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On the Synchronizing Probability Function and the Triple Rendezvous Time: New Approaches to Černý's Conjecture François Gonze Prof. Raphaël Jungers 2 March 2015 1

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Subject: synchronizing automata Deterministic, finite state, complete automata (DFA) Set of states, alphabet of letters and transition function 2

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Synchronizing? One agent on each state Moving according to the letters There exists a word (sequence of letters) regrouping them 3

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abbbabbba Synchronizing word Is it synchronizing? The word abbbabbba regroups the agents 4 Success!

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Applications Computer and system control: applying a sequence of operations to restore control Industry: get a large amount of parts in the same state Theoretical mathematics: primitivity of matrix sets [Blondel et al., 2014] 5

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Plan 6 Černý’s conjecture Approach: the triple rendezvous time A tool: the synchronizing probability function Counter examples on conjectures

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Cerny’s Conjecture Aim: obtain a short synchronizing word Černý: bound on the smallest synchronizing word 7

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State of the art Proven in many particular cases ([Eppstein, 1990], [Dubuc, 1998], [Kari, 2003], [Trahtman, 2007], [Steinberg, 2010], etc.) Open in the general case 8

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Plan 9 Černý’s conjecture Approach: the triple rendezvous time A tool: the synchronizing probability function Counter examples on conjectures

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Intermediate problem Synchronization: mapping n states onto a single one How many letters are needed to map k

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Triple rendezvous time 11 abbba Success !

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First bound For any synchronizing automaton, This gives: 12

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Plan 13 Černý’s conjecture Approach: the triple rendezvous time A tool: the synchronizing probability function Counter examples on conjectures

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Game theoretical formulation Reformulation of synchronization as a two-player game [Jungers, 2012]: 1.Length t is chosen. 2.Player Two chooses the initial state and keep it secret. 3.Player One chooses a word of length at most t. 4.The word is applied to the automaton. 5.Player One guesses what the final state is. If Player One is right, he wins 14

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Where would you hide? If t is equal to 0? If t is equal to 1? If t is equal to 9? 15

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Matrix representation Letters and words : nxn matrices Single state or set of states : 1xn vectors Ones in Columns : states that are regrouped All-ones column : synchronization of the automaton 16

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Matrix of columns A(t) : set of all columns in products of length t Equivalent to all possible choices of Player One In our example : From that, we can obtain linear programs allowing to calculate the SPF 17

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Synchronizing probability function 18 Both players play optimally Best probability of winning for Player One

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SPF evolution 19

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Using the SPF 20 We obtain a better bound on the triple rendezvous time:

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Plan 21 Černý’s conjecture Approach: the triple rendezvous time A tool: the synchronizing probability function Counter examples on conjectures

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Conjectures 22 Conjecture regarding the SPF [Jungers, 2012]: Conjecture on the triple rendezvous time [Jungers, 2012]: This conjecture is stronger than Černý’s conjecture

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Counter example automaton 23 Automaton with 9 states, k(11)=2/9 and T3=12 Contradicts both conjectures

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SPF comparison New automaton SPF in black Černý’s automaton with 9 states in dashed 24

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Family extension 25 Extension of the family to 11 and 13 states. It can be extended to any odd number, keeping the property.

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Conclusion Results : 1.Upper bound on T3 obtained with the SPF. 2.Counter-example to a conjecture on the SPF evolution. 3.Counter-example to a conjecture on T3 which is also a lower bound onthe maximal T3. Further research : 1.Refining the upper bound on T3. 2.Extend the concept to T4, etc. Thanks ! 26

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