Libor Polák

Syntactic Semiring of a Language

A classical construction assigns to any language its (ordered) syntactic monoid. Recently the author defined the so-called syntactic semiring of a language. We discuss here the relationships between those two structures. Pins refinement of Eilenberg theorem gives a one-to-one correspondence between positive varieties of rational languages and pseudovarieties of ordered monoids. The authors modification uses so-called conjunctive varieties of rational languages and pseudovarieties of idempotent semirings. We present here also several examples of our varieties of languages.

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.

Syntactic semiring and language equations

A classical construction assigns to any language its (ordered) syntactic monoid. Recently the author defined the so-called syntactic semiring of a language. We show here that elements of the syntactic semiring of <i>L</i> can be identified with transformations of a certain modification of the minimal automaton for <i>L</i>. The main issue here are the inequalities <i>r</i>(<i>x</i><inf>1</inf>, ..., <i>x<inf>m</inf></i>) ⊆ <i>L</i> and equations <i>r</i>(<i>x</i><inf>1</inf>, ..., <i>x<inf>m</inf></i>) = <i>L</i> where <i>L</i> is a given regular language over a finite alphabet <i>A</i> and <i>r</i> is a given regular expression over <i>A</i> in variables <i>x</i><inf>1</inf>, ..., <i>x<inf>m</inf></i>. We show that the search for maximal solutions can be translated into the (finite) syntactic semiring of the language <i>L</i>. In such a way we are able to decide the solvability and to find all maximal solutions effectively. In fact, the last questions were already solved by Conway using his factors. The first advantage of our method is the complexity and the second one is that we calculate in a transparent algebraic structure.

Polynomial Operators on Classes of Regular Languages

We assign to each positive variety

A solution of the word problem for free ∗-regular semigroups

\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

On Pseudovarieties of Semiring Homomorphisms

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

Operators on classes of regular languages

\mathcal V

Hierarchies of piecewise testable 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.