75,359 research outputs found
On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words
The domain of Combinatorics on Words, first introduced by Axel Thue in 1906, covers by now many subdomains. In this work we are investigating scattered factors as a representation of non-complete information and two measurements for words, namely the locality of a word and prefix normality, which have applications in pattern matching. In the first part of the thesis we investigate scattered factors: A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u1, u2, . . . , un, and v0,v1,...,vn such that u = u1u2 Ì Ì Ìun and w = v0u1v1u2v2 Ì Ì Ìunvn. First, we consider the set of length-k scattered factors of a given word w, called the k-spectrum of w and denoted by ScatFactk(w). We prove a series of properties of the sets ScatFactk(w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two- letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c occurrences more than the other. In particular, we consider the question which cardinalities n = | ScatFactk (w)| are obtainable, for a positive integer k, when w is either a weakly-0- balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k Ì c. Second, we investigate k-spectra that contain all possible words of length k, i.e., k-spectra of so called k-universal words. We present an algorithm deciding whether the k-spectra for given k of two words are equal or not, running in optimal time. Moreover, we present several results regarding k-universal words and extend this notion to circular universality that helps in investigating how the universality of repetitions of a given word can be determined. We conclude the part about scattered factors with results on the reconstruction problem of words from scattered factors that asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w P {a, b} Ì can be reconstructed from the number of occurrences of at most min(|w|a, |w|b) + 1 scattered factors of the form aib, where |w|a is the number of occurrences of the letter a in w. Moreover, we generalise the result to alphabets of the form {1, . . . , q} by showing that at most âq Ì1 |w|i (q Ì i + 1) scattered factors suffices to reconstruct w. Both results i=1 improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here. In the second part we consider patterns, i.e., words consisting of not only letters but also variables, and in particular their locality. A pattern is called k-local if on marking the pattern in a given order never more than k marked blocks occur. We start with the proof that determining the minimal k for a given pattern such that the pattern is k-local is NP- complete. Afterwards we present results on the behaviour of the locality of repetitions and palindromes. We end this part with the proof that the matching problem becomes also NP-hard if we do not consider a regular pattern - for which the matching problem is efficiently solvable - but repetitions of regular patterns. In the last part we investigate prefix normal words which are binary words in which each prefix has at least the same number of 1s as any factor of the same length. First introduced in 2011 by Fici and LiptĂĄk, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems
Undecidable properties of self-affine sets and multi-tape automata
We study the decidability of the topological properties of some objects
coming from fractal geometry. We prove that having empty interior is
undecidable for the sets defined by two-dimensional graph-directed iterated
function systems. These results are obtained by studying a particular class of
self-affine sets associated with multi-tape automata. We first establish the
undecidability of some language-theoretical properties of such automata, which
then translate into undecidability results about their associated self-affine
sets.Comment: 10 pages, v2 includes some corrections to match the published versio
Bubble-Flip---A New Generation Algorithm for Prefix Normal Words
We present a new recursive generation algorithm for prefix normal words.
These are binary strings with the property that no substring has more 1s than
the prefix of the same length. The new algorithm uses two operations on binary
strings, which exploit certain properties of prefix normal words in a smart
way. We introduce infinite prefix normal words and show that one of the
operations used by the algorithm, if applied repeatedly to extend the string,
produces an ultimately periodic infinite word, which is prefix normal.
Moreover, based on the original finite word, we can predict both the length and
the density of an ultimate period of this infinite word.Comment: 30 pages, 3 figures, accepted in Theoret. Comp. Sc.. This is the
journal version of the paper with the same title at LATA 2018 (12th
International Conference on Language and Automata Theory and Applications,
Tel Aviv, April 9-11, 2018
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts in the same volume. Part I is dedicated to information theory and
the mathematical formalization of randomness based on Kolmogorov complexity.
This last application goes back to the 60's and 70's with the work of
Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last
years.Comment: 40 page
Normal, Abby Normal, Prefix Normal
A prefix normal word is a binary word with the property that no substring has
more 1s than the prefix of the same length. This class of words is important in
the context of binary jumbled pattern matching. In this paper we present
results about the number of prefix normal words of length , showing
that for some and
. We introduce efficient
algorithms for testing the prefix normal property and a "mechanical algorithm"
for computing prefix normal forms. We also include games which can be played
with prefix normal words. In these games Alice wishes to stay normal but Bob
wants to drive her "abnormal" -- we discuss which parameter settings allow
Alice to succeed.Comment: Accepted at FUN '1
On inversion sets and the weak order in Coxeter groups
In this article, we investigate the existence of joins in the weak order of
an infinite Coxeter group W. We give a geometric characterization of the
existence of a join for a subset X in W in terms of the inversion sets of its
elements and their position relative to the imaginary cone. Finally, we discuss
inversion sets of infinite reduced words and the notions of biconvex and
biclosed sets of positive roots.Comment: 22 pages; 10 figures; v2 some references were added; v2: final
version, to appear in European Journal of Combinatoric
- âŠ