arXiv subjects

Igor Rystsov

Publications and source records attributed to Igor Rystsov.

4 recordsLinked to original sources

arXiv2026

Shortest Synchronizing Words on Product Automata

We consider some operations on finite automata graphs, in particular various product, union and merge operations, and investigate how they relate to the parameter~$\mathrm{rt}$, the \emph{length of a shortest synchronizing word}. We derive some lower and upper bounds for the operations that we study. We also examine several classes of automata (for instance, partially ordered, commutative or idempotent automata) that are preserved under these operations and show in which way the parameter $\mathrm{rt}$ behaves in these automata classes. Also, we consider automata classes that can be constructed by combining automata from better known classes; in particular, these new classes of automata will also satisfy the bound of the famous Černý conjecture. This also motivates us to study even more closure properties. In particular, we show that certain automata classes related to synchronization form pseudo-varieties. We also commence a similar study concerning the length of shortest homing sequences of Mealy automata.

cs.FL↗

arXiv2025

Reset thresholds of transformation monoids

Motivated by the Černý conjecture for automata, we introduce the concept of monoidal automata, which allows the formulation of the Černý conjecture for monoids. We show upper bounds on the reset threshold of monoids with certain properties. In particular, we obtain a quadratic upper bound if the transformation monoid contains a primitive group of permutations and a singular of maximal rank with only one point of contraction.

cs.FL↗

arXiv2025

Cerny type automata and rank conjecture

The aim of this paper is to prove the Černý conjecture and the rank conjecture for Černý type automata and monoids. A transformation monoid is said to be Černý type if it is generated by a simple idempotent and a regular group of permutations. We prove Černý conjecture for the Černý type synchronizing automata and the rank conjecture for the Černý type transformation monoids. In particular, we obtain the tight bound for the reset threshold of Černý type synchronizing monoids.

cs.FL↗

arXiv2023

Primitive Automata that are Synchronizing

A deterministic finite (semi)automaton is primitive if its transition monoid (semigroup) acting on the set of states has no non-trivial congruences. It is synchronizing if it contains a constant map (transformation). In analogy to synchronizing groups, we study the possibility of characterizing automata that are synchronizing if primitive. We prove that the implication holds for several classes of automata. In particular, we show it for automata whose every letter induce either a permutation or a semiconstant transformation (an idempotent with one point of contraction) unless all letters are of the first type. We propose and discuss two conjectures about possible more general characterizations.

cs.FL↗