Various Aspects of Automaton Synchronization Mikhail V. Berlinkov, - - PowerPoint PPT Presentation

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Various Aspects of Automaton Synchronization Mikhail V. Berlinkov, - - PowerPoint PPT Presentation

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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

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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

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SLIDE 2

Deterministic Finite Automata and their Graphs

By deterministic finite automaton (DFA) A we mean Q, Σ, where Q is the state set and Σ is the alphabet; each a ∈ Σ is a mapping from Q to Q. The underlying graph of each letter a ∈ Σ defined as UG(a) = (Q, {(p, p.a) | p ∈ Q}) consists of one or more connected components called clusters. The underlying graph of A is the edge union of the underlying graphs of its letters. Automata are usually classified by their underlying graphs. Examples: circular, one-cluster, Eulerian, etc.

Mikhail V. Berlinkov Paris, 2015 2 / 22

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SLIDE 3

Deterministic Finite Automata and their Graphs

By deterministic finite automaton (DFA) A we mean Q, Σ, where Q is the state set and Σ is the alphabet; each a ∈ Σ is a mapping from Q to Q. The underlying graph of each letter a ∈ Σ defined as UG(a) = (Q, {(p, p.a) | p ∈ Q}) consists of one or more connected components called clusters. The underlying graph of A is the edge union of the underlying graphs of its letters. Automata are usually classified by their underlying graphs. Examples: circular, one-cluster, Eulerian, etc.

Mikhail V. Berlinkov Paris, 2015 2 / 22

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SLIDE 4

Deterministic Finite Automata and their Graphs

By deterministic finite automaton (DFA) A we mean Q, Σ, where Q is the state set and Σ is the alphabet; each a ∈ Σ is a mapping from Q to Q. The underlying graph of each letter a ∈ Σ defined as UG(a) = (Q, {(p, p.a) | p ∈ Q}) consists of one or more connected components called clusters. The underlying graph of A is the edge union of the underlying graphs of its letters. Automata are usually classified by their underlying graphs. Examples: circular, one-cluster, Eulerian, etc.

Mikhail V. Berlinkov Paris, 2015 2 / 22

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SLIDE 5

Deterministic Finite Automata and their Graphs

By deterministic finite automaton (DFA) A we mean Q, Σ, where Q is the state set and Σ is the alphabet; each a ∈ Σ is a mapping from Q to Q. The underlying graph of each letter a ∈ Σ defined as UG(a) = (Q, {(p, p.a) | p ∈ Q}) consists of one or more connected components called clusters. The underlying graph of A is the edge union of the underlying graphs of its letters. Automata are usually classified by their underlying graphs. Examples: circular, one-cluster, Eulerian, etc.

Mikhail V. Berlinkov Paris, 2015 2 / 22

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SLIDE 6

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 7

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 8

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 9

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 10

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 11

Synchronizing Automata

The set of words Σ∗ corresponds to the transformation monoid. A word v is reset for A if it is a constant mapping, that is, q.v = p.v for each p, q ∈ Q. In other words, each path labeled by v leads to a particular state. A is called synchronizing if it possesses a reset word. The minimum length of reset words for A is called its reset threshold. Applications: coding theory, data transmission, robotics, software verification, dna-computing, symbolic dynamics, etc.

Mikhail V. Berlinkov Paris, 2015 3 / 22

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SLIDE 12

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

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SLIDE 13

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

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SLIDE 14

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

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SLIDE 15

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

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SLIDE 16

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

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SLIDE 17

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

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SLIDE 18

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

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SLIDE 19

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

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SLIDE 20

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-21
SLIDE 21

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-22
SLIDE 22

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-23
SLIDE 23

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-24
SLIDE 24

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-25
SLIDE 25

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-26
SLIDE 26

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-27
SLIDE 27

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-28
SLIDE 28

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-29
SLIDE 29

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-30
SLIDE 30

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-31
SLIDE 31

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-32
SLIDE 32

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-33
SLIDE 33

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-34
SLIDE 34

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-35
SLIDE 35

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

slide-36
SLIDE 36

Greedy compressing algorithm for synchronization

1 2 3 4 b b b a a a 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 ba3ba3b whence rt(A ) = 9 < |v|.

Mikhail V. Berlinkov Paris, 2015 5 / 22

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SLIDE 37

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

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SLIDE 38

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

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SLIDE 39

The ˇ Cerný conjecture

ˇ Cerný, 1964

For each n there is an n-state automaton Cn with rt(Cn) = (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(n3/2) for the reset threshold.

