Completely Reachable Automata: an interplay between semigroups, - - PowerPoint PPT Presentation

April 11, 2019

Completely Reachable Automata: an interplay between semigroups, finite automata, and binary trees

Mikhail Volkov (joint with Evgenija Bondar)

Ural Federal University, Ekaterinburg, Russia

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

A finite automaton is a simple but extremely productive concept that captures the very important idea of an object interacting with its environment.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

A finite automaton is a simple but extremely productive concept that captures the very important idea of an object interacting with its environment. Environment Object

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

A finite automaton is a simple but extremely productive concept that captures the very important idea of an object interacting with its environment. Environment action Object

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

A finite automaton is a simple but extremely productive concept that captures the very important idea of an object interacting with its environment. Environment Object

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

This notion originates in the seminal work by Alan Turing (“On Computable Numbers, With an Application to the Entscheidungsproblem”, Proc. London Math. Soc., Ser. 2, 42 (1936), 230–265). “The behavior of the computer at any moment is determined by the symbols which he is observing, and his state of mind at that moment”. Another important source is the work by neurobiologists Warren McCulloch and Walter Pitts (“A Logical Calculus of the Ideas Immanent in Nervous Activity”, Bull. Math. Biophys. 5 (1943), 115–133).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Finite Automata

This notion originates in the seminal work by Alan Turing (“On Computable Numbers, With an Application to the Entscheidungsproblem”, Proc. London Math. Soc., Ser. 2, 42 (1936), 230–265). “The behavior of the computer at any moment is determined by the symbols which he is observing, and his state of mind at that moment”. Another important source is the work by neurobiologists Warren McCulloch and Walter Pitts (“A Logical Calculus of the Ideas Immanent in Nervous Activity”, Bull. Math. Biophys. 5 (1943), 115–133).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Visualization

Finite automata admit a convenient visual representation.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Visualization

Finite automata admit a convenient visual representation. 1 2 3 a b b b b a a a

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Visualization

Finite automata admit a convenient visual representation. 1 2 3 Here one sees 4 states called 0,1,2,3,

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Visualization

Finite automata admit a convenient visual representation. 1 2 3 a a a a Here one sees 4 states called 0,1,2,3, an action called a

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Visualization

Finite automata admit a convenient visual representation. 1 2 3 a a a a b b b b Here one sees 4 states called 0,1,2,3, an action called a and another action called b.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata (DFA): A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata (DFA): A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the finite state set;
• Σ is the input finite alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the totally defined transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Definitions and Terminology

We consider complete deterministic finite automata: A = Q, Σ, δ. Here

• Q is the state set;
• Σ is the input alphabet;
• δ : Q × Σ → Q is the transition function.

Σ∗ stands for the set of all words over Σ including the empty word. The function δ uniquely extends to a function Q × Σ∗ → Q still denoted by δ. To simplify notation we often write q . w for δ(q, w) and P . w for {δ(q, w) | q ∈ P}. Given a DFA A = Q, Σ, a non-empty subset P ⊆ Q is reachable in A if P = Q . w for some word w ∈ Σ∗. A DFA is completely reachable if every non-empty set of its states is reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Motivation: Synchronizing Automata

A DFA A = Q, Σ is called synchronizing if there is a word w ∈ Σ∗ whose action resets A , that is, leaves A in one particular state no matter at which state it started: q . w = q′ . w for all q, q′ ∈ Q. In short, |Q . w| = 1; that is, a singleton is reachable in A . Hence, a completely reachable automaton is synchronizing. Any w with |Q . w| = 1 is a reset word for A . The minimum length of reset words for A is called the reset threshold of A .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Motivation: Synchronizing Automata

A DFA A = Q, Σ is called synchronizing if there is a word w ∈ Σ∗ whose action resets A , that is, leaves A in one particular state no matter at which state it started: q . w = q′ . w for all q, q′ ∈ Q. In short, |Q . w| = 1; that is, a singleton is reachable in A . Hence, a completely reachable automaton is synchronizing. Any w with |Q . w| = 1 is a reset word for A . The minimum length of reset words for A is called the reset threshold of A .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Motivation: Synchronizing Automata

