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 Černý conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitiv...

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Bibliographic Details
Published in:International journal of foundations of computer science 2020-09, Vol.31 (6), p.777-803
Main Authors: Catalano, Costanza, Jungers, Raphaël M.
Format: Article
Language:English
Subjects:
Convexity
Optimization
Optimization techniques
Statistical analysis
Synchronism
Upper bounds
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Summary:Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding Černý 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.
ISSN:0129-0541
1793-6373
DOI:10.1142/S0129054120410051