Related papers: Between primitive and $2$-transitive: Synchronizat…
A quasi-automatic semigroup is a finitely generated semigroup with a rational set of representatives such that the graph of right multiplication by any generator is a rational relation. A asynchronously automatic semigroup is a…
In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior of the system w.r.t. variety of word transformations performed by the system:…
This paper considers the existence of short synchronizing words in deterministic finite automata (DFAs). We define two general strategies for generating synchronizing words, and we show that each of these strategies can be applied if and…
We survey results in the literature that establish the \v{C}ern\'y conjecture for various classes of finite automata. We also list classes for which the conjecture remains open, but a quadratic (in the number of states) upper bound on the…
The synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state - the synchronized state. Confirming the scenario drawn in 1+1…
Transformation Synchronization is the problem of recovering absolute transformations from a given set of pairwise relative motions. Despite its usefulness, the problem remains challenging due to the influences from noisy and outlier…
We present a new classification of elementary cellular automata. It is based on the structure of the network of states, connected with the transitions between them; the latter are determined by the automaton rule. Recently an algorithm has…
We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of…
An automata network is a finite graph where each node holds a state from some finite alphabet and is equipped with an update function that changes its state according to the configuration of neighboring states. More concisely, it is given…
A word $w$ is called synchronizing (recurrent, reset, magic, directable) word of deterministic finite automaton (DFA) if $w$ sends all states of the automaton to a unique state. In 1964 Jan \v{C}erny found a sequence of n-state complete DFA…
Fast matching of regular expressions with bounded repetition, aka counting, such as (ab){50,100}, i.e., matching linear in the length of the text and independent of the repetition bounds, has been an open problem for at least two decades.…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
A goal of this paper is to introduce the new construction of an automaton with shortest synchronizing word of length $O(d^{\frac{n}{d}})$, where $d \in \mathbb{N}$ and $n$ is the number of states for that automaton. Additionally we…
Let $X$ be a finite set such that $|X|=n$. Let $\trans$ and $\sym$ denote respectively the transformation monoid and the symmetric group on $n$ points. Given $a\in \trans\setminus \sym$, we say that a group $G\leq \sym$ is $a$-normalizing…
The classical subset construction for non-deterministic automata can be generalized to other side-effects captured by a monad. The key insight is that both the state space of the determinized automaton and its semantics---languages over an…
We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the…
Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph…
Synchronisation and pattern formation have been intensely addressed for systems evolving on static networks. Extending the study to include the inherent ability of the network to adjust over time proved cumbersome and led to conclusions…
This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. A corresponding random dynamical system is formulated in a two-step procedure, at…
Minimizing finite automata, proving trace equivalence of labelled transition systems or representing sofic subshifts involve very similar arguments, which suggests the possibility of a unified formalism. We propose finite states…