A new approach to the periodicity lemma on strings with holes

We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a "wild card", a "don't-care" or an "indeterminate letter" in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [J. Berstel, L. Boasson, Partial words and a theorem of Fine and Wilf, Theoret. Comput. Sci. 218 (1999) 135-141]. We then study the two-hole case, where our result agrees with the one given in [F. Blanchet-Sadri, Robert A. Hegstrom, Partial words and a theorem of Fine and Wilf revisited, Theoret. Comput. Sci. 270 (1-2) (2002) 401-419] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally, we extend our result to three or more holes using elementary methods, and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum-length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from that used by Blanchet-Sadri and Hegstrom, and also simpler

Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time

We show that the equivalence of deterministic linear top-down tree-to-word transducers is decidable in polynomial time. Linear tree-to-word transducers are non-copying but not necessarily order-preserving and can be used to express XML and other document transformations. The result is based on a partial normal form that provides a basic characterization of the languages produced by linear tree-to-word transducers.Comment: short version of this paper will be published in the proceedings of the 20th Conference on Developments in Language Theory (DLT 2016), Montreal, Canad

Linear recurrence sequences and periodicity of multidimensional continued fractions

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi--Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions

Periodic elements in Garside groups

Let $G$ be a Garside group with Garside element $\Delta$, and let $\Delta^m$ be the minimal positive central power of $\Delta$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\Delta^a$; every finite subgroup of the quotient group $G/$ is cyclic. By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an $n$-braid is periodic if and only if it is conjugate to a power of one of two specific roots of $\Delta^2$. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of $\Delta^m$. We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type $A_n$, $B_n$, $D_n$, $I_2(e)$ and the braid group of the complex reflection group of type $(e,e,n)$, endowed with the dual Garside structure, we may further assume the precentrality.Comment: The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this version. 27 page