A DFA A = Q, Σ is called synchronizing if there is a word w ∈ Σ∗ whose action resets A , that is, leaves A in one particular state no matter at which state it started: q . w = q′ . w for all q, q′ ∈ Q. In short, |Q . w| = 1; that is, a singleton is reachable in A . Hence, a completely reachable automaton is synchronizing. Any w with |Q . w| = 1 is a reset word for A . The minimum length of reset words for A is called the reset threshold of A .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Motivation: Synchronizing Automata

A DFA A = Q, Σ is called synchronizing if there is a word w ∈ Σ∗ whose action resets A , that is, leaves A in one particular state no matter at which state it started: q . w = q′ . w for all q, q′ ∈ Q. In short, |Q . w| = 1; that is, a singleton is reachable in A . Hence, a completely reachable automaton is synchronizing. Any w with |Q . w| = 1 is a reset word for A . The minimum length of reset words for A is called the reset threshold of A .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Motivation: Synchronizing Automata

A DFA A = Q, Σ is called synchronizing if there is a word w ∈ Σ∗ whose action resets A , that is, leaves A in one particular state no matter at which state it started: q . w = q′ . w for all q, q′ ∈ Q. In short, |Q . w| = 1; that is, a singleton is reachable in A . Hence, a completely reachable automaton is synchronizing. Any w with |Q . w| = 1 is a reset word for A . The minimum length of reset words for A is called the reset threshold of A .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Example

1 2 3 a b b b b a a a A reset word is abbbabbba: applying it at any state brings the automaton to the state 1. In fact, this is the reset word of minimum length for the automaton whence its reset threshold is 9. The automaton belongs to the series Cn found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton Cn has n states, 2 input letters and reset threshold (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Example

1 2 3 a b b b b a a a A reset word is abbbabbba: applying it at any state brings the automaton to the state 1. In fact, this is the reset word of minimum length for the automaton whence its reset threshold is 9. The automaton belongs to the series Cn found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton Cn has n states, 2 input letters and reset threshold (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Example

1 2 3 a b b b b a a a A reset word is abbbabbba: applying it at any state brings the automaton to the state 1. In fact, this is the reset word of minimum length for the automaton whence its reset threshold is 9. The automaton belongs to the series Cn found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton Cn has n states, 2 input letters and reset threshold (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Example

1 2 3 a b b b b a a a A reset word is abbbabbba: applying it at any state brings the automaton to the state 1. In fact, this is the reset word of minimum length for the automaton whence its reset threshold is 9. The automaton belongs to the series Cn found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton Cn has n states, 2 input letters and reset threshold (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Example

1 2 3 a b b b b a a a A reset word is abbbabbba: applying it at any state brings the automaton to the state 1. In fact, this is the reset word of minimum length for the automaton whence its reset threshold is 9. The automaton belongs to the series Cn found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton Cn has n states, 2 input letters and reset threshold (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Series

The states of Cn are the residues modulo n, and the input letters a and b act as follows: 0 . a = 1, m . a = m for 0 < m < n, m . b = m + 1 (mod n). The automaton in the previous slide is C4.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Series

The states of Cn are the residues modulo n, and the input letters a and b act as follows: 0 . a = 1, m . a = m for 0 < m < n, m . b = m + 1 (mod n). The automaton in the previous slide is C4.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Series

The states of Cn are the residues modulo n, and the input letters a and b act as follows: 0 . a = 1, m . a = m for 0 < m < n, m . b = m + 1 (mod n). The automaton in the previous slide is C4. Here is a generic automaton from the ˇ Cern´ y series:

n−2 n−1 1 2

a a a a b b a b b . . . . . .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Series

The states of Cn are the residues modulo n, and the input letters a and b act as follows: 0 . a = 1, m . a = m for 0 < m < n, m . b = m + 1 (mod n). The automaton in the previous slide is C4. Here is a generic automaton from the ˇ Cern´ y series:

n−2 n−1 1 2

a a a a b b a b b . . . . . . ˇ Cern´ y has proved that the shortest reset word for Cn is (abn−1)n−2a of length n(n − 2) + 1 = (n − 1)2.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

The ˇ Cern´ y Conjecture

Define the ˇ Cern´ y function C(n) as the maximum reset threshold

• f all synchronizing automata with n states. The above property
• f the series {Cn}, yields the inequality C(n) ≥ (n − 1)2.

