Easy cases of the D0L sequence equivalence problem

AbstractTo test the equivalence of two binary D0L sequences it suffices to compare the first four terms of the sequences. We introduce a larger class of D0L systems for which sequence equivalence can be decided by considering the first ten initial terms

Towards a proof of the decidability of the momentary stagnation of the growth function of D0L systems

This is the published version of a work that was accepted for publication in Theoretical Computer Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Theoretical Computer Science, 341, 1–3, (2005) DOI: 10.1016/j.tcs.2005.05.002This paper proves the decidability of several problems in the theory of HD 0L, D0L and PD 0L systems, some of which that have been proved before but are now proved in a different way. First, the paper tackles the decidability of the nilpotency of HD 0L systems and the infinitude of PD 0L languages. Then, we prove the decidability of the problem of momentary stagnation of the growth function of PD 0L systems. Finally, we suggest a way to solve the decidability of the momentary stagnation of the growth function of D0L systems, proving the decidability of the infinitude of HD 0L as a trivial consequence.This paper has been partially sponsored by the UPV/EHU project number 9/upv 00003.230-13707/2001. This paper has been partially sponsored by the Spanish Ministry of Science and Technology (MCYT) project number TIC2002-01948

On images of D0L and DT0L power series

AbstractThe D0L and DT0L power series are generalizations of D0L and DT0L languages. We continue the study of these series by investigating various decidability questions concerning the images of D0L and DT0L power series

Decidability of the HD0L ultimate periodicity problem

In this paper we prove the decidability of the HD0L ultimate periodicity problem

A Characterization of Morphic Words with Polynomial Growth

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words