Emanuele Rodaro

• Professor
• Professor (Associate) at Politecnico di Milano

We present obstruction results for self-similar groups regarding the generation of free groups. As a main consequence of our main results, we solve an open problem posed by Grigorchuk by showing that in an automaton group where a co-accessible state acts as the identity, any self-similar subgroup acting transitively on the first level, or any subgr...

We are interested in the generalised word problem (aka subgroup membership problem) for stabiliser subgroups of groups acting on rooted $d$-regular trees. Stabilisers of infinite rays in the tree are not finitely generated in general, and so the problem is not even well posed unless the infinite ray has a finite description, for example, if the ray...

We define a new class of groups arising from context-free inverse graphs. We provide closure properties, prove that their co-word problems are context-free, study the torsion elements, and realize them as subgroups of the asynchronous rational group. Furthermore, we use a generalized version of the free product of graphs and prove that such a produ...

We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. Using the concept of expandability introduced earlier by the current authors, we show that the la...

We extend the characterization of context‐free groups of Muller and Schupp in two ways. We first show that for a quasi‐transitive inverse graph , being quasi‐isometric to a tree, or context‐free in the sense of Muller–Schupp (finitely many end‐cone up to end‐isomorphism), or having the automorphism group that is virtually free, are all equivalent c...

For each p\geq 1 , the star automaton group \mathcal{G}_{S_p} is an automaton group which can be defined starting from a star graph on p+1 vertices. We study Schreier graphs associated with the action of the group \mathcal{G}_{S_p} on the regular rooted tree T_{p+1} of degree p+1 and on its boundary \partial T_{p+1} . With the transitive action on...

In 1991 the first public key protocol involving automaton groups has been proposed. In this paper we give a survey about algorithmic problems around automaton groups which may have potential applications in cryptography. We then present a new public key protocol based on the conjugacy search problem in some families of automaton groups. At the end...

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $\Gamma$, being context-free is equivalent to having the automorphism group $Aut(\Gamma)$ that is virtually free, which in turn is also equivalent to being quasi-isometric to a tree. As a consequence of thi...

We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area.

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is isomorphic to the partially commutative monoid associated with the complement of the line graph of the defining tree. A...

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal i...

In this paper we prove that a uniformly distributed random circular automaton An of order n synchronizes with high probability (w.h.p.). More precisely, we prove thatP[An synchronizes]=1−O(1n). The main idea of the proof is to translate the synchronization problem into a problem concerning properties of a random matrix; these properties are then es...

For each $p\geq 1$, the star automaton group $\mathcal{G}_{S_p}$ is an automaton group which can be defined starting from a star graph on $p+1$ vertices. We study Schreier graphs associated with the action of the group $\mathcal{G}_{S_p}$ on the regular rooted tree $T_{p+1}$ of degree $p+1$ and on its boundary $\partial T_{p+1}$. With the transitiv...

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 specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups genera...

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 elements and is amenable of exponential growth. We also highlight a connection between our construction and the...

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal i...

We develop the theory of fragile words by introducing the concept of eraser morphism and extending the concept to more general contexts such as (free) inverse monoids. We characterize the image of the eraser morphism in the free group case, and show that it has decidable membership problem. We establish several algorithmic properties of the class o...

We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which ω-words admit infinite orbits: for such a word, every prefix...

We show that an automaton group or semigroup is infinite if and only if it admits an ω-word (i.e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups...

We show that if a semisimple synchronizing automaton with n states has a minimal reachable non-unary subset of cardinality r≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docum...

We study the dynamics of the action of an automaton group on the set of infinite words, and more precisely the discontinuous points of the map which associates to a point its set of stabilizers — the singular points. We show that, for any Mealy automaton, the set of singular points has measure zero. Then we focus our attention on several classes of...

We develop the theory of fragile words by introducing the concept of eraser morphism and extending the concept to more general contexts such as (free) inverse monoids. We characterize the image of the eraser morphism in the free group case, and show that it has decidable membership problem. We establish several algorithmic properties of the class o...

In this paper we prove that a uniformly distributed random circular automaton $\mathcal{A}_n$ of order $n$ synchronizes with high probability (whp). More precisely, we prove that $$ \mathbb{P}\left[\mathcal{A}_n \text{ synchronizes}\right] = 1- O\left(\frac{1}{n}\right). $$ The main idea of the proof is to translate the synchronization problem into...

