631 research outputs found
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
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
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
Computing exponentially faster: Implementing a nondeterministic universal Turing machine using DNA
The theory of computer science is based around Universal Turing Machines
(UTMs): abstract machines able to execute all possible algorithms. Modern
digital computers are physical embodiments of UTMs. The nondeterministic
polynomial (NP) time complexity class of problems is the most significant in
computer science, and an efficient (i.e. polynomial P) way to solve such
problems would be of profound economic and social importance. By definition
nondeterministic UTMs (NUTMs) solve NP complete problems in P time. However,
NUTMs have previously been believed to be physically impossible to construct.
Thue string rewriting systems are computationally equivalent to UTMs, and are
naturally nondeterministic. Here we describe the physical design for a NUTM
that implements a universal Thue system. The design exploits the ability of DNA
to replicate to execute an exponential number of computational paths in P time.
Each Thue rewriting step is embodied in a DNA edit implemented using a novel
combination of polymerase chain reactions and site-directed mutagenesis. We
demonstrate that this design works using both computational modelling and in
vitro molecular biology experimentation. The current design has limitations,
such as restricted error-correction. However, it opens up the prospect of
engineering NUTM based computers able to outperform all standard computers on
important practical problems
Max Dehn, Axel Thue, and the Undecidable
This is a short essay on the roles of Max Dehn and Axel Thue in the
formulation of the word problem for (semi)groups, and the story of the proofs
showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve
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