213 research outputs found
On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs
We tackle the problem of the randomized generation of slowly synchronizing
deterministic automata (DFAs) by generating random primitive sets of matrices.
We show that when the randomized procedure is too simple the exponent of the
generated sets is O(n log n) with high probability, thus the procedure fails to
return DFAs with large reset threshold. We extend this result to random
nondeterministic automata (NDFAs) by showing, in particular, that a uniformly
sampled NDFA has both a 2-directing word and a 3-directing word of length O(n
log n) with high probability. We then present a more involved randomized
algorithm that manages to generate DFAs with large reset threshold and we
finally leverage this finding for exhibiting new families of DFAs with reset
threshold of order .Comment: 31 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1805.0672
Synchronizing weighted automata
We introduce two generalizations of synchronizability to automata with
transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently,
to finite sets of matrices in K^nxn.) Let us call a matrix A
location-synchronizing if there exists a column in A consisting of nonzero
entries such that all the other columns of A are filled by zeros. If
additionally all the entries of this designated column are the same, we call A
synchronizing. Note that these notions coincide for stochastic matrices and
also in the Boolean semiring. A set M of matrices in K^nxn is called
(location-)synchronizing if M generates a matrix subsemigroup containing a
(location-)synchronizing matrix. The K-(location-)synchronizability problem is
the following: given a finite set M of nxn matrices with entries in K, is it
(location-)synchronizing?
Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient
conditions for the semiring K when the problems are PSPACE-complete and show
several undecidability results as well, e.g. synchronizability is undecidable
if 1 has infinite order in (K,+,0) or when the free semigroup on two generators
can be embedded into (K,*,1).Comment: In Proceedings AFL 2014, arXiv:1405.527
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
Distributed graph problems through an automata-theoretic lens
The locality of a graph problem is the smallest distance such that each
node can choose its own part of the solution based on its radius-
neighborhood. In many settings, a graph problem can be solved efficiently with
a distributed or parallel algorithm if and only if it has a small locality.
In this work we seek to automate the study of solvability and locality: given
the description of a graph problem , we would like to determine if
is solvable and what is the asymptotic locality of as a function of the
size of the graph. Put otherwise, we seek to automatically synthesize efficient
distributed and parallel algorithms for solving .
We focus on locally checkable graph problems; these are problems in which a
solution is globally feasible if it looks feasible in all constant-radius
neighborhoods. Prior work on such problems has brought primarily bad news:
questions related to locality are undecidable in general, and even if we focus
on the case of labeled paths and cycles, determining locality is
-hard (Balliu et al., PODC 2019).
We complement prior negative results with efficient algorithms for the cases
of unlabeled paths and cycles and, as an extension, for rooted trees. We
introduce a new automata-theoretic perspective for studying locally checkable
graph problems. We represent a locally checkable problem as a
nondeterministic finite automaton over a unary alphabet. We
identify polynomial-time-computable properties of the automaton
that near-completely capture the solvability and locality of in cycles
and paths, with the exception of one specific case that is
\mbox{co-\mathsf{NP}}-complete
Computational Complexity of Synchronization under Regular Constraints
Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata
Tight Upper Bounds for Streett and Parity Complementation
Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound and upper
bound , where is the state size, is the number of
Streett pairs, and can be as large as . Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound for and for ,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011
- …