synchronizing words for probabilistic automata laurent doyen lsv, ens cachan & cnrs thierry...
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![Page 1: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/1.jpg)
Synchronizing Words for Probabilistic Automata
Laurent Doyen
LSV, ENS Cachan & CNRS
Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles
5th Gasics meeting
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Example [AV04]
Block factory
conveyor belt storage
![Page 3: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/3.jpg)
Block factory
conveyor belt
High
Low
storage
Example [AV04]
![Page 4: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/4.jpg)
Example
Block factory
conveyor belt
High
Low
H
HH
H
![Page 5: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/5.jpg)
Example
Block factory
conveyor belt
High
Low
L
L
LL
![Page 6: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/6.jpg)
Example
Block factory
conveyor belt
High
Low
H,L
HH
H
L
LL
Deterministic finite
automaton
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{ , , , }
Example
Block factoryH,L
HH
H
L
LL
• no sensor
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{ , , , }
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
{ }
w ∈
{H,L}*
∈
{H,L}*
![Page 9: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/9.jpg)
{ , , , }
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
{ }
The word w is synchronizing: no matter the initial state, the automaton ends up in a singleton
w ∈
{H,L}*
Reachability in subset construction
∈
{H,L}*
![Page 10: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/10.jpg)
Example
Block factory
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H,L
HH
H
L
LL
H
![Page 11: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/11.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 12: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/12.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 13: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/13.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 14: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/14.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 15: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/15.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 16: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/16.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 17: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/17.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 18: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/18.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing word
H
![Page 19: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/19.jpg)
Example
Block factoryH,L
HH
H
L
LL
• no sensor
• robust control: w
w = L H H H L H H H L
synchronizing wordExistence of a synchronising word can be decided in PTIMECerny’64
H
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Applications
• Discrete-event systems
• Planning
• Biocomputing
• Robotics
Robust control, reset from unknown state
See [Vol08]
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Probabilistic systems
Block factory
conveyor belt storage
.5 .5
High
Low
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Probabilistic automata
H,L
HH
H
L
LL
.5 .5
Probabilistic automaton
What is a synchronizing word for probabilistic automata ?
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Probabilistic automata
Outcome of a word:
w = aaba …
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Probabilistic automata
Outcome of a word:
w = aaba …
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Probabilistic automata
Outcome of a word:
w = aaba …
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Probabilistic automata
Outcome of a word:
w = aaba …
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Probabilistic automata
Outcome of a word:
w = aaba …
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Probabilistic automata
An infinite word
What is a synchronizing word for probabilistic automata ?
The probability mass tends to accumulate in a single state.
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Probabilistic automata
Outcome of a word:
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Probabilistic automata
Outcome of a word:
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Probabilistic automata
Outcome of a word:
![Page 32: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/32.jpg)
Probabilistic automata
Outcome of a word:
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Probabilistic automata
What is a synchronizing word for probabilistic automata ?
Outcome of a word:
where
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Probabilistic automata
What is a synchronizing word for probabilistic automata ?
Outcome of a word:
is synchronizing if
where
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Synchronizing words
Two variants:
strongly synchronizing
weakly synchronizing
![Page 36: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/36.jpg)
Decision Problems
![Page 37: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/37.jpg)
Decision problems
• Emptiness
• Universality
Does there exist a synchronizing word ?
Are all words synchronizing ?
Note: we consider randomized words
Words are called pure words.
![Page 38: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/38.jpg)
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
H,L
HH
H
L
LL
H,L
HH
H
L
LL
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Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-complete
![Page 40: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/40.jpg)
Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-complete
![Page 41: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/41.jpg)
Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-complete
witness sequence
![Page 42: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/42.jpg)
Emptiness problem
States in the witness sequence have exactly one successor
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Emptiness problem
States in the witness sequence have exactly one successor
All other states have to inject some probability in the witness sequence
![Page 44: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/44.jpg)
Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-completePSPACE upper bound: emptiness of a Büchi automaton
subset sonctructionobligation setwitness sequence
Büchi condition: o is empty infinitely often
![Page 45: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/45.jpg)
Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-completePSPACE upper bound: emptiness of a Büchi automaton
subset sonctructionobligation setwitness sequence
Büchi condition: o is empty infinitely often
![Page 46: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/46.jpg)
Emptiness problem
Does there exist a synchronizing word ?• Pure words are sufficient
• The emptiness problem is PSPACE-completePSPACE lower bound: universality of NFA
![Page 47: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/47.jpg)
Decision problems
• Emptiness
• Universality
Does there exist a synchronizing word ?
