We associate with a word $w$ on a finite alphabet $A$ an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of $w$. Then when $\left|A\right|=2$ we deduce, using the sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties...

In this article, a new class of the epoch-incremental reinforcement learning algorithm is proposed. In the incremental mode, the fundamental TD(0) or TD(λ) algorithm is performed and an environment model is created. In the epoch mode, on the basis of the environment model, the distances of past-active states to the terminal state are computed. These distances and the reinforcement terminal state signal are used to improve the agent policy.

We consider shifted equality sets of the form $E_G(a,g_1,g_2)=\{w\mid g_1\left(w\right)=a{g}_2\left(w\right)\}$, where $g_1$ and $g_2$ are nonerasing morphisms and $a$ is a letter. We are interested in the family consisting of the languages $h\left(E_G\left(J\right)\right)$, where $h$ is a coding and $E_G\left(J\right)$ is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language $L\subseteq A^{*}$ is a projection of a shifted equality set, that is, $L={\pi }_A\left(E_G(a,g_1,g_2)\right)$ for some (nonerasing) morphisms $g_1$ and $g_2$ and a letter $a$, where ${\pi }_A$ deletes the letters not in $A$. Then we deduce...

We consider shifted equality sets of the form EG(a,g1,g2) = {ω | g1(ω) = ag2(ω)}, where g1 and g2 are nonerasing morphisms and a is a letter. We are interested in the family consisting of the languages h(EG(J)), where h is a coding and (EG(J)) is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language L ⊆ A* is a projection of a shifted equality set, that is, L = πA(EG(a,g1,g2)) for some (nonerasing) morphisms g1...

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities....

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As...

It is well-known that some of the most basic properties of words, like the commutativity ($xy=yx$) and the conjugacy ($xz=zy$), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation $x^m{y}^n=z^p$ has only periodic solutions in a free monoid, that is, if $x^m{y}^n=z^p$ holds with integers $m,n,p\ge 2$, then there exists a word $w$ such that $x,y,z$ are powers of $w$. This result, which received a lot of attention, was first proved by Lyndon and...

It is well-known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation xMyN = zP has only periodic solutions in a free monoid, that is, if xMyN = zP holds with integers m,n,p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot...

We generalize Jiroušek’s (right) composition operator in such a way that it can be applied to distribution functions with values in a “semifield“, and introduce (parenthesized) compositional expressions, which in some sense generalize Jiroušek’s “generating sequences” of compositional models. We say that two compositional expressions are equivalent if their evaluations always produce the same results whenever they are defined. Our first result is that a set system $\mathscr{H}$ is star-like with centre $X$ if...