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A note on Thue games
In this work we improve on a result from [1]. In particular,
we investigate the situation where a word is constructed jointly by two
players who alternately append letters to the end of an existing word.
One of the players (Ann) tries to avoid (non-trivial) repetitions, while
the other one (Ben) tries to enforce them. We show a construction that
is closer to the lower bound showed in [2] using entropy compression, and
building on the probabilistic arguments based on a version of the Lov´asz
Local Lemma from [3]. We provide an explicit strategy for Ann to avoid
(non-trivial) repetitions over a 7-letter alphabet
Star-Free Languages are Church-Rosser Congruential
The class of Church-Rosser congruential languages has been introduced by
McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser
congruential (belongs to CRCL), if there is a finite, confluent, and
length-reducing semi-Thue system S such that L is a finite union of congruence
classes modulo S. To date, it is still open whether every regular language is
in CRCL. In this paper, we show that every star-free language is in CRCL. In
fact, we prove a stronger statement: For every star-free language L there
exists a finite, confluent, and subword-reducing semi-Thue system S such that
the total number of congruence classes modulo S is finite and such that L is a
union of congruence classes modulo S. The construction turns out to be
effective
On winning shifts of marked uniform substitutions
The second author introduced with I. T\"orm\"a a two-player word-building
game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has
a predetermined (possibly finite) choice sequence , ,
of integers such that on round the player chooses a subset
of size of some fixed finite alphabet and the player picks
a letter from the set . The outcome is determined by whether the word
obtained by concatenating the letters picked lies in a prescribed target
set (a win for player ) or not (a win for player ). Typically, we
consider to be a subshift. The winning shift of a subshift is
defined as the set of choice sequences for which has a winning strategy
when the target set is the language of . The winning shift mirrors
some properties of . For instance, and have the same entropy.
Virtually nothing is known about the structure of the winning shifts of
subshifts common in combinatorics on words. In this paper, we study the winning
shifts of subshifts generated by marked uniform substitutions, and show that
these winning shifts, viewed as subshifts, also have a substitutive structure.
Particularly, we give an explicit description of the winning shift for the
generalized Thue-Morse substitutions. It is known that and have the
same factor complexity. As an example application, we exploit this connection
to give a simple derivation of the first difference and factor complexity
functions of subshifts generated by marked substitutions. We describe these
functions in particular detail for the generalized Thue-Morse substitutions.Comment: Extended version of a paper presented at RuFiDiM I
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
A new approach to nonrepetitive sequences
A sequence is nonrepetitive if it does not contain two adjacent identical
blocks. The remarkable construction of Thue asserts that 3 symbols are enough
to build an arbitrarily long nonrepetitive sequence. It is still not settled
whether the following extension holds: for every sequence of 3-element sets
there exists a nonrepetitive sequence with
. Applying the probabilistic method one can prove that this is true
for sufficiently large sets . We present an elementary proof that sets of
size 4 suffice (confirming the best known bound). The argument is a simple
counting with Catalan numbers involved. Our approach is inspired by a new
algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its
interpretations by Fortnow and Tao. The presented method has further
applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with
arXiv:1103.381
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