167,430 research outputs found
There Exist some Omega-Powers of Any Borel Rank
Omega-powers of finitary languages are languages of infinite words
(omega-languages) in the form V^omega, where V is a finitary language over a
finite alphabet X. They appear very naturally in the characterizaton of regular
or context-free omega-languages. Since the set of infinite words over a finite
alphabet X can be equipped with the usual Cantor topology, the question of the
topological complexity of omega-powers of finitary languages naturally arises
and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It
has been recently proved that for each integer n > 0, there exist some
omega-powers of context free languages which are Pi^0_n-complete Borel sets,
that there exists a context free language L such that L^omega is analytic but
not Borel, and that there exists a finitary language V such that V^omega is a
Borel set of infinite rank. But it was still unknown which could be the
possible infinite Borel ranks of omega-powers. We fill this gap here, proving
the following very surprising result which shows that omega-powers exhibit a
great topological complexity: for each non-null countable ordinal alpha, there
exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete
omega-powers.Comment: To appear in the Proceedings of the 16th EACSL Annual Conference on
Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11-15,
2007, Lecture Notes in Computer Science, (c) Springer, 200
On the insertion of n-powers
In algebraic terms, the insertion of -powers in words may be modelled at
the language level by considering the pseudovariety of ordered monoids defined
by the inequality . We compare this pseudovariety with several other
natural pseudovarieties of ordered monoids and of monoids associated with the
Burnside pseudovariety of groups defined by the identity . In
particular, we are interested in determining the pseudovariety of monoids that
it generates, which can be viewed as the problem of determining the Boolean
closure of the class of regular languages closed under -power insertions. We
exhibit a simple upper bound and show that it satisfies all pseudoidentities
which are provable from in which both sides are regular elements
with respect to the upper bound
On the Expressive Power of Regular Expressions with Backreferences
A rewb is a regular expression extended with a feature called backreference. It is broadly known that backreference is a practical extension of regular expressions, and is supported by most modern regular expression engines, such as those in the standard libraries of Java, Python, and more. Meanwhile, indexed languages are the languages generated by indexed grammars, a formal grammar class proposed by A.V.Aho. We show that these two models\u27 expressive powers are related in the following way: every language described by a rewb is an indexed language. As the smallest formal grammar class previously known to contain rewbs is the class of context sensitive languages, our result strictly improves the known upper-bound. Moreover, we prove the following two claims: there exists a rewb whose language does not belong to the class of stack languages, which is a proper subclass of indexed languages, and the language described by a rewb without a captured reference is in the class of nonerasing stack languages, which is a proper subclass of stack languages. Finally, we show that the hierarchy investigated in a prior study, which separates the expressive power of rewbs by the notion of nested levels, is within the class of nonerasing stack languages
On the Expressive Power of Regular Expressions with Backreferences
A rewb is a regular expression extended with a feature called backreference.
It is broadly known that backreference is a practical extension of regular
expressions, and is supported by most modern regular expression engines, such
as those in the standard libraries of Java, Python, and more. Meanwhile,
indexed languages are the languages generated by indexed grammars, a formal
grammar class proposed by A.V.Aho. We show that these two models' expressive
powers are related in the following way: every language described by a rewb is
an indexed language. As the smallest formal grammar class previously known to
contain rewbs is the class of context sensitive languages, our result strictly
improves the known upper-bound. Moreover, we prove the following two claims:
there exists a rewb whose language does not belong to the class of stack
languages, which is a proper subclass of indexed languages, and the language
described by a rewb without a captured reference is in the class of nonerasing
stack languages, which is a proper subclass of stack languages. Finally, we
show that the hierarchy investigated in a prior study, which separates the
expressive power of rewbs by the notion of nested levels, is within the class
of nonerasing stack languages.Comment: 20 pages, the full version of the paper to appear in MFCS 202
On Pansiot Words Avoiding 3-Repetitions
The recently confirmed Dejean's conjecture about the threshold between
avoidable and unavoidable powers of words gave rise to interesting and
challenging problems on the structure and growth of threshold words. Over any
finite alphabet with k >= 5 letters, Pansiot words avoiding 3-repetitions form
a regular language, which is a rather small superset of the set of all
threshold words. Using cylindric and 2-dimensional words, we prove that, as k
approaches infinity, the growth rates of complexity for these regular languages
tend to the growth rate of complexity of some ternary 2-dimensional language.
The numerical estimate of this growth rate is about 1.2421.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Topological Complexity of omega-Powers : Extended Abstract
This is an extended abstract presenting new results on the topological
complexity of omega-powers (which are included in a paper "Classical and
effective descriptive complexities of omega-powers" available from
arXiv:0708.4176) and reflecting also some open questions which were discussed
during the Dagstuhl seminar on "Topological and Game-Theoretic Aspects of
Infinite Computations" 29.06.08 - 04.07.08
Detecting palindromes, patterns, and borders in regular languages
Given a language L and a nondeterministic finite automaton M, we consider
whether we can determine efficiently (in the size of M) if M accepts at least
one word in L, or infinitely many words. Given that M accepts at least one word
in L, we consider how long a shortest word can be. The languages L that we
examine include the palindromes, the non-palindromes, the k-powers, the
non-k-powers, the powers, the non-powers (also called primitive words), the
words matching a general pattern, the bordered words, and the unbordered words.Comment: Full version of a paper submitted to LATA 2008. This is a new version
with John Loftus added as a co-author and containing new results on
unbordered word
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