Pin, 1983 (based on a combinatorial result of Frankl, 1982)

Each n-state automaton has a reset word of length (n3 − n)/6. Quadratic upper bounds on the reset threshold?

Mikhail V. Berlinkov Paris, 2015 7 / 22

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SLIDE 40

The ˇ Cerný conjecture

ˇ Cerný, 1964

For each n there is an n-state automaton Cn with rt(Cn) = (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(n3/2) for the reset threshold.

Pin, 1983 (based on a combinatorial result of Frankl, 1982)

Each n-state automaton has a reset word of length (n3 − n)/6. Quadratic upper bounds on the reset threshold?

Mikhail V. Berlinkov Paris, 2015 7 / 22

slide-41
SLIDE 41

The ˇ Cerný conjecture

ˇ Cerný, 1964

For each n there is an n-state automaton Cn with rt(Cn) = (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(n3/2) for the reset threshold.

Pin, 1983 (based on a combinatorial result of Frankl, 1982)

Each n-state automaton has a reset word of length (n3 − n)/6. Quadratic upper bounds on the reset threshold?

Mikhail V. Berlinkov Paris, 2015 7 / 22

slide-42
SLIDE 42

The ˇ Cerný conjecture

ˇ Cerný, 1964

For each n there is an n-state automaton Cn with rt(Cn) = (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(n3/2) for the reset threshold.

Pin, 1983 (based on a combinatorial result of Frankl, 1982)

Each n-state automaton has a reset word of length (n3 − n)/6. Quadratic upper bounds on the reset threshold?

Mikhail V. Berlinkov Paris, 2015 7 / 22

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SLIDE 43

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

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SLIDE 44

Example from the Italian Job Movie

Mikhail V. Berlinkov Paris, 2015 9 / 22

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SLIDE 45

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

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SLIDE 46

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-47
SLIDE 47

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-48
SLIDE 48

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-49
SLIDE 49

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-50
SLIDE 50

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-51
SLIDE 51

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-52
SLIDE 52

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-53
SLIDE 53

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-54
SLIDE 54

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-55
SLIDE 55

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-56
SLIDE 56

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-57
SLIDE 57

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-58
SLIDE 58

Kari Automaton and Greedy Extension Method

1 2 5 4 3 a a a a a a b b b b b b A reset word is the reverse to v = baabbbabbaab... Augmenting sequence is v1 = b, v2 = aabb, v3 = babbaab, v4 = . . .. 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

slide-59
SLIDE 59

Random Walk Synchronization

1 2 3 4 b b b a a a b a The probability of catching is Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-60
SLIDE 60

Random Walk Synchronization

1 2 3 4 b | π1(b) b | π2(b) b | π3(b) a | π1(a) a | π2(a) a | π3(a) b | π4(b) a | π4(a) The probability of catching is Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-61
SLIDE 61

Random Walk Synchronization

1 2 3 4

1 2 1 2 1 2 1 2 1 2 1 2

1 The probability of catching is Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-62
SLIDE 62

Random Walk Synchronization

2 7 2 7 2 7 1 7 1 2 1 2 1 2 1 2 1 2 1 2

1 The probability of catching is Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-63
SLIDE 63

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-64
SLIDE 64

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is 2

7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-65
SLIDE 65

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

3 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-66
SLIDE 66

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

3 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-67
SLIDE 67

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

3 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-68
SLIDE 68

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

4 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-69
SLIDE 69

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

4 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-70
SLIDE 70

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

5 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-71
SLIDE 71

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

5 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-72
SLIDE 72

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

5 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-73
SLIDE 73

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is

6 7

Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-74
SLIDE 74

Random Walk Synchronization

2 7 2 7 2 7 1 7

b b b a a a b a The probability of catching is 1 Augmenting sequence w.r.t. α is b.aaa.ba.a.a.b 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

slide-75
SLIDE 75

Synchronizing Automata and Markov Chains

Let A = (Q, Σ) be a s.c. automaton.

  • B. IJFCS 2012

The following are equivalent

1

There is a p.d. π : Σn−1 → R+ with the stationary distribution α of the Markov chain M (A n−1, π);

2

A is synchronizing and for each x / ∈< 1n > there is a word u ∈ Σn−1 such that (αu, 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 log3 n bound for the reset threshold of Prefix Code Automata. The ˇ Cerný conjecture for automata with a letter of rank

3

√ 6n − 6. The previous bound is 1 + log2 n.

