On Bounded Linear Codes and the Commutative Equivalence

The problem of the commutative equivalence of semigroups generated by semi-linear languages is studied. In particular conditions ensuring that the Kleene closure of a bounded semi-linear code is commutatively equivalent to a regular language are investigated

On the Commutative Equivalence of Algebraic Formal Series and Languages

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

Universal Lyndon Words

A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon words}, which are words over an $n$-letter alphabet that have length $n!$ and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every $n$ and exhibit combinatorial and structural properties of these words. We then define particular prefix codes, which we call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in bijection with the set of the shortest unrepeated prefixes of the conjugates of a universal Lyndon word. This allows us to give an algorithm for constructing all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201

Harmonic and gold Sturmian words

AbstractIn the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p+q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand=A∪PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p<q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1), where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold∩Harm contain continuously many elements. Finally, some conjectures are formulated

Relationships Between Bounded Languages, Counter Machines, Finite-Index Grammars, Ambiguity, and Commutative Equivalence

It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (DCM). This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index — such as finite-index matrix grammars (Mfin) and finite-index ET0L (ET0Lfin) — have identical bounded languages as DCM. Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or com- mutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous Mfin has a rational char- acteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that the grammar systems Mfin and ET0Lfin can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on Mfin and ET0Lfin languages implying commutative regularity are obtained. In particular, it is shown that every finite-index ED0L language is commutatively regular

Codes of central Sturmian words

AbstractA central Sturmian word, or simply central word, is a word having two coprime periods p and q and length equal to p+q-2. We consider sets of central words which are codes. Some general properties of central codes are shown. In particular, we prove that a non-trivial maximal central code is infinite. Moreover, it is not maximal as a code. A central code is called prefix central code if it is a prefix code. We prove that a central code is a prefix (resp., maximal prefix) central code if and only if the set of its ‘generating words’ is a prefix (resp., maximal prefix) code. A suitable arithmetization of the theory is obtained by considering the bijection θ, called ratio of periods, from the set of all central words to the set of all positive irreducible fractions defined as: θ(ε)=1/1 and θ(w)=p/q (resp., θ(w)=q/p) if w begins with the letter a (resp., letter b), p is the minimal period of w, and q=|w|-p+2. We prove that a central code X is prefix (resp., maximal prefix) if and only if θ(X) is an independent (resp., independent and full) set of fractions. Finally, two interesting classes of prefix central codes are considered. One is the class of Farey codes which are naturally associated with the Farey series; we prove that Farey codes are maximal prefix central codes. The other is given by uniform central codes. A noteworthy property related to the number of occurrences of the letter a in the words of a maximal uniform central code is proved

Reduction et synchronisation d'automates non ambigus

SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Independent sets of words and the synchronization problem

The synchronization problem is investigated for the class of locally strongly transitive automata introduced in Carpi and D'Alessandro (2009) [9]. Some extensions of this problem related to the notions of stable set and word of minimal rank of an automaton are studied. An application to synchronizing colorings of aperiodic graphs with a Hamiltonian path is also considered. (C) 2012 Elsevier Inc. All rights reserved

The synchronization problem for strongly transitive automata

The synchronization problem is investigated for a new class of deterministic automata called strongly transitive. An extension to unambiguous automata is also considered. © 2008 Springer-Verlag Berlin Heidelberg