We study the orbits of right infinite or $\omega$-words under the action of semigroups and groups generated by automata. We see that an automaton group or semigroup is infinite if and only if it admits an $\omega$-word with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization...

We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which {\omega}-words admit infinite orbits: for such a word, every...

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 specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generat...

We address the problem of finding examples of non-bireversible transducers defining free groups, we show examples of transducers with sink accessible from every state which generate free groups, and, in general, we link this problem to the non-existence of certain words with interesting combinatorial and geometrical properties that we call fragile...

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group ge...

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group ge...

In dealing with monoids, the natural notion of kernel of a monoid morphism \(f:M\rightarrow N\) between two monoids M and N is that of the congruence \(\sim _f\) on M defined, for every \(m,m'\in M\), by \(m\sim _fm'\) if \(f(m)=f(m')\). In this paper, we study kernels and equalizers of monoid morphisms in the categorical sense. We consider the cas...

We show that if a semisimple synchronizing automaton with $n$ states has a minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the $2$-packing number for families of $r$-subsets of $[1,n]$.

We study a connection between synchronizing automata and its set $M$ of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of factors does not contain a word of length at most $\frac{1}{4}\min\{|u|: u\in I\}+\frac{1}{16}$ has a reset w...

We study a connection between synchronizing automata and its set $M$ of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of factors does not contain a word of length at most $\frac{1}{4}\min\{|u|: u\in I\}+\frac{1}{16}$ has a reset w...

In this paper, we study the word problem of automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup...

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup...

We introduce the notion of a reset left regular decomposition of an ideal regular language, and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata. We show that every ideal regular language has at least one reset left regular decompo...

The solvability of the word problem for Yamamura’s HNN-extensions [S;A1,A2;φ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[S;A_{1},A_{2};\varphi ]$$\end{document} ha...

We prove that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group. We specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to the existence of a finite Schreier graph in th...

We study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk. We show t...

We present a ring theoretic approach to Černý's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý's series. Semisimplicity giv...

This paper is a survey of some recent results on the word problem for amalgams of inverse semigroups. Some decidability results for special types of amalgams are summarized pointing out where and how the conditions posed on amalgams are used to guarantee the decidability of the word problem. Then a recent result on undecidability is shortly illustr...

We construct a bireversible self-dual automaton with $3$ states over an alphabet with $3$ letters which generates the lamplighter group $\mathbb{Z}_3\wr\mathbb{Z}$.

We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we s...

We follow language theoretic approach to synchronizing automata and Černý’s conjecture initiated in a series of recent papers. We find a precise lower bound for the reset complexity of a principal ideal language. Also we show a strict connection between principal left ideals and synchronizing automata. Actually, it is proved that all strongly conne...

We link the algebraic property of being free to some infiniteness property regarding the dynamics of the (semi)group generated by the enriched dual transducer. We specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to have a finite Schreier graphs in the boundary of the enric...

We approach Černý’s conjecture using the Wedderburn- Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý’s series. Furthermore, semisimplicity gives the advan...

We give a geometric approach to groups defined by automata via the notion of enriched dual of an inverse transducer. Using this geometric correspondence we first provide some finiteness results, then we consider groups generated by the dual of Cayley type of machines. Lastly, we address the problem of the study of the action of these groups in the...

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the cor...

We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential...

Traffic models based on cellular automata have high computational efficiency because of their simplicity in describing unrealistic vehicular behavior and the versatility of cellular automata to be implemented on parallel processing. On the other hand, the other microscopic traffic models such as car-following models are computationally more expensi...

We introduce the notion of reset left regular decomposition of an ideal regular language and we prove that there is a one-to-one correspondence between these decompositions and strongly connected synchronizing automata. We show that each ideal regular language has at least a reset left regular decomposition. As a consequence each ideal regular lang...

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free...

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. Considering the Foata normal form metric on trace monoids and uniformly continuous endomorphisms, a finiteness theorem is proved for the infinite fixed points of the continuous extension to real traces.

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms, if the dependence graph of $G$...

We propose a new stochastic traffic flow model for highways (freeways), which is a hybrid between the classical cellular automata and the other microscopic traffic models, using continuous cellular automata. We combine the computational efficiency of cellular automata models with the accuracy of the microscopic models by introducing continuity in s...

