333 research outputs found
On Commutation and Conjugacy of Rational Languages and the Fixed Point Method
The research on language equations has been active during last decades. Compared to the equations on words the equations on languages are much more difficult to solve. Even very simple equations that are easy to solve for words can be very hard for languages. In this thesis we study two of such equations, namely commutation and conjugacy equations. We study these equations on some limited special cases and compare some of these results to the solutions of corresponding equations on words. For both equations we study the maximal solutions, the centralizer and the conjugator. We present a fixed point method that we can use to search these maximal solutions and analyze the reasons why this method is not successful for all languages. We give also several examples to illustrate the behaviour of this method.Siirretty Doriast
On the simplest centralizer of a language
Given a finite alphabet Σ and a language
L ⊆ ∑+,
the centralizer of L is defined as the maximal language commuting with it.
We prove that if the primitive root of the smallest word of L (with respect to a lexicographic order) is prefix distinguishable in L then the centralizer of L
is as simple as possible, that is, the submonoid
L*.
This lets us obtain a simple proof of a known result concerning the centralizer of nonperiodic three-word languages
Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes
In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis.
The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares.
From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations.
The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast
Qudit Colour Codes and Gauge Colour Codes in All Spatial Dimensions
Two-level quantum systems, qubits, are not the only basis for quantum
computation. Advantages exist in using qudits, d-level quantum systems, as the
basic carrier of quantum information. We show that color codes, a class of
topological quantum codes with remarkable transversality properties, can be
generalized to the qudit paradigm. In recent developments it was found that in
three spatial dimensions a qubit color code can support a transversal
non-Clifford gate, and that in higher spatial dimensions additional
non-Clifford gates can be found, saturating Bravyi and K\"onig's bound [Phys.
Rev. Lett. 110, 170503 (2013)]. Furthermore, by using gauge fixing techniques,
an effective set of Clifford gates can be achieved, removing the need for state
distillation. We show that the qudit color code can support the qudit analogues
of these gates, and show that in higher spatial dimensions a color code can
support a phase gate from higher levels of the Clifford hierarchy which can be
proven to saturate Bravyi and K\"onig's bound in all but a finite number of
special cases. The methodology used is a generalisation of Bravyi and Haah's
method of triorthogonal matrices [Phys. Rev. A 86 052329 (2012)], which may be
of independent interest. For completeness, we show explicitly that the qudit
color codes generalize to gauge color codes, and share the many of the
favorable properties of their qubit counterparts.Comment: Authors' final cop
Playing with Conway’s problem
AbstractThe centralizer of a language is the maximal language commuting with it. The question, raised by Conway in [J.H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971], whether the centralizer of a rational language is always rational, recently received a lot of attention. In Kunc [M. Kunc, The power of commuting with finite sets of words, in: Proc. of STACS 2005, in: LNCS, vol. 3404, Springer, 2005, pp. 569–580], a strong negative answer to this problem was given by showing that even complete co-recursively enumerable centralizers exist for finite languages. Using a combinatorial game approach, we give here an incremental construction of rational languages embedding any recursive computation in their centralizers
The Small-Is-Very-Small Principle
The central result of this paper is the small-is-very-small principle for
restricted sequential theories. The principle says roughly that whenever the
given theory shows that a property has a small witness, i.e. a witness in every
definable cut, then it shows that the property has a very small witness: i.e. a
witness below a given standard number.
We draw various consequences from the central result. For example (in rough
formulations): (i) Every restricted, recursively enumerable sequential theory
has a finitely axiomatized extension that is conservative w.r.t. formulas of
complexity . (ii) Every sequential model has, for any , an extension
that is elementary for formulas of complexity , in which the
intersection of all definable cuts is the natural numbers. (iii) We have
reflection for -sentences with sufficiently small witness in any
consistent restricted theory . (iv) Suppose is recursively enumerable
and sequential. Suppose further that every recursively enumerable and
sequential that locally inteprets , globally interprets . Then,
is mutually globally interpretable with a finitely axiomatized sequential
theory.
The paper contains some careful groundwork developing partial satisfaction
predicates in sequential theories for the complexity measure depth of
quantifier alternations
Low-complexity quantum codes designed via codeword-stabilized framework
We consider design of the quantum stabilizer codes via a two-step,
low-complexity approach based on the framework of codeword-stabilized (CWS)
codes. In this framework, each quantum CWS code can be specified by a graph and
a binary code. For codes that can be obtained from a given graph, we give
several upper bounds on the distance of a generic (additive or non-additive)
CWS code, and the lower Gilbert-Varshamov bound for the existence of additive
CWS codes. We also consider additive cyclic CWS codes and show that these codes
correspond to a previously unexplored class of single-generator cyclic
stabilizer codes. We present several families of simple stabilizer codes with
relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl
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