The ˇ Cern´ y conjecture is the claim that in fact the equality C(n) = (n − 1)2 holds true. This simply looking conjecture is arguably the most longstanding open problem in the combinatorial theory of finite automata. Up to recently, everything we knew about the conjecture in general could be summarized in one line: (n − 1)2 ≤ C(n) ≤ n3 − n 6 . A small improvement on this bound has been found by Marek Szyku la (published in STACS 2018): the new bound is still cubic in n but improves the coefficient 1

6 at n3 by 125 511104 ≈ 0.000245.

The new bound is (85059n3 + 90024n2 + 196504n − 10648)/511104.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Approaching the ˇ Cern´ y Conjecture

Since the ˇ Cern´ y Conjecture has proved to be hard in general, a natural strategy consists in considering its restrictions to some special classes of DFAs. The conjecture has been proved for many important special cases. This includes for instance:

• Louis Dubuc’s result for automata in which a letter acts on the

state set Q as a cyclic permutation of order |Q| (Sur le automates circulaires et la conjecture de ˇ Cern´ y, RAIRO Inform. Theor. Appl., 32 (1998) 21–34 [in French]).

• Jarkko Kari’s result for automata with Eulerian digraphs

(Synchronizing finite automata on Eulerian digraphs, Theoret.

• Comput. Sci., 295 (2003) 223–232).
• Avraam Trahtman result for automata whose transition monoid

contains no non-trivial subgroups (The ˇ Cern´ y conjecture for aperiodic automata, Discrete Math. Theoret. Comp. Sci., 9, no.2 (2007), 3–10).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Approaching the ˇ Cern´ y Conjecture

Since the ˇ Cern´ y Conjecture has proved to be hard in general, a natural strategy consists in considering its restrictions to some special classes of DFAs. The conjecture has been proved for many important special cases. This includes for instance:

• Louis Dubuc’s result for automata in which a letter acts on the

state set Q as a cyclic permutation of order |Q| (Sur le automates circulaires et la conjecture de ˇ Cern´ y, RAIRO Inform. Theor. Appl., 32 (1998) 21–34 [in French]).

• Jarkko Kari’s result for automata with Eulerian digraphs

(Synchronizing finite automata on Eulerian digraphs, Theoret.

• Comput. Sci., 295 (2003) 223–232).
• Avraam Trahtman result for automata whose transition monoid

contains no non-trivial subgroups (The ˇ Cern´ y conjecture for aperiodic automata, Discrete Math. Theoret. Comp. Sci., 9, no.2 (2007), 3–10).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Approaching the ˇ Cern´ y Conjecture

Since the ˇ Cern´ y Conjecture has proved to be hard in general, a natural strategy consists in considering its restrictions to some special classes of DFAs. The conjecture has been proved for many important special cases. This includes for instance:

• Louis Dubuc’s result for automata in which a letter acts on the

state set Q as a cyclic permutation of order |Q| (Sur le automates circulaires et la conjecture de ˇ Cern´ y, RAIRO Inform. Theor. Appl., 32 (1998) 21–34 [in French]).

• Jarkko Kari’s result for automata with Eulerian digraphs

(Synchronizing finite automata on Eulerian digraphs, Theoret.

• Comput. Sci., 295 (2003) 223–232).
• Avraam Trahtman result for automata whose transition monoid

contains no non-trivial subgroups (The ˇ Cern´ y conjecture for aperiodic automata, Discrete Math. Theoret. Comp. Sci., 9, no.2 (2007), 3–10).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Approaching the ˇ Cern´ y Conjecture

Since the ˇ Cern´ y Conjecture has proved to be hard in general, a natural strategy consists in considering its restrictions to some special classes of DFAs. The conjecture has been proved for many important special cases. This includes for instance:

• Louis Dubuc’s result for automata in which a letter acts on the

state set Q as a cyclic permutation of order |Q| (Sur le automates circulaires et la conjecture de ˇ Cern´ y, RAIRO Inform. Theor. Appl., 32 (1998) 21–34 [in French]).

• Jarkko Kari’s result for automata with Eulerian digraphs

(Synchronizing finite automata on Eulerian digraphs, Theoret.

