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Various Aspects of Automaton Synchronization Mikhail V. Berlinkov, Institute of Mathematics and Computer Science, Ural Federal University (Ekaterinburg, Russia), berlm@mail.ru Paris, 2015 Mikhail V. Berlinkov Paris, 2015 1 / 22

  1. The History The notion was formalized in a paper by Jan ˇ Cerný (Poznámka k homogénnym eksperimentom s koneˇ cnými automatami, Matematicko-fyzikalny ˇ Casopis Slovensk. Akad. Vied 14, no.3 (1964) 208–216 [in Slovak]) though implicitly it had been around since at least 1956. The idea of synchronization is pretty natural and of obvious importance: we aim to restore control over a device whose current state is not known. Think of a satellite which loops around the Moon and cannot be controlled from the Earth while “behind” the Moon ( ˇ Cerný’s original motivation). Independently, the same notion was discovered in coding theory by Shimon Even (Test for synchronizability of finite automata and variable length codes, IEEE Trans. Inform. Theory 10 (1964) 185–189). The name synchronizing seems to have originated from Even’s paper. Mikhail V. Berlinkov Paris, 2015 4 / 22

  2. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  3. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  4. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 2 , 3 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  5. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 2 , 3 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  6. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 2 , 3 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  7. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 2 , 3 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  8. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 3 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  9. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 3 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  10. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 3 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  11. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 3 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  12. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 2 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  13. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 2 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  14. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 2 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  15. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 2 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  16. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 2 , 3 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  17. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 3 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  18. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  19. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 , 4 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  20. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  21. Greedy compressing algorithm for synchronization 1 2 a b b a a b 4 3 b a A reset word is v = baababaaab . δ ( Q , v ) = { 1 } The word v is reset whence rt ( A ) ≤ | v | = 10. The shortest reset word for A is ba 3 ba 3 b whence rt ( A ) = 9 < | v | . Mikhail V. Berlinkov Paris, 2015 5 / 22

  22. Various Settings for Synchronization and Outline Whether or not a given automaton is synchronizing? If it is synchronizing, how hard is to synchronize it? 1 Deterministic Setting ˇ Cerný conjecture and Markov Chains Testing for Synchronization Random Case Expected Reset Threshold Computing Reset Thresholds 2 Modifiable Setting Road Coloring Problem Computing Synchronizing Colorings 3 Stochastic Setting Synchronization and Prediction Rates Markov Chain Convergence vs Reset Threshold Mikhail V. Berlinkov Paris, 2015 6 / 22

  23. Various Settings for Synchronization and Outline Whether or not a given automaton is synchronizing? If it is synchronizing, how hard is to synchronize it? 1 Deterministic Setting ˇ Cerný conjecture and Markov Chains Testing for Synchronization Random Case Expected Reset Threshold Computing Reset Thresholds 2 Modifiable Setting Road Coloring Problem Computing Synchronizing Colorings 3 Stochastic Setting Synchronization and Prediction Rates Markov Chain Convergence vs Reset Threshold Mikhail V. Berlinkov Paris, 2015 6 / 22

  24. The ˇ Cerný conjecture ˇ Cerný, 1964 For each n there is an n -state automaton C n with rt ( C n ) = ( n − 1 ) 2 . The ˇ Cerný conjecture, 1964 Each n -state synchronizing automaton has a reset word of length ( n − 1 ) 2 , i.e. rt ( A ) ≤ ( n − 1 ) 2 . Greedy compression algorithm yields the cubic upper bound Θ( n 3 / 2 ) for the reset threshold. Pin, 1983 (based on a combinatorial result of Frankl, 1982) Each n -state automaton has a reset word of length ( n 3 − n ) / 6. Quadratic upper bounds on the reset threshold? Mikhail V. Berlinkov Paris, 2015 7 / 22

  25. The ˇ Cerný conjecture ˇ Cerný, 1964 For each n there is an n -state automaton C n with rt ( C n ) = ( n − 1 ) 2 . The ˇ Cerný conjecture, 1964 Each n -state synchronizing automaton has a reset word of length ( n − 1 ) 2 , i.e. rt ( A ) ≤ ( n − 1 ) 2 . Greedy compression algorithm yields the cubic upper bound Θ( n 3 / 2 ) for the reset threshold. Pin, 1983 (based on a combinatorial result of Frankl, 1982) Each n -state automaton has a reset word of length ( n 3 − n ) / 6. Quadratic upper bounds on the reset threshold? Mikhail V. Berlinkov Paris, 2015 7 / 22