Mikhail V. Berlinkov Paris, 2015 12 / 22

slide-76
SLIDE 76

Synchronizing Automata and Markov Chains

Let A = (Q, Σ) be a s.c. automaton.

  • B. IJFCS 2012

The following are equivalent

1

There is a p.d. π : Σn−1 → R+ with the stationary distribution α of the Markov chain M (A n−1, π);

2

A is synchronizing and for each x / ∈< 1n > there is a word u ∈ Σn−1 such that (αu, 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 log3 n bound for the reset threshold of Prefix Code Automata. The ˇ Cerný conjecture for automata with a letter of rank

3

√ 6n − 6. The previous bound is 1 + log2 n.

Mikhail V. Berlinkov Paris, 2015 12 / 22

slide-77
SLIDE 77

Synchronizing Automata and Markov Chains

Let A = (Q, Σ) be a s.c. automaton.

  • B. IJFCS 2012

The following are equivalent

1

There is a p.d. π : Σn−1 → R+ with the stationary distribution α of the Markov chain M (A n−1, π);

2

A is synchronizing and for each x / ∈< 1n > there is a word u ∈ Σn−1 such that (αu, 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 log3 n bound for the reset threshold of Prefix Code Automata. The ˇ Cerný conjecture for automata with a letter of rank

3

√ 6n − 6. The previous bound is 1 + log2 n.

Mikhail V. Berlinkov Paris, 2015 12 / 22

slide-78
SLIDE 78

Synchronizing Automata and Markov Chains

Let A = (Q, Σ) be a s.c. automaton.

  • B. IJFCS 2012

The following are equivalent

1

There is a p.d. π : Σn−1 → R+ with the stationary distribution α of the Markov chain M (A n−1, π);

2

A is synchronizing and for each x / ∈< 1n > there is a word u ∈ Σn−1 such that (αu, 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 log3 n bound for the reset threshold of Prefix Code Automata. The ˇ Cerný conjecture for automata with a letter of rank

3

√ 6n − 6. The previous bound is 1 + log2 n.

Mikhail V. Berlinkov Paris, 2015 12 / 22

slide-79
SLIDE 79

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(n2) 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

slide-80
SLIDE 80

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(n2) 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

slide-81
SLIDE 81

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(n2) 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

slide-82
SLIDE 82

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 nn mappings. The probability is 1 − Θ( 1

n) for k = 2? [Cameron, 2011].

  • B. 2013 in ArXiv

The probability for automata of being synchronizable is 1 − O(

1 nk/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).

1

Connected case? Supposed bound is 1 − αn for some α < 1.

2

k-ary alphabet? Supposed bound is 1 − Θ(1/nk−1).

Mikhail V. Berlinkov Paris, 2015 14 / 22

slide-83
SLIDE 83

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 nn mappings. The probability is 1 − Θ( 1

n) for k = 2? [Cameron, 2011].

  • B. 2013 in ArXiv

The probability for automata of being synchronizable is 1 − O(

1 nk/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).

1

Connected case? Supposed bound is 1 − αn for some α < 1.

2

k-ary alphabet? Supposed bound is 1 − Θ(1/nk−1).

Mikhail V. Berlinkov Paris, 2015 14 / 22

slide-84
SLIDE 84

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 nn mappings. The probability is 1 − Θ( 1

n) for k = 2? [Cameron, 2011].

  • B. 2013 in ArXiv

The probability for automata of being synchronizable is 1 − O(

1 nk/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).

1

Connected case? Supposed bound is 1 − αn for some α < 1.

2

k-ary alphabet? Supposed bound is 1 − Θ(1/nk−1).

Mikhail V. Berlinkov Paris, 2015 14 / 22

slide-85
SLIDE 85

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 nn mappings. The probability is 1 − Θ( 1

n) for k = 2? [Cameron, 2011].

  • B. 2013 in ArXiv

The probability for automata of being synchronizable is 1 − O(

1 nk/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).

1

Connected case? Supposed bound is 1 − αn for some α < 1.

2

k-ary alphabet? Supposed bound is 1 − Θ(1/nk−1).

Mikhail V. Berlinkov Paris, 2015 14 / 22

slide-86
SLIDE 86

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 nn mappings. The probability is 1 − Θ( 1

n) for k = 2? [Cameron, 2011].

  • B. 2013 in ArXiv

The probability for automata of being synchronizable is 1 − O(

1 nk/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).