• Comput. Sci., 295 (2003) 223–232).
• Avraam Trahtman result for automata whose transition monoid

contains no non-trivial subgroups (The ˇ Cern´ y conjecture for aperiodic automata, Discrete Math. Theoret. Comp. Sci., 9, no.2 (2007), 3–10).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Approaching the ˇ Cern´ y Conjecture

Since the ˇ Cern´ y Conjecture has proved to be hard in general, a natural strategy consists in considering its restrictions to some special classes of DFAs. The conjecture has been proved for many important special cases. This includes for instance:

• Louis Dubuc’s result for automata in which a letter acts on the

state set Q as a cyclic permutation of order |Q| (Sur le automates circulaires et la conjecture de ˇ Cern´ y, RAIRO Inform. Theor. Appl., 32 (1998) 21–34 [in French]).

• Jarkko Kari’s result for automata with Eulerian digraphs

(Synchronizing finite automata on Eulerian digraphs, Theoret.

• Comput. Sci., 295 (2003) 223–232).
• Avraam Trahtman result for automata whose transition monoid

contains no non-trivial subgroups (The ˇ Cern´ y conjecture for aperiodic automata, Discrete Math. Theoret. Comp. Sci., 9, no.2 (2007), 3–10).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Observation

Observation (Marina Maslennikova, arXiv:1404.2816; Henk Don, Electronic J. Combinatorics 23 (2016) #P3.12) The ˇ Cern´ y automata Cn are completely reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Observation

Observation (Marina Maslennikova, arXiv:1404.2816; Henk Don, Electronic J. Combinatorics 23 (2016) #P3.12) The ˇ Cern´ y automata Cn are completely reachable. For an illustration, consider the power-set automaton

• f the ˇ

Cern´ y automaton C4.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Observation

1 2 3 a, b b b b a a a

Mikhail Volkov Completely Reachable Automata

April 11, 2019

An Observation

1 2 3 a, b b b b a a a 03 01 12 23 02 13 a a a b b b b a 012 013 123 023 0123 b a a b a b b b a a a b b a

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Restricting to Completely Reachable Automata

Recall that every completely reachable automaton is synchronizing. On the other hand, the above observation ensures that the lower bound (n − 1)2 for the ˇ Cern´ y function C(n) is attained by a family

• f completely reachable automata.

Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Restricting to Completely Reachable Automata

Recall that every completely reachable automaton is synchronizing. On the other hand, the above observation ensures that the lower bound (n − 1)2 for the ˇ Cern´ y function C(n) is attained by a family

• f completely reachable automata.

Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Restricting to Completely Reachable Automata

Recall that every completely reachable automaton is synchronizing. On the other hand, the above observation ensures that the lower bound (n − 1)2 for the ˇ Cern´ y function C(n) is attained by a family

• f completely reachable automata.

Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Syntactic Complexity

The transition monoid M(A ) of a given DFA A = Q, Σ is the monoid of all transformations of Q induced by the words in Σ∗. If one thinks of a DFA as a computational device, its transition monoid can be thought of as the device’s ‘software library’, and measuring the complexity of an automaton by the size

• f its ‘software library’ appears to be fairly natural.

Define the syntactic complexity of a DFA A as the size

• f its transition monoid M(A ).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Syntactic Complexity

The transition monoid M(A ) of a given DFA A = Q, Σ is the monoid of all transformations of Q induced by the words in Σ∗. If one thinks of a DFA as a computational device, its transition monoid can be thought of as the device’s ‘software library’, and measuring the complexity of an automaton by the size

• f its ‘software library’ appears to be fairly natural.

Define the syntactic complexity of a DFA A as the size

• f its transition monoid M(A ).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Syntactic Complexity

The transition monoid M(A ) of a given DFA A = Q, Σ is the monoid of all transformations of Q induced by the words in Σ∗. If one thinks of a DFA as a computational device, its transition monoid can be thought of as the device’s ‘software library’, and measuring the complexity of an automaton by the size

• f its ‘software library’ appears to be fairly natural.

Define the syntactic complexity of a DFA A as the size

• f its transition monoid M(A ).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Maximal Completely Reachable Automata

Fran¸ cois Gonze, Vladimir Gusev, Bal´ azs Gerencs´ er, Rapha¨ el Jungers, and MV (Developments in Language Theory, Lect. Notes

• Comp. Sci. 10396 (2017) 185–197) studied completely reachable

automata of maximal syntactic complexity, that is, automata whose transition monoid is equal to the full monoid

• f transformations of the state set.

