Related papers: Automata generating free products of groups of ord…
Classification of groups generated by 3-state automata over a 2-letter alphabet started in the first paper (see http://www.arxiv.org/abs/math/0612178) is continued.
We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we…
In this paper, we construct countably many isolated circular orders on the free products $G = F_{2n} \ast \mathbb{Z}_{m_1} \ast \cdots \ast \mathbb{Z}_{m_k}$ of cyclic groups. Moreover, we prove that these isolated circular orders are not…
We present logically based methods for constructing XP and FPT graph algorithms, parametrized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make possible to check properties that are not…
It is examined finite state automorphisms of regular rooted trees constructed to represent groups $GL(n,\mathbb{Z})$. The number of states of automorphisms that correspond to elementary matrices is computed. Using the representation of…
We prove, for various important classes of Mealy automata, that almost all generated groups have an element of infinite order. In certain cases, it also implies other results such as exponential growth.
For every natural number $n$, we classify abelian groups generated by an $n$-state time-varying automaton over the binary alphabet, as well as by an $n$-state Mealy automaton over the binary alphabet.
We study trees where each successor set is equipped with some additional structure. We introduce a family of automaton models for such trees and prove their equivalence to certain fixed-point logics. As a consequence we obtain…
Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n^2/4+n/2-1.
In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two…
We give a new example of an automata group of intermediate growth. It is generated by an automaton with 4 states on an alphabet with 8 letters. This automata group has exponential activity and its limit space is not simply connected.
The basis for most of the ideas mentioned in this paper is the theory of cellular automata. A cellular automata contains a regular grid of cells, with each cell having a pre-defined set of finite states. The initial state is determined at…
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit…
In this paper, we describe a new method for constructing minimal, deterministic, acyclic finite-state automata from a set of strings. Traditional methods consist of two phases: the first to construct a trie, the second one to minimize it.…
Algorithms for (nondeterministic) finite-state tree automata (FTAs) are often tested on random FTAs, in which all internal transitions are equiprobable. The run-time results obtained in this manner are usually overly optimistic as most such…
A finitely generated group is said to be an automata group if it admits a faithful self-similar finite-state representation on some regular $m$-tree. We prove that if $G$ is a subgroup of an automata group, then for each finitely generated…
For any nontrivial abelian group $\mathbb{X}$ we construct a reversible (bireversible in case the order of $\mathbb{X}$ is odd) automaton such that its set of states and alphabet are identified with $\mathbb{X}$, transition and output…
The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.
A cellular automaton with $n$ states may be used for construction of reversible second-order cellular automaton with $n^2$ states. Reversible cellular automata with hidden parameters discussed in this paper are generalization of such…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be…