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 Primitivity of Sets of Matrices

A nonnegative matrix $A$ is called primitive if $A^k$ is positive for some integer $k>0$. A generalization of this concept to finite sets of matrices is as follows: a set of matrices $\mathcal M = \{A_1, A_2, \ldots, A_m \}$ is primitive if $A_{i_1} A_{i_2} \ldots A_{i_k}$ is positive for some indices $i_1, i_2, ..., i_k$. The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product. We show that while primitivity is algorithmically decidable, unless $P=NP$ it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices. On the other hand, defining ${\mathcal P}$ to be the set of matrices with no zero rows or columns, we give a simple combinatorial proof of a previously-known characterization of primitivity for matrices in ${\mathcal P}$ which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Cerny on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in ${\mathcal P}$. In particular, any primitive set of $n \times n$ matrices in ${\mathcal P}$ has a positive product of length $O(n^3)$

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

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

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 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

Primitive digraphs with large exponents and slowly synchronizing automata

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a slightly updated bibliography) of the authors' paper published in Russian in: Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol. 402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf Version 2: a few typos are correcte

Attainable Values of Reset Thresholds

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and [(p-1)(q-1), p(q-2)+n-q+1], where 2 n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n). This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers. We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case. Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n)