8 research outputs found
Reachability of Consensus and Synchronizing Automata
We consider the problem of determining the existence of a sequence of
matrices driving a discrete-time consensus system to consensus. We transform
this problem into one of the existence of a product of the transition
(stochastic) matrices that has a positive column. We then generalize some
results from automata theory to sets of stochastic matrices. We obtain as a
main result a polynomial-time algorithm to decide the existence of a sequence
of matrices achieving consensus.Comment: Update after revie
A linear bound on the k-rendezvous time for primitive sets of NZ matrices
A set of nonnegative matrices is called primitive if there exists a product
of these matrices that is entrywise positive. Motivated by recent results
relating synchronizing automata and primitive sets, we study the length of the
shortest product of a primitive set having a column or a row with k positive
entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices
having no zero rows and no zero columns. We prove that the k-RT is at most
linear w.r.t. the matrix size n for small k, while the problem is still open
for synchronizing automata. We provide two upper bounds on the k-RT: the second
is an improvement of the first one, although the latter can be written in
closed form. We then report numerical results comparing our upper bounds on the
k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata
Cerny's conjecture is a longstanding open problem in automata theory. We
study two different concepts, which allow to approach it from a new angle. The
first one is the triple rendezvous time, i.e., the length of the shortest word
mapping three states onto a single one. The second one is the synchronizing
probability function of an automaton, a recently introduced tool which
reinterprets the synchronizing phenomenon as a two-player game, and allows to
obtain optimal strategies through a Linear Program.
Our contribution is twofold. First, by coupling two different novel
approaches based on the synchronizing probability function and properties of
linear programming, we obtain a new upper bound on the triple rendezvous time.
Second, by exhibiting a family of counterexamples, we disprove a conjecture on
the growth of the synchronizing probability function. We then suggest natural
follow-ups towards Cernys conjecture.Comment: A preliminary version of the results has been presented at the
conference LATA 2015. The current ArXiv version includes the most recent
improvement on the triple rendezvous time upper bound as well as formal
proofs of all the result
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
On the interplay between Babai and Černý’s conjectures
Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017
Reachability of Consensus and Synchronizing Automata
We consider the problem of determining the existence of a sequence of matrices driving a discrete-time multi-agent consensus system to consensus. We transform this
problem into the problem of the existence of a product of the (stochastic) transition matrices that has a positive column. This allows us to make use of results from automata theory to sets of stochastic matrices. Our main result is a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus