Abstract
We push further a recently proposed approach for studying synchronizing automata and Černý’s conjecture, namely, the synchronizing probability function. In this approach, the synchronizing phenomenon is reinterpreted as a Two-Player game, in which the optimal strategies of the players can be obtained through a Linear Program. Our analysis mainly focuses on the concept of triple rendezvous time, the length of the shortest word mapping three states onto a single one. It represents an intermediate step in the synchronizing process, and is a good proxy of its overall length. Our contribution is twofold. First, using the synchronizing probability function and properties of linear programming, we provide a new upper bound on the triple rendezvous time. Second, we disprove a conjecture on the synchronizing probability function by exhibiting a family of counterexamples. We discuss the game theoretic approach and possible further work in the light of our results.
This is a short conference version. For a long version with more details and examples see [12].
R.M. Jungers is a F.R.S.-FNRS Research Associate.
This work was also supported by the communauté francaise de Belgique - Actions de Recherche Concertées and by the Belgian Program on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office.
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Gonze, F., Jungers, R.M. (2015). On the Synchronizing Probability Function and the Triple Rendezvous Time. In: Dediu, AH., Formenti, E., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2015. Lecture Notes in Computer Science(), vol 8977. Springer, Cham. https://doi.org/10.1007/978-3-319-15579-1_16
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