. 1998
4. Meduna, A.: Grammars with context conditions and their applications, 2005
5. Matoušek, J., Nešetřil, J.: Kapitoly z diskrétní matematiky. 3., upr. a dopl. vyd. V Praze: Karolinum, 2007.
Literature - recommended:
1. Jungnickel, D: Graphs, networks and algorithms, 2008
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-AIŘ-P full-time study --- no specialisation -- Cr,Ex 6 Compulsory 2 1 W