  26. The ˇ Cerný conjecture ˇ Cerný, 1964 For each n there is an n -state automaton C n with rt ( C n ) = ( n − 1 ) 2 . The ˇ Cerný conjecture, 1964 Each n -state synchronizing automaton has a reset word of length ( n − 1 ) 2 , i.e. rt ( A ) ≤ ( n − 1 ) 2 . Greedy compression algorithm yields the cubic upper bound Θ( n 3 / 2 ) for the reset threshold. Pin, 1983 (based on a combinatorial result of Frankl, 1982) Each n -state automaton has a reset word of length ( n 3 − n ) / 6. Quadratic upper bounds on the reset threshold? Mikhail V. Berlinkov Paris, 2015 7 / 22

  27. The ˇ Cerný conjecture ˇ Cerný, 1964 For each n there is an n -state automaton C n with rt ( C n ) = ( n − 1 ) 2 . The ˇ Cerný conjecture, 1964 Each n -state synchronizing automaton has a reset word of length ( n − 1 ) 2 , i.e. rt ( A ) ≤ ( n − 1 ) 2 . Greedy compression algorithm yields the cubic upper bound Θ( n 3 / 2 ) for the reset threshold. Pin, 1983 (based on a combinatorial result of Frankl, 1982) Each n -state automaton has a reset word of length ( n 3 − n ) / 6. Quadratic upper bounds on the reset threshold? Mikhail V. Berlinkov Paris, 2015 7 / 22

  28. Particular Cases Quadratic bounds were approved for various classes: Circular automata with prime number of states [Pin, 1978]; Orientable automata [Eppstein, 1990]; Circular automata [Dubuc, 1998]; Eulerian automata [Kari, 2003]; Aperiodic automata [Trahtman, 2007]; Weakly-monotonic automata [Volkov, 2009]; With monoids belonging to DS class automata [Almeida, Margolis, Steinberg, Volkov, 2009]; One-cluster automata [Béal M., Perrin D., 2009]; One-cluster with prime number of states [Steinberg, 2011]; Respecting intervals of a directed graph automata [Grech, Kisielewicz, 2012]; ... Linear Algebra, Group and Semigroup theories, theory of Markov chains, ... Mikhail V. Berlinkov Paris, 2015 8 / 22

  29. Example from the Italian Job Movie Mikhail V. Berlinkov Paris, 2015 9 / 22

  30. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  31. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  32. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  33. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  34. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  35. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  36. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  37. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  38. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  39. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  40. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  41. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  42. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  43. Kari Automaton and Greedy Extension Method a 0 1 2 b a a b b b b 3 4 5 b a a a A reset word is the reverse to v = baabbbabbaab ... Augmenting sequence is v 1 = b , v 2 = aabb , v 3 = babbaab , v 4 = . . . . This method is optimal for the ˇ Cerný series but returns a reset word of length more than 25 = ( 6 − 1 ) 2 for this automaton. Mikhail V. Berlinkov Paris, 2015 10 / 22

  44. Random Walk Synchronization a b b 1 2 a a b b 4 3 a The probability of catching is b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  45. Random Walk Synchronization a | π 1 ( a ) b | π 1 ( b ) b | π 2 ( b ) 1 2 b | π 4 ( b ) a | π 4 ( a ) a | π 2 ( a ) b | π 3 ( b ) 4 3 a | π 3 ( a ) The probability of catching is b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  46. Random Walk Synchronization 1 2 1 1 1 2 2 2 1 1 2 1 4 3 2 1 2 The probability of catching is b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  47. Random Walk Synchronization 1 2 2 2 1 1 7 7 2 2 1 1 2 1 2 1 7 7 2 1 2 The probability of catching is b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  48. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 The probability of catching is b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  49. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 The probability of catching is 2 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  50. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 3 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  51. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 3 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  52. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 3 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  53. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 4 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  54. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 4 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  55. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 5 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  56. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 5 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  57. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 5 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  58. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 6 The probability of catching is 7 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  59. Random Walk Synchronization a 2 2 b b 7 7 a a b 1 2 b a 7 7 The probability of catching is 1 b . aaa . ba . a . a . b Augmenting sequence w.r.t. α is The lengths of words in the augmenting sequence w.r.t. α is always at most n − 1 but there can be a-priori even exponential. The method can be extended to sets of words uW where u is a ‘’compressing words” and W is “complete” for < Q . u > keeping | uW | bound for augmenting words. Mikhail V. Berlinkov Paris, 2015 11 / 22

  60. Synchronizing Automata and Markov Chains Let A = ( Q , Σ) be a s.c. automaton. B. IJFCS 2012 The following are equivalent There is a p.d. π : Σ n − 1 �→ R + with the stationary distribution α of 1 the Markov chain M ( A n − 1 , π ) ; A is synchronizing and for each x / 2 ∈ < 1 n > there is a word u ∈ Σ n − 1 such that ( α u , x ) > ( α, x ) ; Corollary: Renew and generalize quadratic bounds on the r.t. for Eulerian and one-cluster case. Berlinkov, M; Szykuła, M; 2015 (submitted to MFCS) n log 3 n bound for the reset threshold of Prefix Code Automata. √ 6 n − 6. The ˇ 3 Cerný conjecture for automata with a letter of rank The previous bound is 1 + log 2 n . Mikhail V. Berlinkov Paris, 2015 12 / 22

