1,908 research outputs found
Shortest Repetition-Free Words Accepted by Automata
We consider the following problem: given that a finite automaton of
states accepts at least one -power-free (resp., overlap-free) word, what is
the length of the shortest such word accepted? We give upper and lower bounds
which, unfortunately, are widely separated.Comment: 12 pages, conference pape
Strong inapproximability of the shortest reset word
The \v{C}ern\'y conjecture states that every -state synchronizing
automaton has a reset word of length at most . We study the hardness
of finding short reset words. It is known that the exact version of the
problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and
complete for the DP class, and that approximating the length of the shortest
reset word within a factor of is NP-hard [Gerbush and Heeringa,
CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly
improve on these results by showing that, for every , it is NP-hard
to approximate the length of the shortest reset word within a factor of
. This is essentially tight since a simple -approximation
algorithm exists.Comment: extended abstract to appear in MFCS 201
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
Foliations for solving equations in groups: free, virtually free, and hyperbolic groups
We give an algorithm for solving equations and inequations with rational
constraints in virtually free groups. Our algorithm is based on Rips
classification of measured band complexes. Using canonical representatives, we
deduce an algorithm for solving equations and inequations in hyperbolic groups
(maybe with torsion). Additionnally, we can deal with quasi-isometrically
embeddable rational constraints.Comment: 70 pages, 7 figures, revised version. To appear in Journal of
Topolog
On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups
We show that the isomorphism problem is solvable in the class of central
extensions of word-hyperbolic groups, and that the isomorphism problem for
biautomatic groups reduces to that for biautomatic groups with finite centre.
We describe an algorithm that, given an arbitrary finite presentation of an
automatic group , will construct explicit finite models for the skeleta
of and hence compute the integral homology and cohomology of
.Comment: 21 pages, 4 figure
Learning to Prove Safety over Parameterised Concurrent Systems (Full Version)
We revisit the classic problem of proving safety over parameterised
concurrent systems, i.e., an infinite family of finite-state concurrent systems
that are represented by some finite (symbolic) means. An example of such an
infinite family is a dining philosopher protocol with any number n of processes
(n being the parameter that defines the infinite family). Regular model
checking is a well-known generic framework for modelling parameterised
concurrent systems, where an infinite set of configurations (resp. transitions)
is represented by a regular set (resp. regular transducer). Although verifying
safety properties in the regular model checking framework is undecidable in
general, many sophisticated semi-algorithms have been developed in the past
fifteen years that can successfully prove safety in many practical instances.
In this paper, we propose a simple solution to synthesise regular inductive
invariants that makes use of Angluin's classic L* algorithm (and its variants).
We provide a termination guarantee when the set of configurations reachable
from a given set of initial configurations is regular. We have tested L*
algorithm on standard (as well as new) examples in regular model checking
including the dining philosopher protocol, the dining cryptographer protocol,
and several mutual exclusion protocols (e.g. Bakery, Burns, Szymanski, and
German). Our experiments show that, despite the simplicity of our solution, it
can perform at least as well as existing semi-algorithms.Comment: Full version of FMCAD'17 pape
Groups and semigroups with a one-counter word problem
We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian
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