11 research outputs found
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
Synchronization Problems in Automata without Non-trivial Cycles
We study the computational complexity of various problems related to
synchronization of weakly acyclic automata, a subclass of widely studied
aperiodic automata. We provide upper and lower bounds on the length of a
shortest word synchronizing a weakly acyclic automaton or, more generally, a
subset of its states, and show that the problem of approximating this length is
hard. We investigate the complexity of finding a synchronizing set of states of
maximum size. We also show inapproximability of the problem of computing the
rank of a subset of states in a binary weakly acyclic automaton and prove that
several problems related to recognizing a synchronizing subset of states in
such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889.
Conference version was published at CIAA 2017, LNCS vol. 10329, pages
188-200, 201
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 Randomized Generation of Slowly Synchronizing Automata
Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank n-1 . We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of Omega(n^2/4) . We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a 0 entry changed into a 1 is primitive and has exponent of O(n log n) with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to 1 with probability p and equal to 0 with probability 1-pwhen np-log n - > infty as n - > infty
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
The Synchronizing Probability Function for Primitive Sets of Matrices
Motivated by recent results relating synchronizing DFAs and primitive sets,
we tackle the synchronization process and the related longstanding
\v{C}ern\'{y} conjecture by studying the primitivity phenomenon for sets of
nonnegative matrices having neither zero-rows nor zero-columns. We formulate
the primitivity process in the setting of a two-player probabilistic game and
we make use of convex optimization techniques to describe its behavior. We
develop a tool for approximating and upper bounding the exponent of any
primitive set and supported by numerical results we state a conjecture that, if
true, would imply a quadratic upper bound on the reset threshold of a new class
of automata.Comment: 24 pages, 9 figures. Submitted to DLT 2018 Special Issu
Algorithmic optimization and parallelization of eppstein's synchronizing heuristic
Testing is the most expensive and time consuming phase in the development of complex systems. Model–based testing is an approach that can be used to automate the generation of high quality test suites, which is the most challenging part of testing. Formal models, such as finite state machines or automata, have been used as specifications from which the test suites can be automatically generated. The tests are applied after the system is synchronized to a particular state, which can be accomplished by using a synchronizing word. Computing a shortest synchronizing word is of interest for practical purposes, e.g. for a shorter testing time. However, computing a shortest synchronizing word is an NP–hard problem. Therefore, heuristics are used to compute short synchronizing words. GREEDY is one of the fastest synchronizing heuristics currently known. In this thesis, we present approaches to accelerate GREEDY algorithm. Firstly, we focus on parallelization of GREEDY. Second, we propose a lazy execution of the preprocessing phase of the algorithm, by postponing the preparation of the required information until it is to be used in the reset word generation phase. We suggest other algorithmic enhancements as well for the implementation of the heuristics. Our experimental results show that depending on the automata size, GREEDY can be made 500⇥ faster. The suggested improvements become more effective as the size of the automaton increases