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further the connection between union-closed families and upward-closed families started in Reimer (2003) using rising o...

Let S be the amalgamated free product of two finite inverse semigroups. We prove that the Schützenberger graph of an element of S with respect to a standard presentation of S is a context-free graph in the sense of Müller and Schupp (Theor. Comput. Sci. 37:51–75, 1985), showing that the language L recognized by the Schützenberger automaton is conte...

We investigate the connections between amalgams and Yamamura's HNN-extensions of inverse semigroups. In particular, we prove that amalgams of inverse semigroups with an identity adjoint are quotient semigroups of some special Yamamura's HNN-extensions. As a consequence, we show how to obtain the Schützenberger graph of a word w with respect to the...

We consider five operators on a regular language. Each of them is a tool for constructing a code (respectively prefix, suffix, bifix, infix) and a hypercode out of a given regular language. We give the precise values of the (deterministic) state complexity of these operators: over a constant-size alphabet for the first four of them and over a quadr...

Never minimal automata, introduced in [4], are strongly connected automata which are not minimal for any choice of their final states. In [4] the authors raise the question whether recognizing such automata is a polynomial time task or not. In this paper, we show that the complement of this problem is equivalent to the problem of checking whether o...

A deterministic finite-state automaton A is said to be synchronizing if there is a synchronizing word, i.e. a word that takes all the states of the automaton A to a particular one. We consider synchronizing automata whose language of synchronizing words is finitely generated as a two-sided ideal in Σ*. Answering a question stated in [1], here we pr...

We show that the word problem for an amalgam $[S_1,S_2;U,\omega_1,\omega_2]$ of inverse semigroups may be undecidable even if we assume $S_1$ and $S_2$ (and therefore $U$) to have finite $\mathcal{R}$-classes and $\omega_1,\omega_2$ to be computable functions, interrupting a series of positive decidability results on the subject. This is achieved b...

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG). Using this characterization we prove that any such automaton possesses a synchro...

Let S = S1 *U S2 = Inv〈X; R〉 be the free amalgamated product of the finite inverse semigroups S1, S2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations wL ≡ wR with wL, wR ∈ (X ∪ X-1 ∪ Ξ ∪ Ξ-1)+ such that each x ∈ Ξ labels at most one edge in the Schützenberger automaton of...

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2 whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of fini...

We consider four operators on a regular language. Each of them is a tool for constructing a code (respectively prefix, suffix, bifix and infix) out of a given regular language. We give the precise values of the (deterministic) state complexity of each of these operators.

A synchronizing word w for a given synchronizing DFA is called minimal if no proper prefix or suffix of w is synchronizing. We characterize the class of synchronizing automata having finite language of minimal synchronizing words (such automata are called finitely generated). Using this characterization we prove that any such automaton possesses a...

We consider four operators on a regular language. Each of them is a tool for constructing a code (respectively prefix, suffix, bifix and infix) out of a given regular language. We give the precise values of the (deterministic) state complexity of each of these operators.

It is known that the satisfiability problem for equations over free partially commutative monoids is decidable but computationally hard. In this paper we consider the satisfiability problem for equations over free partially commutative monoids under the constraint that the solution is a subset of the alphabet. We prove that this problem is NP-compl...

We prove that the word problem is decidable in Yamamura’s HNN extensions of finite inverse semigroups, by providing an iterative construction of approximate automata of the Schützenberger automata of words relative to the standard presentation of Yamamura’s HNN-extensions.

A given set F of n × n matrices is said to be mortal if the n × n null matrix belongs to the free semigroup generated by F. It is known that the mortality problem for 3×3 matrices with integer entries is undecidable [7],[3]. In this paper we prove that the mortality problem is decidable for any set of 2 × 2 integer matrices whose determinants assum...

Let S be a finite semigroup and A be a generating set of S; let’s associate with A the natural number Δ(A)=min{m:A m =S}. A. Cherubini, J. M. Howie and B. Piochi [Commun. Algebra 32, No. 7, 2783-2801 (2004; Zbl 1071.20050)] defined the status Stat(S) of a finite semigroup S as the minimum of the products |A|Δ(A): 〈A〉=S. In the present paper some bo...