Observe that the ˇ Cern´ y automata Cn with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid Tn requires at least 3 generators. We constructed a series of n-state automata in this class with the reset threshold n(n−1)

2

, thus establishing a lower bound for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2n2 − 6n + 5.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Maximal Completely Reachable Automata

Fran¸ cois Gonze, Vladimir Gusev, Bal´ azs Gerencs´ er, Rapha¨ el Jungers, and MV (Developments in Language Theory, Lect. Notes

• Comp. Sci. 10396 (2017) 185–197) studied completely reachable

automata of maximal syntactic complexity, that is, automata whose transition monoid is equal to the full monoid

• f transformations of the state set.

Observe that the ˇ Cern´ y automata Cn with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid Tn requires at least 3 generators. We constructed a series of n-state automata in this class with the reset threshold n(n−1)

2

, thus establishing a lower bound for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2n2 − 6n + 5.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Maximal Completely Reachable Automata

Fran¸ cois Gonze, Vladimir Gusev, Bal´ azs Gerencs´ er, Rapha¨ el Jungers, and MV (Developments in Language Theory, Lect. Notes

• Comp. Sci. 10396 (2017) 185–197) studied completely reachable

automata of maximal syntactic complexity, that is, automata whose transition monoid is equal to the full monoid

• f transformations of the state set.

Observe that the ˇ Cern´ y automata Cn with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid Tn requires at least 3 generators. We constructed a series of n-state automata in this class with the reset threshold n(n−1)

2

, thus establishing a lower bound for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2n2 − 6n + 5.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Maximal Completely Reachable Automata

Fran¸ cois Gonze, Vladimir Gusev, Bal´ azs Gerencs´ er, Rapha¨ el Jungers, and MV (Developments in Language Theory, Lect. Notes

• Comp. Sci. 10396 (2017) 185–197) studied completely reachable

automata of maximal syntactic complexity, that is, automata whose transition monoid is equal to the full monoid

• f transformations of the state set.

Observe that the ˇ Cern´ y automata Cn with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid Tn requires at least 3 generators. We constructed a series of n-state automata in this class with the reset threshold n(n−1)

2

, thus establishing a lower bound for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2n2 − 6n + 5.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Minimal Completely Reachable Automata

Here we focus on automata being in a sense the extreme opposites

• f those studied in the DLT paper, namely, on completely

reachable automata of minimal syntactic complexity. Clearly, if a completely reachable automaton A has n states, the size of M(A ) is at least 2n − 1 since each non-empty subset must occur as the image of a transformation from M(A ). Is this lower bound tight? In other words, can one select 2n − 1 transformations

• f an n-element set Q such that: 1) each non-empty subset of Q

is the image of one of these transformations; 2) the selected transformations form a monoid? If so, can one somehow classify such monoids?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Algebraic Viewpoint

These questions are far from being obvious and are of interest also from the viewpoint of algebra, where they have been stated as the questions of the existence and the classification of L-cross-sections in the full transformation monoid Tn, see, e.g., O. Ganyushkin,

• V. Mazorchuk, Classical Finite Transformation Semigroups:

An Introduction. Springer, 2009. For a comparison, we mention the dual questions of the existence and the classification of R-cross-sections in Tn. Here one wants to find Bn (where Bn stands for the Bell number) transformations

• f an n-element set Q such that: 1) each partition of Q is

the kernel of one of these transformations; 2) the selected transformations form a monoid. It is known that there are n! ways to build such a monoid, and moreover, all such monoids turn out to be isomorphic as abstract monoids (V. Pekhterev, 2003, see also the above monograph).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Trees

Our construction produces minimal completely reachable automata from full binary trees satisfying certain subordination conditions. A binary tree is full if each its vertex v either is a leaf or has exactly two children. We refer to the left/right child of v as the son/daughter of v. If Γ is a tree and v is its vertex, Γv is the subtree of Γ rooted at v. A homomorphism between trees is a map between their vertex sets that preserves the roots, the parent–child relation and the genders

• f non-root vertices. If u and v are vertices of a tree Γ, we say that

u subordinates v if there is a 1-1 homomorphism Γu → Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Respectful Trees

A respectful tree is such that: (S1) if a male vertex has a nephew, he subordinates his uncle; (S2) if a female vertex has a niece, she subordinates her aunt.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Respectful Trees

A respectful tree is such that: (S1) if a male vertex has a nephew, he subordinates his uncle; (S2) if a female vertex has a niece, she subordinates her aunt. This tree is not respectful as it fails to satisfy (S2):

