Ondřej Klíma

Alternative Automata Characterization of Piecewise Testable Languages

We present a transparent condition on a minimal automaton which is equivalent to piecewise testability of the corresponding regular language. The condition simplifies the original Simon’s condition on the minimal automaton in a different way than conditions of Stern and Trahtman. Secondly, we prove that every piecewise testable language L is k-piecewise testable for k equal to the depth of the minimal DFA of L. This result improves all previously known estimates of such k.

Piecewise testable languages via combinatorics on words

A regular language L over an alphabet A is called piecewise testable if it is a finite Boolean combination of languages of the form A ? a 1 A ? a 2 A ? ? A ? a ? A ? , where a 1 , ? , a ? ? A , ? ? 0 . An effective characterization of piecewise testable languages was given in 1972 by Simon who proved that a language L is piecewise testable if and only if its syntactic monoid is J -trivial. Nowadays, there exist several proofs of this result based on various methods from algebraic theory of regular languages. Our contribution adds a new purely combinatorial proof. Highlights? Simons effective characterization of piecewise testable languages is studied. ? A new purely combinatorial proof of Simons theorem is presented. ? The original estimate of Simon is slightly improved.

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.

Polynomial Operators on Classes of Regular Languages

We assign to each positive variety

Descriptional complexity of biautomata

\mathcal V

Biautomata for k -piecewise testable languages

and each natural number k the class of all (positive) Boolean combinations of the restricted polynomials, i.e. the languages of the form

Hierarchies of Piecewise Testable Languages

L_0a_1 L_1a_2\dots a_\ell L_\ell, \text{ where } \ell\leq k

Systems of equations over finite semigroups and the #CSP dichotomy conjecture

, a 1 ,...,a *** are letters and L 0 ,...,L *** are languages from the variety

Hierarchies of piecewise testable languages

\mathcal V

On varieties of literally idempotent languages

. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of languages considered. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions to particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies. 2000 Classification: 68Q45 Formal languages and automata.