1. 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

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

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

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

5. 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

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

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

8. 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

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

10. Intermediate problem Synchronization: mapping n states onto a single one How many letters are needed to map k<n states onto a single one? For k=2, in a synchronizing automaton, one letter 10

11. Triple rendezvous time 11 abbba Success !

12. First bound For any synchronizing automaton, This gives: 12

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

14. 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

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

16. 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

17. 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

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

19. SPF evolution 19

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

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

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

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

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

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

26. 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