Andrzej Kisielewicz

On the number of operations in a clone

A clone C on a set A is a set of operations on A containing the projection operations and closed under composition. A combinatorial invariant of a clone is its pn-sequence (po0(C), PI (C), . . . ) , where p,(C) is the number of essentially n-ary operations in C. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the pn-sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for.a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded pn-sequence and that it is decidable if a clone is totally symmetric.

Rainbow Induced Subgraphs in Proper Vertex Colorings

Given a graph H we define ρ(H) to be the minimum order of a graph G such that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H. We give upper and lower bounds for ρ(H), compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of ρ(H) for paths. A part of this research has been guided by a computer search and, accordingly, some computational results are presented. A special motivation comes from research in on-line coloring.

Varieties of commutative semigroups

In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.

Computing the shortest reset words of synchronizing automata

In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to

Generating small automata and the Černý conjecture

n=350

COMPLEXITY OF SEMIGROUP IDENTITY CHECKING

n=350 states and up to

On the problem of freeness of multiplicative matrix semigroups

k=10