Synchronizing Words for Probabilistic Automata

1. Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles 5th Gasics meeting

2. Example [AV04] Block factory conveyor belt storage

3. Example [AV04] Block factory High Low conveyor belt storage

4. Example Block factory H High Low H H H conveyor belt

5. Example L Block factory L High Low L L conveyor belt

6. Example L Block factory H,L High Low H H H L L Deterministic finite automaton conveyor belt

7. Example L Block factory H,L • no sensor H H { , , , } H L L

8. Example L Block factory H,L • no sensor • robust control: w H ∈ {H,L}* H { , , , } H L L w ∈ {H,L}* { }

9. Example L Block factory H,L • no sensor • robust control: w H ∈ {H,L}* H { , , , } H L L w ∈ {H,L}* The word w is synchronizing: no matter the initial state, the automaton ends up in a singleton { } Reachability in subset construction

10. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

11. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

12. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

13. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

14. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

15. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

16. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

17. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

18. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL synchronizing word

19. Example L Block factory H,L • no sensor • robust control: w H H H L L H w =L HHHL HHHL Existence of a synchronising word can be decided in PTIME synchronizing word Cerny’64

20. Applications Robust control, reset from unknown state • Discrete-event systems • Planning • Biocomputing • Robotics See [Vol08]

21. Probabilistic systems Block factory High Low .5 .5 conveyor belt storage

22. Probabilistic automata L H,L H .5 H .5 H L L Probabilistic automaton What is a synchronizing word for probabilistic automata ?

23. Probabilistic automata Outcome of a word: w = aaba …

24. Probabilistic automata Outcome of a word: w = aaba …

25. Probabilistic automata Outcome of a word: w = aaba …

26. Probabilistic automata Outcome of a word: w = aaba …

27. Probabilistic automata Outcome of a word: w = aaba …

28. Probabilistic automata What is a synchronizing word for probabilistic automata ? An infinite word The probability mass tends to accumulate in a single state.

29. Probabilistic automata Outcome of a word:

30. Probabilistic automata Outcome of a word:

31. Probabilistic automata Outcome of a word:

32. Probabilistic automata Outcome of a word:

33. Probabilistic automata What is a synchronizing word for probabilistic automata ? Outcome of a word: where

34. Probabilistic automata What is a synchronizing word for probabilistic automata ? Outcome of a word: is synchronizing if where

35. Synchronizing words Two variants: strongly synchronizing weakly synchronizing

36. Decision Problems

37. Decision problems • Emptiness Does there exist a synchronizing word ? • Universality Are all words synchronizing ? Note: we consider randomized words Words are called pure words.

38. Synchronizing words for DFA If we view DFA as special case of probabilistic automata: there exists a synchronizing (finite) word for DFA A iff there exists a synchronizing (infinite) word for A with . uniform initial distribution L L H,L H,L H H H H H H L L L L

39. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete

40. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete

41. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete witness sequence

42. Emptiness problem States in the witness sequence have exactly one successor

43. Emptiness problem States in the witness sequence have exactly one successor All other states have to inject some probability in the witness sequence

44. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete PSPACE upper bound: emptiness of a Büchi automaton subset sonctruction obligation set witness sequence Büchi condition: o is empty infinitely often

45. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete PSPACE upper bound: emptiness of a Büchi automaton subset sonctruction obligation set witness sequence Büchi condition: o is empty infinitely often

46. Emptiness problem Does there exist a synchronizing word ? • Pure words are sufficient • The emptiness problem is PSPACE-complete PSPACE lower bound: universality of NFA

47. Decision problems • Emptiness Does there exist a synchronizing word ? • Universality Are all words synchronizing ? Note: we consider randomized words Words are called pure words.

48. Universality problem Are all words synchronizing ? • Pure words are not sufficient All pure words are synchronizing, not all randomized words.

49. Universality problem Are all words synchronizing ? • Pure words are not sufficient All pure words are synchronizing, not all randomized words. • The uniformly randomized word is not sufficient is not synchronizing

50. Universality problem Are all words synchronizing ? No, if there are two absorbing components. (from which there is a word to stay inside).