105 research outputs found
Complexity and growth for polygonal billiards
We establish a relationship between the word complexity and the number of
generalized diagonals for a polygonal billiard. We conclude that in the
rational case the complexity function has cubic upper and lower bounds. In the
tiling case the complexity has cubic asymptotic growth.Comment: 12 pages, 4 figure
Nivat's conjecture holds for sums of two periodic configurations
Nivat's conjecture is a long-standing open combinatorial problem. It concerns
two-dimensional configurations, that is, maps where is a finite set of symbols. Such configurations are often
understood as colorings of a two-dimensional square grid. Let denote
the number of distinct block patterns occurring in a configuration
. Configurations satisfying for some
are said to have low rectangular complexity. Nivat conjectured that such
configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can
be decomposed into a sum of periodic configurations. In this paper we show that
if there are at most two components, Nivat's conjecture holds. As a corollary
we obtain an alternative proof of a result of Cyr and Kra: If there exist such that , then is periodic. The
technique used in this paper combines the algebraic approach of Kari and the
author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with
proofs. 12 pages + references + appendi
On growth and fluctuation of k-abelian complexity
An extension of abelian complexity, so called k-abelian complexity, has been considered recently in a number of articles. This paper considers two particular aspects of this extension: First, how much the complexity can increase when moving from a level k to the next one. Second, how much the complexity of a given word can fluctuate. For both questions we give optimal solutions. (C) 2017 Elsevier Ltd. All rights reserved
The complexity of tangent words
In a previous paper, we described the set of words that appear in the coding
of smooth (resp. analytic) curves at arbitrary small scale. The aim of this
paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
Garside and quadratic normalisation: a survey
Starting from the seminal example of the greedy normal norm in braid monoids,
we analyse the mechanism of the normal form in a Garside monoid and explain how
it extends to the more general framework of Garside families. Extending the
viewpoint even more, we then consider general quadratic normalisation
procedures and characterise Garside normalisation among them.Comment: 30 page
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
A circular order on edge-coloured trees and RNA m-diagrams
We study a circular order on labelled, m-edge-coloured trees with k vertices,
and show that the set of such trees with a fixed circular order is in bijection
with the set of RNA m-diagrams of degree k, combinatorial objects which can be
regarded as RNA secondary structures of a certain kind. We enumerate these sets
and show that the set of trees with a fixed circular order can be characterized
as an equivalence class for the transitive closure of an operation which, in
the case m=3, arises as an induction in the context of interval exchange
transformations.Comment: 15 pages, 7 figures. New title. Shortened version, presenting the
results more efficientl
Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages
There has been much research into freeness properties of
finitely generated matrix semigroups under various constraints, mainly
related to the dimensions of the generator matrices and the semiring over
which the matrices are defined. A recent paper has also investigated freeness
properties of matrices within a bounded language of matrices, which
are of the form M1M2 · · · Mk ⊆ F
n×n
for some semiring F [9]. Most freeness
problems have been shown to be undecidable starting from dimension
three, even for upper-triangular matrices over the natural numbers.
There are many open problems still remaining in dimension two.
We introduce a notion of freeness and ambiguity for scalar reachability
problems in matrix semigroups and bounded languages of matrices.
Scalar reachability concerns the set {ρ
TMτ |M ∈ S}, where ρ, τ ∈ F
n
are vectors and S is a finitely generated matrix semigroup. Ambiguity
and freeness problems are defined in terms of uniqueness of factorizations
leading to each scalar. We show various undecidability results
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