1

Connected case? Supposed bound is 1 − αn for some α < 1.

2

k-ary alphabet? Supposed bound is 1 − Θ(1/nk−1).

Mikhail V. Berlinkov Paris, 2015 14 / 22

slide-87
SLIDE 87

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-88
SLIDE 88

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-89
SLIDE 89

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-90
SLIDE 90

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-91
SLIDE 91

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-92
SLIDE 92

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 n1+ǫ with probability 1 − O(n− 1

8 +ǫ).

This yields O(n2.875) upper bound on the expected reset threshold.

Corollary; B., Szykuła, 2015 (submitted to MFCS)

The expected value of the reset threshold is at most n7/4+o(1). We guess the bound can be improved to n1+o(1).

Mikhail V. Berlinkov Paris, 2015 15 / 22

slide-93
SLIDE 93

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-94
SLIDE 94

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-95
SLIDE 95

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-96
SLIDE 96

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-97
SLIDE 97

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-98
SLIDE 98

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-99
SLIDE 99

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-100
SLIDE 100

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 0.5c log n for k = 2 [B. 2013] within nǫ for k = 2 and certain ǫ > 0 [Gawrychovski, 2014]. What is the minimum of ǫ ≤ 1 for which nǫ-approximation is possible?

Mikhail V. Berlinkov Paris, 2015 16 / 22

slide-101
SLIDE 101

Finding Reset Words of Prescribed Lengths

Given a k-letter n-state synchronizing automaton A such that rt(A ) ≤ L, return a reset word of length at most L. Greedy compression algorithm for the general case; Particular classes for which the proof is constructive and polynomial;

B., Szykuła, 2015 (submitted)

Polynomial algorithms for (Quasi-)Eulerian, (Quasi-)One-Cluster and Prefix code automata. Given a k-letter n-state circular synchronizing automaton, return a reset word of length at most (n − 1)2.

Mikhail V. Berlinkov Paris, 2015 17 / 22

slide-102
SLIDE 102

Finding Reset Words of Prescribed Lengths

Given a k-letter n-state synchronizing automaton A such that rt(A ) ≤ L, return a reset word of length at most L. Greedy compression algorithm for the general case; Particular classes for which the proof is constructive and polynomial;

B., Szykuła, 2015 (submitted)

Polynomial algorithms for (Quasi-)Eulerian, (Quasi-)One-Cluster and Prefix code automata. Given a k-letter n-state circular synchronizing automaton, return a reset word of length at most (n − 1)2.

Mikhail V. Berlinkov Paris, 2015 17 / 22

slide-103
SLIDE 103

Finding Reset Words of Prescribed Lengths

Given a k-letter n-state synchronizing automaton A such that rt(A ) ≤ L, return a reset word of length at most L. Greedy compression algorithm for the general case; Particular classes for which the proof is constructive and polynomial;

B., Szykuła, 2015 (submitted)

Polynomial algorithms for (Quasi-)Eulerian, (Quasi-)One-Cluster and Prefix code automata. Given a k-letter n-state circular synchronizing automaton, return a reset word of length at most (n − 1)2.

Mikhail V. Berlinkov Paris, 2015 17 / 22

slide-104
SLIDE 104

Finding Reset Words of Prescribed Lengths

Given a k-letter n-state synchronizing automaton A such that rt(A ) ≤ L, return a reset word of length at most L. Greedy compression algorithm for the general case; Particular classes for which the proof is constructive and polynomial;

B., Szykuła, 2015 (submitted)

Polynomial algorithms for (Quasi-)Eulerian, (Quasi-)One-Cluster and Prefix code automata. Given a k-letter n-state circular synchronizing automaton, return a reset word of length at most (n − 1)2.

Mikhail V. Berlinkov Paris, 2015 17 / 22

slide-105
SLIDE 105

Finding Reset Words of Prescribed Lengths

Given a k-letter n-state synchronizing automaton A such that rt(A ) ≤ L, return a reset word of length at most L. Greedy compression algorithm for the general case; Particular classes for which the proof is constructive and polynomial;

B., Szykuła, 2015 (submitted)

Polynomial algorithms for (Quasi-)Eulerian, (Quasi-)One-Cluster and Prefix code automata. Given a k-letter n-state circular synchronizing automaton, return a reset word of length at most (n − 1)2.

