Cyclic rewriting and conjugacy problems

Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.Comment: 37 pages, 1 figure, submitted. Changes to introductio

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.Comment: 16 + vii page

Thue's 1914 paper: a translation

This paper includes notes to accompany a reading of Thue's 1914 paper "Probleme uber Veranderungen von Zeichenreihen nach gegebenen Reglen", along with a translation of that paper. Thue's 1914 paper is mainly famous for proving an early example of an undecidable problem, cited prominently by Post. However, Post's paper principally makes use of the definition of Thue systems, described on the first two pages of Thue's paper, and does not depend on the more specific results in the remainder of Thue's paper. A closer study of the remaining parts of that paper highlight a number of important themes in the history of computing: the transition from algebra to formal language theory, the analysis of the "computational power" (in a pre-1936 sense) of rules, and the development of algorithms to generate rule-sets

The monoid of queue actions

We investigate the monoid of transformations that are induced by sequences of writing to and reading from a queue storage. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties, e.g., conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is NL-complete. Furthermore, we present an algebraic characterization of this monoid's recognizable subsets. Finally, we prove that it is not Thurston-automatic

Geodesic rewriting systems and pregroups

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well-behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We describe a Knuth-Bendix completion process to construct such systems, show how they may be found with the help of Stallings' pregroups and conversely may be used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory, Dortmund and Carleton Conferences". Series: Trends in Mathematics. Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009, Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause

On One-Rule Grid Semi-Thue Systems

International audienceThe family of one-rule grid semi-Thue systems, introduced by Alfons Geser, is the family of one-rule semi-Thue systems such that there exists a letter c that occurs as often in the left-hand side as the right-hand side of the rewriting rule. We prove that for any one-rule grid semi-Thue system S, the set S(w) of all words obtainable from w using repeatedly the rewriting rule of S is a constructible context-free language. We also prove the regularity of the set Loop(S) of all words that start a loop in a one-rule grid semi-Thue systems S.La famille des systèmes de semi-Thue à une seule règle "en grille", introduite par Alfons Geser, est la famille des systèmes de réécriture de mots pour lesquels il existe une lettre apparaissant autant de fois dans la partie gauche et dans la partie droite de leur unique règle. Nous prouvons que, pour tout système S de cette famille, l'ensemble S(w) des mots obtenus à partir du mot w en appliquant itérativement la règle de réécriture de S est un langage algébrique constructible. Nous prouvons également que l'ensemble Loop(S) des mots qui sont à l'origine d'une boucle de réécriture pour un systèmes de semi-Thue à une seule règle "en grille" S est un langage régulier

The complexity of the word problems for commutative semigroups and polynomial ideals

AbstractAny decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied. This bound is achieved by a simple procedure for the semigroup problem