48 research outputs found
Knuth-Bendix algorithm and the conjugacy problems in monoids
We present an algorithmic approach to the conjugacy problems in monoids,
using rewriting systems. We extend the classical theory of rewriting developed
by Knuth and Bendix to a rewriting that takes into account the cyclic
conjugates.Comment: This is a new version of the paper 'The conjugacy problems in monoids
and semigroups'. This version will appear in the journal 'Semigroup forum
Reconstruction of Binary Image Using Techniques of Discrete Tomography
Discrete tomography deals with the reconstruction of images, in particular binary images, from their projections. A number of binary image reconstruction methods have been considered in the literature, using different projection models or additional constraints. Here, we will consider reconstruction of a binary image with some prescribed numerical information on the rows of the binary image treated as a binary matrix of 0's and 1's. The problem involves information, referred to as row projection, on the number of 1's and the number of subword 01's in the rows of the binary image to be constructed. The algorithm proposed constructs one among the many binary images having the same numerical information on the number of 1's and the number of subword 01. This proposed algorithm will also construct the image uniquely for a special kind of a binary image with its rows in some specific form
Word Equations and Related Topics. Independence, Decidability and Characterizations
The three main topics of this work are independent systems and chains of
word equations, parametric solutions of word equations on three unknowns,
and unique decipherability in the monoid of regular languages.
The most important result about independent systems is a new method
giving an upper bound for their sizes in the case of three unknowns. The
bound depends on the length of the shortest equation. This result has
generalizations for decreasing chains and for more than three unknowns.
The method also leads to shorter proofs and generalizations of some old
results.
Hmelevksii’s theorem states that every word equation on three unknowns
has a parametric solution. We give a significantly simplified proof for this
theorem. As a new result we estimate the lengths of parametric solutions
and get a bound for the length of the minimal nontrivial solution and for
the complexity of deciding whether such a solution exists.
The unique decipherability problem asks whether given elements of some
monoid form a code, that is, whether they satisfy a nontrivial equation. We
give characterizations for when a collection of unary regular languages is a
code. We also prove that it is undecidable whether a collection of binary
regular languages is a code.Siirretty Doriast
Extended-cycle integrals of modular functions for badly approximable numbers
Cycle integrals of modular functions are expected to play a role in real
quadratic analogue of singular moduli. In this paper, we extend the definition
of cycle integrals of modular functions from real quadratic numbers to badly
approximable numbers. We also give explicit representations of values of
extended-cycle integrals for some cases.Comment: 19 pages, 4 figure