Mikhail V. Berlinkov Paris, 2015 17 / 22

slide-106
SLIDE 106

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-107
SLIDE 107

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-108
SLIDE 108

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-109
SLIDE 109

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-110
SLIDE 110

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-111
SLIDE 111

Road Coloring Problem

Let A be a (non-synchronizing) automaton. Is there a synchronizing automaton B with the same underlying graph as A ?

Road Coloring Problem [Adler, Goodwin, Weiss, 1977]

Does each strongly-connected aperiodic graph (AGW-graph) have a synchronizing coloring?

Trahtman, 2007

Each AGW-graph has a synchronizing coloring. Proof sketch: Find a coloring s.t. one letter has a unique highest tree; Find a stable pair of states at the bottom of this tree. Consider the factor automaton with respect to the stability relation.

Mikhail V. Berlinkov Paris, 2015 18 / 22

slide-112
SLIDE 112

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-113
SLIDE 113

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-114
SLIDE 114

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-115
SLIDE 115

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-116
SLIDE 116

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-117
SLIDE 117

Computing Synchronizing Coloring

Let A be an n-state automaton with AGW-graph. How complicated to find a synchronizing coloring?

Trahtman, 2008

Cubic time algorithm.

Béal, Perrin, 2008

Quadratic time algorithm. How complicated to find an optimal synchronizing coloring?

  • B. 2009, Applied Discrete Mathematics J. (in Russian)

No polynomial time algorithm can approximate this problem within a constant factor less than 2. Complexity of approximation within factor 2?

Mikhail V. Berlinkov Paris, 2015 19 / 22

slide-118
SLIDE 118

Synchronization and Prediction Rates

Let A be a s.c. automaton equipped with transition probabilities defined for each state independently. If there are no pairs with equivalent probability future, A is called an ǫ-machine.

Travers, N.; Crutchfield, P; 2011

Let pj(u) be the probability of the most probable state if u ∈ Σj is generated by A . Then for some 0 < a, b < 1 If A is synchronizing then Pr(pj < 1) ≤ O(aL) - exact; If A is not synchronizing, Pr(pj < 1 − bL) → 0 - asymptotic. The infinum of such a and b are called synchronization rate and prediction rate constants resp.

  • B. 2014 (in ArXiv)

The synchronization and prediction rate constants can be approximated in polynomial time with any given precision.

Mikhail V. Berlinkov Paris, 2015 20 / 22

slide-119
SLIDE 119

Synchronization and Prediction Rates

Let A be a s.c. automaton equipped with transition probabilities defined for each state independently. If there are no pairs with equivalent probability future, A is called an ǫ-machine.

Travers, N.; Crutchfield, P; 2011

Let pj(u) be the probability of the most probable state if u ∈ Σj is generated by A . Then for some 0 < a, b < 1 If A is synchronizing then Pr(pj < 1) ≤ O(aL) - exact; If A is not synchronizing, Pr(pj < 1 − bL) → 0 - asymptotic. The infinum of such a and b are called synchronization rate and prediction rate constants resp.

  • B. 2014 (in ArXiv)

The synchronization and prediction rate constants can be approximated in polynomial time with any given precision.

Mikhail V. Berlinkov Paris, 2015 20 / 22

slide-120
SLIDE 120

Synchronization and Prediction Rates

Let A be a s.c. automaton equipped with transition probabilities defined for each state independently. If there are no pairs with equivalent probability future, A is called an ǫ-machine.

Travers, N.; Crutchfield, P; 2011

Let pj(u) be the probability of the most probable state if u ∈ Σj is generated by A . Then for some 0 < a, b < 1 If A is synchronizing then Pr(pj < 1) ≤ O(aL) - exact; If A is not synchronizing, Pr(pj < 1 − bL) → 0 - asymptotic. The infinum of such a and b are called synchronization rate and prediction rate constants resp.

  • B. 2014 (in ArXiv)

The synchronization and prediction rate constants can be approximated in polynomial time with any given precision.

Mikhail V. Berlinkov Paris, 2015 20 / 22

slide-121
SLIDE 121

Synchronization and Prediction Rates

Let A be a s.c. automaton equipped with transition probabilities defined for each state independently. If there are no pairs with equivalent probability future, A is called an ǫ-machine.

Travers, N.; Crutchfield, P; 2011

Let pj(u) be the probability of the most probable state if u ∈ Σj is generated by A . Then for some 0 < a, b < 1 If A is synchronizing then Pr(pj < 1) ≤ O(aL) - exact; If A is not synchronizing, Pr(pj < 1 − bL) → 0 - asymptotic. The infinum of such a and b are called synchronization rate and prediction rate constants resp.

