11,075 research outputs found
The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable
Monadic second order logic can be used to express many classical notions of
sets of vertices of a graph as for instance: dominating sets, induced
matchings, perfect codes, independent sets or irredundant sets. Bounds on the
number of sets of any such family of sets are interesting from a combinatorial
point of view and have algorithmic applications. Many such bounds on different
families of sets over different classes of graphs are already provided in the
literature. In particular, Rote recently showed that the number of minimal
dominating sets in trees of order is at most and that
this bound is asymptotically sharp up to a multiplicative constant. We build on
his work to show that what he did for minimal dominating sets can be done for
any family of sets definable by a monadic second order formula.
We first show that, for any monadic second order formula over graphs that
characterizes a given kind of subset of its vertices, the maximal number of
such sets in a tree can be expressed as the \textit{growth rate of a bilinear
system}. This mostly relies on well known links between monadic second order
logic over trees and tree automata and basic tree automata manipulations. Then
we show that this "growth rate" of a bilinear system can be approximated from
above.We then use our implementation of this result to provide bounds on the
number of independent dominating sets, total perfect dominating sets, induced
matchings, maximal induced matchings, minimal perfect dominating sets, perfect
codes and maximal irredundant sets on trees. We also solve a question from D.
Y. Kang et al. regarding -matchings and improve a bound from G\'orska and
Skupie\'n on the number of maximal matchings on trees. Remark that this
approach is easily generalizable to graphs of bounded tree width or clique
width (or any similar class of graphs where tree automata are meaningful)
Limit sets of stable Cellular Automata
We study limit sets of stable cellular automata standing from a symbolic
dynamics point of view where they are a special case of sofic shifts admitting
a steady epimorphism. We prove that there exists a right-closing
almost-everywhere steady factor map from one irreducible sofic shift onto
another one if and only if there exists such a map from the domain onto the
minimal right-resolving cover of the image. We define right-continuing
almost-everywhere steady maps and prove that there exists such a steady map
between two sofic shifts if and only if there exists a factor map from the
domain onto the minimal right-resolving cover of the image. In terms of
cellular automata, this translates into: A sofic shift can be the limit set of
a stable cellular automaton with a right-closing almost-everywhere dynamics
onto its limit set if and only if it is the factor of a fullshift and there
exists a right- closing almost-everywhere factor map from the sofic shift onto
its minimal right- resolving cover. A sofic shift can be the limit set of a
stable cellular automaton reaching its limit set with a right-continuing
almost-everywhere factor map if and only if it is the factor of a fullshift and
there exists a factor map from the sofic shift onto its minimal right-resolving
cover. Finally, as a consequence of the previous results, we provide a
characterization of the Almost of Finite Type shifts (AFT) in terms of a
property of steady maps that have them as range.Comment: 18 pages, 3 figure
Nondeterministic State Complexity for Suffix-Free Regular Languages
We investigate the nondeterministic state complexity of basic operations for
suffix-free regular languages. The nondeterministic state complexity of an
operation is the number of states that are necessary and sufficient in the
worst-case for a minimal nondeterministic finite-state automaton that accepts
the language obtained from the operation. We consider basic operations
(catenation, union, intersection, Kleene star, reversal and complementation)
and establish matching upper and lower bounds for each operation. In the case
of complementation the upper and lower bounds differ by an additive constant of
two.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Revisiting Waiting Times in DNA evolution
Transcription factors are short stretches of DNA (or -mers) mainly located
in promoters sequences that enhance or repress gene expression. With respect to
an initial distribution of letters on the DNA alphabet, Behrens and Vingron
consider a random sequence of length that does not contain a given -mer
or word of size . Under an evolution model of the DNA, they compute the
probability that this -mer appears after a unit time of 20
years. They prove that the waiting time for the first apparition of the -mer
is well approximated by . Their work relies on the
simplifying assumption that the -mer is not self-overlapping. They observe
in particular that the waiting time is mostly driven by the initial
distribution of letters.
Behrens et al. use an approach by automata that relaxes the assumption
related to words overlaps. Their numerical evaluations confirms the validity of
Behrens and Vingron approach for non self-overlapping words, but provides up to
44% corrections for highly self-overlapping words such as . We
devised an approach of the problem by clump analysis and generating functions;
this approach leads to prove a quasi-linear behaviour of for a
large range of values of , an important result for DNA evolution. We present
here this clump analysis, first by language decomposition, and next by an
automaton construction; finally, we describe an equivalent approach by
construction of Markov automata.Comment: 19 pages, 3 Figures, 2 Table
On Nonnegative Integer Matrices and Short Killing Words
Let be a natural number and a set of -matrices
over the nonnegative integers such that the joint spectral radius of
is at most one. We show that if the zero matrix is a product
of matrices in , then there are with . This result has applications in
automata theory and the theory of codes. Specifically, if
is a finite incomplete code, then there exists a word of
length polynomial in such that is not a factor of any
word in . This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code
Learning Moore Machines from Input-Output Traces
The problem of learning automata from example traces (but no equivalence or
membership queries) is fundamental in automata learning theory and practice. In
this paper we study this problem for finite state machines with inputs and
outputs, and in particular for Moore machines. We develop three algorithms for
solving this problem: (1) the PTAP algorithm, which transforms a set of
input-output traces into an incomplete Moore machine and then completes the
machine with self-loops; (2) the PRPNI algorithm, which uses the well-known
RPNI algorithm for automata learning to learn a product of automata encoding a
Moore machine; and (3) the MooreMI algorithm, which directly learns a Moore
machine using PTAP extended with state merging. We prove that MooreMI has the
fundamental identification in the limit property. We also compare the
algorithms experimentally in terms of the size of the learned machine and
several notions of accuracy, introduced in this paper. Finally, we compare with
OSTIA, an algorithm that learns a more general class of transducers, and find
that OSTIA generally does not learn a Moore machine, even when fed with a
characteristic sample
Entanglement Generation of Clifford Quantum Cellular Automata
Clifford quantum cellular automata (CQCAs) are a special kind of quantum
cellular automata (QCAs) that incorporate Clifford group operations for the
time evolution. Despite being classically simulable, they can be used as basic
building blocks for universal quantum computation. This is due to the
connection to translation-invariant stabilizer states and their entanglement
properties. We will give a self-contained introduction to CQCAs and investigate
the generation of entanglement under CQCA action. Furthermore, we will discuss
finite configurations and applications of CQCAs.Comment: to appear in the "DPG spring meeting 2009" special issue of Applied
Physics
On the Minimal Uncompletable Word Problem
Let S be a finite set of words over an alphabet Sigma. The set S is said to
be complete if every word w over the alphabet Sigma is a factor of some element
of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are
interested in finding bounds on the minimal length of words in Sigma* which are
not elements of Fact(S*) in terms of the maximal length of words in S.Comment: 5 pages; added references, corrected typo
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