Michal Kunc

The Power of Commuting with Finite Sets of Words

We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conways conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities.

What do we know about language equations

In the talk we give an overview of recent developments in the area of language equations, with an emphasis on methods for dealing with non-classical types of equations whose theory has not been successfully developed already in the previous decades, and on results forming the current borderline of our knowledge. This abstract is in particular meant to provide the interested listener with references to the material discussed in the talk.

Equational description of pseudovarieties of homomorphisms

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in AC0 is given.

The power of commuting with finite sets of words

We show that one can construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conways conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities.

Regular solutions of language inequalities and well quasi-orders

By means of constructing suitable well quasi-orders of free monoids we prove that all maximal solutions of certain systems of language inequalities are regular. This way we deal with a wide class of systems of inequalities where all constants are languages recognized by finite simple semigroups. In a similar manner we also demonstrate that the largest solution of the inequality XK ⊆ LX is regular provided the language L is regular.

Describing periodicity in two-way deterministic finite automata using transformation semigroups

A framework for the study of periodic behaviour of two-way deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x+. The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n+1 states, and transforming it to a one-way automaton requires exactly max0≤l≤n G(n - l) + l + 1 states, where G(k) is the maximum order of a permutation of k elements.

On Decidability of Intermediate Levels of Concatenation Hierarchies

It is proved that if definability of regular languages in the \(\Sigma _n\) fragment of the first-order logic on finite words is decidable, then it is decidable also for the \(\Delta _{n+1}\) fragment. In particular, the decidability for \(\Delta _5\) is obtained. More generally, for every concatenation hierarchy of regular languages, it is proved that decidability of one of its half levels implies decidability of the intersection of the following half level with its complement.

On language inequalities XK ⊆ LX

It is known that for a regular language L and an arbitrary language K the largest solution of the inequality XK⊆LX is regular. Here we show that there exist finite languages K and P and star-free languages L, M and R such that the largest solutions of the systems

Computing by commuting

\{XK\subseteq LX,\ X\subseteq M\}

State Complexity of Union and Intersection for Two-way Nondeterministic Finite Automata

and