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 $\Omega(n^2/4)$.Comment: 31 pages, 9 figures. arXiv admin note: text overlap with arXiv:1805.0672

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

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 Ω(n 2/4)