  61. Synchronizing Automata and Markov Chains Let A = ( Q , Σ) be a s.c. automaton. B. IJFCS 2012 The following are equivalent There is a p.d. π : Σ n − 1 �→ R + with the stationary distribution α of 1 the Markov chain M ( A n − 1 , π ) ; A is synchronizing and for each x / 2 ∈ < 1 n > there is a word u ∈ Σ n − 1 such that ( α u , x ) > ( α, x ) ; Corollary: Renew and generalize quadratic bounds on the r.t. for Eulerian and one-cluster case. Berlinkov, M; Szykuła, M; 2015 (submitted to MFCS) n log 3 n bound for the reset threshold of Prefix Code Automata. √ 6 n − 6. The ˇ 3 Cerný conjecture for automata with a letter of rank The previous bound is 1 + log 2 n . Mikhail V. Berlinkov Paris, 2015 12 / 22

  62. Synchronizing Automata and Markov Chains Let A = ( Q , Σ) be a s.c. automaton. B. IJFCS 2012 The following are equivalent There is a p.d. π : Σ n − 1 �→ R + with the stationary distribution α of 1 the Markov chain M ( A n − 1 , π ) ; A is synchronizing and for each x / 2 ∈ < 1 n > there is a word u ∈ Σ n − 1 such that ( α u , x ) > ( α, x ) ; Corollary: Renew and generalize quadratic bounds on the r.t. for Eulerian and one-cluster case. Berlinkov, M; Szykuła, M; 2015 (submitted to MFCS) n log 3 n bound for the reset threshold of Prefix Code Automata. √ 6 n − 6. The ˇ 3 Cerný conjecture for automata with a letter of rank The previous bound is 1 + log 2 n . Mikhail V. Berlinkov Paris, 2015 12 / 22

  63. Synchronizing Automata and Markov Chains Let A = ( Q , Σ) be a s.c. automaton. B. IJFCS 2012 The following are equivalent There is a p.d. π : Σ n − 1 �→ R + with the stationary distribution α of 1 the Markov chain M ( A n − 1 , π ) ; A is synchronizing and for each x / 2 ∈ < 1 n > there is a word u ∈ Σ n − 1 such that ( α u , x ) > ( α, x ) ; Corollary: Renew and generalize quadratic bounds on the r.t. for Eulerian and one-cluster case. Berlinkov, M; Szykuła, M; 2015 (submitted to MFCS) n log 3 n bound for the reset threshold of Prefix Code Automata. √ 6 n − 6. The ˇ 3 Cerný conjecture for automata with a letter of rank The previous bound is 1 + log 2 n . Mikhail V. Berlinkov Paris, 2015 12 / 22

  64. Testing for Synchronization ˇ Cerný, 1964 A is synchronizing if and only if each pair of states p , q can be synchronized, i.e. p . v = q . v for some v ∈ Σ ∗ . The criterion yields O ( n 2 ) algorithm (basically due to Eppstein) which verifies whether or not A is synchronizing. Are there more effective (on average) algorithms? Mikhail V. Berlinkov Paris, 2015 13 / 22

  65. Testing for Synchronization ˇ Cerný, 1964 A is synchronizing if and only if each pair of states p , q can be synchronized, i.e. p . v = q . v for some v ∈ Σ ∗ . The criterion yields O ( n 2 ) algorithm (basically due to Eppstein) which verifies whether or not A is synchronizing. Are there more effective (on average) algorithms? Mikhail V. Berlinkov Paris, 2015 13 / 22

  66. Testing for Synchronization ˇ Cerný, 1964 A is synchronizing if and only if each pair of states p , q can be synchronized, i.e. p . v = q . v for some v ∈ Σ ∗ . The criterion yields O ( n 2 ) algorithm (basically due to Eppstein) which verifies whether or not A is synchronizing. Are there more effective (on average) algorithms? Mikhail V. Berlinkov Paris, 2015 13 / 22

  67. The probability of being synchronizable Let A = ( Q , Σ) be an n -state random automaton, that is, the actions of all k letters are chosen u.a.r. and independently from the set of all n n mappings. n ) for k = 2? [Cameron, 2011]. The probability is 1 − Θ( 1 B. 2013 in ArXiv The probability for automata of being synchronizable is 1 − O ( 1 n k / 2 ) and the bound is tight for the 2-letter alphabet case. B. 2013 in ArXiv Given a random n -state automaton, testing for synchronization can be done in O ( n ) expected time (and it is optimal). Connected case? Supposed bound is 1 − α n for some α < 1. 1 k -ary alphabet? Supposed bound is 1 − Θ( 1 / n k − 1 ) . 2 Mikhail V. Berlinkov Paris, 2015 14 / 22