  • B. 2014 (in ArXiv)

The synchronization and prediction rate constants can be approximated in polynomial time with any given precision.

Mikhail V. Berlinkov Paris, 2015 20 / 22

slide-122
SLIDE 122

Markov Chain Convergence vs Reset Threshold

Let u ∈ Σj be a randomly generated word by ǫ-machine A and p ∈ Q and j ≥ n − 1. Then rt(A ) ≤ j if either Pr(u is reset ) > 0 or

  • q∈Q Pr(p.u = q.u) < 1 or

Pr(q1.u = p; q2.u = p) ≥ Pr(q1.u = p)Pr(q2.u = p). Suppose A has the AGW-graph; Then The corresponding Markov chain M is mixing. Due to the RCP solution, we can define a synchronizing automaton within the probability distribution on the alphabet such that the induced Markov chain is M [Kouji Yano, Kenji Yasutom]. Does the condition that a graph is the AGW-graph imply faster convergence of M?

Mikhail V. Berlinkov Paris, 2015 21 / 22

slide-123
SLIDE 123

Markov Chain Convergence vs Reset Threshold

Let u ∈ Σj be a randomly generated word by ǫ-machine A and p ∈ Q and j ≥ n − 1. Then rt(A ) ≤ j if either Pr(u is reset ) > 0 or

  • q∈Q Pr(p.u = q.u) < 1 or

Pr(q1.u = p; q2.u = p) ≥ Pr(q1.u = p)Pr(q2.u = p). Suppose A has the AGW-graph; Then The corresponding Markov chain M is mixing. Due to the RCP solution, we can define a synchronizing automaton within the probability distribution on the alphabet such that the induced Markov chain is M [Kouji Yano, Kenji Yasutom]. Does the condition that a graph is the AGW-graph imply faster convergence of M?

Mikhail V. Berlinkov Paris, 2015 21 / 22

slide-124
SLIDE 124

Markov Chain Convergence vs Reset Threshold

Let u ∈ Σj be a randomly generated word by ǫ-machine A and p ∈ Q and j ≥ n − 1. Then rt(A ) ≤ j if either Pr(u is reset ) > 0 or

  • q∈Q Pr(p.u = q.u) < 1 or

Pr(q1.u = p; q2.u = p) ≥ Pr(q1.u = p)Pr(q2.u = p). Suppose A has the AGW-graph; Then The corresponding Markov chain M is mixing. Due to the RCP solution, we can define a synchronizing automaton within the probability distribution on the alphabet such that the induced Markov chain is M [Kouji Yano, Kenji Yasutom]. Does the condition that a graph is the AGW-graph imply faster convergence of M?

Mikhail V. Berlinkov Paris, 2015 21 / 22

slide-125
SLIDE 125

Markov Chain Convergence vs Reset Threshold

Let u ∈ Σj be a randomly generated word by ǫ-machine A and p ∈ Q and j ≥ n − 1. Then rt(A ) ≤ j if either Pr(u is reset ) > 0 or

  • q∈Q Pr(p.u = q.u) < 1 or

Pr(q1.u = p; q2.u = p) ≥ Pr(q1.u = p)Pr(q2.u = p). Suppose A has the AGW-graph; Then The corresponding Markov chain M is mixing. Due to the RCP solution, we can define a synchronizing automaton within the probability distribution on the alphabet such that the induced Markov chain is M [Kouji Yano, Kenji Yasutom]. Does the condition that a graph is the AGW-graph imply faster convergence of M?

Mikhail V. Berlinkov Paris, 2015 21 / 22

slide-126
SLIDE 126

Synchronization of Random Automata Marne-la-Vallée (LIGM), May, 26 Toward the solution of the ˇ Cerný Conjecture Université Paris Diderot (LIAFA), May, 29

Merci!

Mikhail V. Berlinkov Paris, 2015 22 / 22

slide-127
SLIDE 127

Synchronization of Random Automata Marne-la-Vallée (LIGM), May, 26 Toward the solution of the ˇ Cerný Conjecture Université Paris Diderot (LIAFA), May, 29

Merci!

Mikhail V. Berlinkov Paris, 2015 22 / 22

slide-128
SLIDE 128

Synchronization of Random Automata Marne-la-Vallée (LIGM), May, 26 Toward the solution of the ˇ Cerný Conjecture Université Paris Diderot (LIAFA), May, 29

Merci!

Mikhail V. Berlinkov Paris, 2015 22 / 22