Modern trends in discrete mathematics and computer science

Compulsory course for master students, 2-nd year, groups 618/1 and 618/2.

Lectures — 16 h.

Responsible lecturer Vladimir Zakharov.

This course is intended to introduce the audience to some topics in discrete mathematics and its applications in software engineering, VLSI designing and biochemistry which are in line with the modern development of computer science and applied mathematics.

Preliminary program and materials

Lecture 1. Theory and practice of software obfuscation.

Presentation.

This lecture is devoted to the software obfuscation problem, its applications in software protection, information hiding and cryptography, to the practical techniques of program obfuscation with the help of structures and techniques from discrete mathematics, automata theory, coding theory. Much attention is payed to formal definitions of obfuscation security. It wil be shown that a secure obfuscation of some programs is impossible in the framework of a "black box" model, but , nevertheless, certain classes of simple programs admit secure obfuscation.

Lecturer: Vladimir Zakharov.

Lecture 2. Formal correctness proofs for sequential programs.

Presentation.

This lecture is devoted to formal verification of sequential imperative programs. The main topic of the lecture is the notion of formal system (logical calculus) applied to correctness proof for imperative programs. Several valuable properties of such formal systems and proofs for simple-but-useful programs are to be discussed.

Lecturer: Vladislav Podymov

Lectures 3,4. Constraint satisfaction problem: an algebraic approach.

Presentation.

The topic of the lecture is the constraint satisfaction problem (CSP). CSP is a universal formalization for a wide variety of problems in discrete mathematics and computer science. A characterization of the CSP complexity has recently been obtained by an algebraic approach. The aim of the lecture is to describe the algebraic approach to CSP and the results established by its applying.

Lecturer: Svetlana N. Selezneva

Lecture 5. Closed classes in k-valued and partial k-valued logics.

This lecture gives a survey of results on the description of some parts of the lattice of closed classes in partial k-valued logic. The results on cardinality of the set of all closed classes in partial k-valued logic containing given precomplete class of k-valued logic are considered.

Lecture 6. Algebras of minimal multiplicative complexity.

Presentation.

We prove that an associative algebra A has minimal rank if and only if the Alder–Strassen bound is also tight for the multiplicative complexity of A, that is, the multiplicative complexity of A is 2 dimA - t(A) where t(A) denotes the number of maximal twosided ideals of A. This generalizes a result by E. Feig who proved this for division algebras. Furthermore,we show that if A is local or superbasic, then every optimal quadratic computation for A is almost bilinear.

Lecturer: Bekhan Chokaev.

Literature

1. Варновский Н.П., Захаров В.А., Кузюрин Н.П., Шокуров А.В. Современное состояние исследований в области обфускации программ: определения стойкости обфускации // Труды института системного программирования, 2014б т. 26, вып. 3, с. 167-191.
2. Cappaert J. Code Obfuscation Techniques for Software Protection. PhD Thesis, 2012, Leuven University.
3. Apt K.R., Boer F.S., Olderog E.-R. Verification of sequential and concurrent programs. Third, extended edition. Springer, 2010.
4. Kleene S.C. Mathematical logic. 1967.
5. Creignou N., Khanna S., Sudan M. Complexity classifications of Boolean constraint satisfaction problems. 2001.
6. Bulatov A., Jeavons P., Krokhin A. Classifying the complexity of constraints using finite algebras // SIAM J. Comput. 2005. V. 34, N 3. P. 720-742.
7. Peter B¨urgisser, Michael Clausen, and M. Amin Shokrollahi. Algebraic Complexity Theory. Springer, 1997.
8. Yurij A. Drozd and Vladimir V. Kirichenko. Finite Dimensional Algebras. Springer, 1994.
9. A. Alder and V. Strassen. On the algorithmic complexity of associative algebras. Theoret. Comput. Sci., 15:201–211, 1981.
10. Valery B. Alekseyev. On the Complexity of Some Algorithms of Matrix Multiplication. J. Algorithms, 6(1):71-85, 1985.
11. Markus Bl¨aser. Lower bounds for the multiplicative complexity of matrix multiplication. Comput. Complexity, 8:203-226, 1999.
12. Markus Bl¨aser. Lower bounds for the bilinear complexity of associative algebras. Comput. Complexity, 9:73-112, 2000.
13. Markus Bl¨aser. A 2:5n2-lower bound for the multiplicative complexity of n х n-matrix multiplication. In Proc. 18th Int. Symp. on Theoret. Aspects of Comput. Sci. (STACS), Lectures Notes in Comput. Sci. 2010, 99-110, 2001.