Are all words synchronizing ?
Note: we consider randomized words
Words are called pure words.
![Page 48: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/48.jpg)
Universality problem
Are all words synchronizing ?
All pure words are synchronizing,
not all randomized words.
• Pure words are not sufficient
![Page 49: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/49.jpg)
Universality problem
Are all words synchronizing ?
is not synchronizing
All pure words are synchronizing,
not all randomized words.
• Pure words are not sufficient
• The uniformly randomized word is not sufficient
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Universality problem
Are all words synchronizing ?
No, if there are two absorbing components.
(from which there is a word to stay inside).
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Universality problem
Are all words synchronizing ?
If there is only one absorbing component, then it is sufficient to check whether the uniformly randomized word is synchronizing.
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Universality problem
Are all words synchronizing ?
If there is only one absorbing component, then it is sufficient to check whether the uniformly randomized word is synchronizing.
Uniformly randomized word is synchronizing.
![Page 53: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/53.jpg)
Universality problem
Are all words synchronizing ?
If there is only one absorbing component, then it is sufficient to check whether the uniformly randomized word is synchronizing.
Uniformly randomized word is synchronizing.
universal
not universal
![Page 54: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/54.jpg)
Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
• existence of absorbing component, check in PSPACE
• whether unif. rand. word is synchronizing, check in PTIME
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Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
• existence of absorbing component, check in PSPACE
• whether unif. rand. word is synchronizing, check in PTIME
- Guess component U Q
- Guess state q U and finite word w
- Check that all runs from q on w stay in U
U
![Page 56: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/56.jpg)
Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
• existence of absorbing component, check in PSPACE
• whether unif. rand. word is synchronizing, check in PTIME
- Guess component U Q
- Guess state q U and finite word w
- Check that all runs from q on w stay in U
q
U
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Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
• existence of absorbing component, check in PSPACE
• whether unif. rand. word is synchronizing, check in PTIME
- Guess component U Q
- Guess state q U and finite word w
- Check that all runs from q on w stay in U
q
U
w
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Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
Towards PSPACE-hardness ?
• Nondeterm. Probabilistic
• Accepting prob. ½ to leave
There exists a state q and a word w such that all runs of from q on w avoid accepting states iff is not universal.
![Page 59: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/59.jpg)
Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
Towards PSPACE-hardness ?
• Nondeterm. Probabilistic
• Accepting prob. ½ to leave
There exists a state q and a word w such that all runs of from q on w avoid accepting states iff is not universal.
Existential blind safety game
Positive coBüchi automaton
![Page 60: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/60.jpg)
Universality problem
Are all words synchronizing ?The universality problem is in PSPACE.
Towards PSPACE-hardness ?
• Nondeterm. Probabilistic
• Accepting prob. ½ to leave
There exists a state q and a word w such that all runs of from q on w avoid accepting states iff is not universal.
Existential blind safety game
PSPACE-hard ?Positive coBüchi
automaton
PSPACE-hard ?
![Page 61: Synchronizing Words for Probabilistic Automata Laurent Doyen LSV, ENS Cachan & CNRS Thierry Massart, Mahsa Shirmohammadi Université Libre de Bruxelles](https://reader036.vdocuments.us/reader036/viewer/2022062409/56649cfa5503460f949cbea5/html5/thumbnails/61.jpg)
Summary
• Infinite synchronizing words for PA
• Generalizes finite sync. words for DFA
• Emptiness is PSPACE-complete
• Universality is in PSPACE – lower bound ?
Outlook
• Labeled automata, MDPs
• Universality in pure words
• Optimal synchronization
• Stochastic games
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Thank you !
Questions ?
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Probabilistic automata
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References
[AV04] D. S. Ananichev and M. V. Volkov. Synchronizing Monotonic Automata. Theor. Comput. Sci. 327(3): 225-239 (2004)
[Vol08] M. V. Volkov. Synchronizing Automata and the Cerny Conjecture. LATA 2008: 11-27