  68. The probability of being synchronizable Let A = ( Q , Σ) be an n -state random automaton, that is, the actions of all k letters are chosen u.a.r. and independently from the set of all n n mappings. n ) for k = 2? [Cameron, 2011]. The probability is 1 − Θ( 1 B. 2013 in ArXiv The probability for automata of being synchronizable is 1 − O ( 1 n k / 2 ) and the bound is tight for the 2-letter alphabet case. B. 2013 in ArXiv Given a random n -state automaton, testing for synchronization can be done in O ( n ) expected time (and it is optimal). Connected case? Supposed bound is 1 − α n for some α < 1. 1 k -ary alphabet? Supposed bound is 1 − Θ( 1 / n k − 1 ) . 2 Mikhail V. Berlinkov Paris, 2015 14 / 22

  69. The probability of being synchronizable Let A = ( Q , Σ) be an n -state random automaton, that is, the actions of all k letters are chosen u.a.r. and independently from the set of all n n mappings. n ) for k = 2? [Cameron, 2011]. The probability is 1 − Θ( 1 B. 2013 in ArXiv The probability for automata of being synchronizable is 1 − O ( 1 n k / 2 ) and the bound is tight for the 2-letter alphabet case. B. 2013 in ArXiv Given a random n -state automaton, testing for synchronization can be done in O ( n ) expected time (and it is optimal). Connected case? Supposed bound is 1 − α n for some α < 1. 1 k -ary alphabet? Supposed bound is 1 − Θ( 1 / n k − 1 ) . 2 Mikhail V. Berlinkov Paris, 2015 14 / 22

  70. The probability of being synchronizable Let A = ( Q , Σ) be an n -state random automaton, that is, the actions of all k letters are chosen u.a.r. and independently from the set of all n n mappings. n ) for k = 2? [Cameron, 2011]. The probability is 1 − Θ( 1 B. 2013 in ArXiv The probability for automata of being synchronizable is 1 − O ( 1 n k / 2 ) and the bound is tight for the 2-letter alphabet case. B. 2013 in ArXiv Given a random n -state automaton, testing for synchronization can be done in O ( n ) expected time (and it is optimal). Connected case? Supposed bound is 1 − α n for some α < 1. 1 k -ary alphabet? Supposed bound is 1 − Θ( 1 / n k − 1 ) . 2 Mikhail V. Berlinkov Paris, 2015 14 / 22

  71. The probability of being synchronizable Let A = ( Q , Σ) be an n -state random automaton, that is, the actions of all k letters are chosen u.a.r. and independently from the set of all n n mappings. n ) for k = 2? [Cameron, 2011]. The probability is 1 − Θ( 1 B. 2013 in ArXiv The probability for automata of being synchronizable is 1 − O ( 1 n k / 2 ) and the bound is tight for the 2-letter alphabet case. B. 2013 in ArXiv Given a random n -state automaton, testing for synchronization can be done in O ( n ) expected time (and it is optimal). Connected case? Supposed bound is 1 − α n for some α < 1. 1 k -ary alphabet? Supposed bound is 1 − Θ( 1 / n k − 1 ) . 2 Mikhail V. Berlinkov Paris, 2015 14 / 22

  72. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  73. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  74. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  75. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  76. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  77. Expected Reset Threshold Let A be a random n -state synchronizing automaton. What is the expected reset threshold of A ? Experiments show that the expected reset threshold is in Ω( 2 . 5 √ n ) [Kisielewicz, Kowalski, Szykuła 2012]. Nycaud, 2014 For each 0 < ǫ < 1 / 8 a random binary n -state automaton has a reset word of length at most n 1 + ǫ with probability 1 − O ( n − 1 8 + ǫ ) . This yields O ( n 2 . 875 ) upper bound on the expected reset threshold. Corollary; B., Szykuła, 2015 (submitted to MFCS) The expected value of the reset threshold is at most n 7 / 4 + o ( 1 ) . We guess the bound can be improved to n 1 + o ( 1 ) . Mikhail V. Berlinkov Paris, 2015 15 / 22

  78. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  79. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  80. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  81. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  82. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  83. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  84. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

  85. Hardness of Computing a Reset Threshold Given a k -letter n -state synchronizing automaton A , compute its reset threshold. Unless P = NP , there are no polynomial-time algorithm for the following approximation. exactly [Rystsov, 1980; Eppstein, 1990], within any constant factor for k = 2 [B. CSR, 2010], within c log n for k ↑ [Gerbush, Heeringa, 2011], within 0 . 5 c log n for k = 2 [B. 2013] within n ǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which n ǫ -approximation is possible? Mikhail V. Berlinkov Paris, 2015 16 / 22

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