Reachability of consensus and synchronizing automata

Chevalier, Pierre-Yves; Hendrickx, Julien M.; Jungers, Raphaël M.

doi:10.1109/cdc.2015.7402864

Cited by 15 publications

(32 citation statements)

References 21 publications

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“…Let A j ∈ M ≤tj for t j < t: we claim that there exists a product L ∈ M t such that A i e ≤ Le. In this case we can erase the column A i e from H t as for any optimal solution p of program (7), p T A i e ≤ p T Le ≤ k. Let M ∈ M t−tj be a product that dominates a permutation matrix (it always exists by hypothesis) and L = A j M ; it holds that for every column a of A j there exists a column l of L such that a ≤ l, which implies A i e ≤ Le. Since L is a product of length t, the claim is proven.…”

Section: The Linear Programming Formulationmentioning

confidence: 99%

“…17. In this case, if we denote by K = (t) the optimal solution of program (7) with M ≤t replaced by M t , K = (t) can still provide an approximation of K(t). Indeed, if s is the first time such that M ≤s contains a matrix that dominates a permutation matrix (s must exist if the set is primitive), then for every t > s it holds that…”

Section: The Linear Programming Formulationmentioning

confidence: 99%

“…The primitivity property of nonnegative matrix sets has lately found applications in various fields as in consensus of discrete-time multi-agent systems [7], in cryptography [11] and in automata theory [4,5,13]. Primitivity can also be seen as one of the simplest reachability problems for nonnegative discrete-time switched systems, as it provides a necessary and sufficient condition for the system to reach the interior of the nonnegative orthant independently on the initial state [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Synchronizing Probability Function for Primitive Sets of Matrices

Catalano

Jungers

2018

Developments in Language Theory

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Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstandingČerný 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.

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Section: The Linear Programming Formulationmentioning

confidence: 99%

Section: The Linear Programming Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Synchronizing Probability Function for Primitive Sets of Matrices

Catalano

Jungers

2018

Developments in Language Theory

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“…The concept of primitive set was just recently formalized by Protasov and Voynov [28], but has been appearing before in different fields as in stochastic switching systems [18,27] and time-inhomogeneous Markov chains [17,31]. It has lately gained more importance due to its applications in consensus of discretetime multi-agent systems [8], cryptography [11] and automata theory [3,13,5]. ⋆ R. M. Jungers is a FNRS Research Associate.…”

Section: Introductionmentioning

confidence: 99%

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

Azfar

Catalano

Charlier

et al. 2019

Developments in Language Theory

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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 (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods.Keywords: Primitive set of matrices, synchronizing automaton, Černý conjecture.Deciding whether a set is primitive is an NP-hard problem if the set contains at least three matrices (while it is still of unknown complexity for sets of two matrices) [3], and so is computing its exponent. For sets of matrices having at least a positive entry in every row and every column (called NZ [13] or allowable matrices [16,18]), the primitivity problem becomes decidable in polynomial-time [28], although computing the exponent remains NP-hard [13]. Methods for approximating the exponent have been proposed [6] as well as upper bounds that depend just on the matrix size; in particular, if we denote with exp N Z (n) the maximal exponent among all the primitive sets of n × n NZ matrices, it is known that exp N Z (n) ≤ (15617n 3 + 7500n 2 + 56250n − 78125)/46875 [3,33]. Better upper bounds have been found for some classes of primitive sets [13,17]. The NZ condition is often met in applications and in particular in the connection with synchronizing automata.

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“…In the 80s and 90s, this subject found applications in robotics and in the industry. More recently, it has been used to model consensus theory and linked with primitivity of matrix sets (see [6,12]). …”

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confidence: 99%

On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata

Gonze

Jungers²

2016

SIAM J. Discrete Math.

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Abstract.Černý'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 followups towardsČerný's conjecture.

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Reachability of consensus and synchronizing automata

Cited by 15 publications

References 21 publications

The Synchronizing Probability Function for Primitive Sets of Matrices

The Synchronizing Probability Function for Primitive Sets of Matrices

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata

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