77 research outputs found
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
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
A Note on a Recent Attempt to Improve the Pin-Frankl Bound
We provide a counterexample to a lemma used in a recent tentative improvement
of the the Pin-Frankl bound for synchronizing automata. This example naturally
leads us to formulate an open question, whose answer could fix the line of
proof, and improve the bound.Comment: Short note presenting a counterexample and the resulting open
questio
Synchronizing Times for -sets in Automata
An automaton is synchronizing if there is a word that maps all states onto
the same state. \v{C}ern\'{y}'s conjecture on the length of the shortest such
word is probably the most famous open problem in automata theory. We consider
the closely related question of determining the minimum length of a word that
maps states onto a single state. For synchronizing automata, we improve the
upper bound on the minimum length of a word that sends some triple to a a
single state from to . We further extend this to an
improved bound on the length of such a word for 4 states and 5 states. In the
case of non-synchronizing automata, we give an example to show that the minimum
length of a word that sends states to a single state can be as large as
.Comment: 19 pages, 4 figure
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
On the synchronization of finite state automata
Abstract: We study some problems related to the synchronization of finite state automata and the Cˇerny’s conjecture. We focus on the synchronization of small sets of states, and more specifically on the synchronization of triples. We argue that it is the most simple synchronization scenario that exhibits the intricacies of the original Cˇerny’s scenario (all states synchronization). Thus, we argue that it is complex enough to be interesting, and tractable enough to be studied via algo- rithmic tools. We use those tools to establish a long list of facts related to those issues. We observe that planar automata seems to be representative of the synchroniz- ing behavior of deterministic finite state automata. Moreover, we present strong evidence suggesting the importance of planar automata in the study of Cˇerny’s conjecture. We also study synchronization games played on planar automata. We prove that recognizing the planar games that can be won by the synchronizer is a co-NP hard problem. We prove some additional results indicating that pla- nar games are as hard as nonplanar games. Those results amount to show that planar automata are representative of the intricacies of automata synchronization.Doctorad
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
Some remarks on synchronization, games and planar automata
Abstract—We study synchronization games on planar automata.
We prove that recognizing the planar games that can be won by the synchronizer is a co-NP hard problem. We prove some additional results indicating that planar games are as hard as nonplanar games. Those results amount to show that planar automata are representative of the intricacies of automata synchronization.Sociedad Argentina de Informática e Investigación Operativa (SADIO
Some remarks on synchronization, games and planar automata
Abstract—We study synchronization games on planar automata.
We prove that recognizing the planar games that can be won by the synchronizer is a co-NP hard problem. We prove some additional results indicating that planar games are as hard as nonplanar games. Those results amount to show that planar automata are representative of the intricacies of automata synchronization.Sociedad Argentina de Informática e Investigación Operativa (SADIO
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