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Respectful Trees

A respectful tree is such that: (S1) if a male vertex has a nephew, he subordinates his uncle; (S2) if a female vertex has a niece, she subordinates her aunt. This tree is not respectful as it fails to satisfy (S2): N A

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Respectful Trees

A respectful tree is such that: (S1) if a male vertex has a nephew, he subordinates his uncle; (S2) if a female vertex has a niece, she subordinates her aunt. This tree is respectful:

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Respectful Trees

A respectful tree is such that: (S1) if a male vertex has a nephew, he subordinates his uncle; (S2) if a female vertex has a niece, she subordinates her aunt. This tree is respectful: Here dotted and dashed arrows show the uncle–nephew and the aunt–niece relations respectively.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Enumeration?

It is easy to show that there exist respectful trees with any number

• f leaves.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Enumeration?

It is easy to show that there exist respectful trees with any number

• f leaves.

The number of respectful trees quickly grows with the number of leaves, but we are not aware of any closed formula for the former number nor of its growth rate.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Enumeration?

It is easy to show that there exist respectful trees with any number

• f leaves.

The number of respectful trees quickly grows with the number of leaves, but we are not aware of any closed formula for the former number nor of its growth rate. # of leaves 1 2 3 4 5 6 7 8 9 10 # of respectful trees 1 1 2 3 6 10 18 32 58 101

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata

For each respectful tree Γ, we construct a DFA A (Γ). The state set of A (Γ) is the set of all leaves of Γ, denoted Λ(Γ). The input letters of A (Γ) are the non-root vertices of Γ so that if A (Γ) has n leaves, it has 2n − 2 input letters. We define the action of the letters by induction on n = |Λ(Γ)|. For n = 1, Γ is the trivial tree with one vertex r and no edges and A (Γ) is the trivial automaton with one state and no transitions. Now suppose that n > 1 and take any non-root vertex v of Γ; we have to define the action of v on the set Λ(Γ). If s and d are respectively the son and the daughter of the root r, the set Λ(Γ) is the disjoint union of Λ(Γs) and Λ(Γd). If v = s and v = d, then v is a non-root vertex in one of the subtrees Γs or Γd; WLOG assume that v belongs to Γs. By the induction assumption applied to Γs, the action of v on the set Λ(Γs) is already defined; we extend this action to the set Λ(Γd) by setting y . v := y for each y ∈ Λ(Γd).

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration

For an illustration consider the respectful tree shown before. s d r

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration

For an illustration consider the respectful tree shown before. v s d r

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration

For an illustration consider the respectful tree shown before. v s d r v v

• defined by the induction assumption

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Continued

It remains to define the actions of s and d. By symmetry, it suffices to define the action of s. If s has no nephew, then d is a leaf and we let x . s := d for each x ∈ Λ(Γ). Otherwise let t be the nephew of s. Recall that Γ is respectful, so by (S1) there is a 1-1 homomorphism ξ : Γt → Γs. The sets Λ(Γℓξ), where ℓ runs over Λ(Γt), partition the set Λ(Γs). Now, if a leaf x ∈ Λ(Γs) belongs to Λ(Γℓξ) for some leaf ℓ of Γt, we let x . s := ℓ. By the induction assumption applied to Γd, the action of t on the set Λ(Γd) is already defined and we extend the action of s to Λ(Γd) by setting y . s := y . t for all y ∈ Λ(Γd). This completes our construction.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration-2

For an illustration, we use another respectful tree. s d r t ℓ1 ℓ2

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration-2

For an illustration, we use another respectful tree. s d r t ℓ1 ℓ2 ξ

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration-2

For an illustration, we use another respectful tree. ℓ1ξ ℓ2ξ s d r t ℓ1 ℓ2 ξ Every leaf in Γs is under either ℓ1ξ or ℓ2ξ.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration-2

For an illustration, we use another respectful tree. ℓ1ξ ℓ2ξ s d r t ℓ1 ℓ2 ξ

• leaves sent to ℓ1
• leaves sent to ℓ2

Every leaf in Γs is under either ℓ1ξ or ℓ2ξ.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata, Illustration-2

For an illustration, we use another respectful tree. ℓ1ξ ℓ2ξ s d r t ℓ1 ℓ2 ξ

• leaves sent to ℓ1
• leaves sent to ℓ2

s s s Every leaf in Γs is under either ℓ1ξ or ℓ2ξ.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv. 1 2 3

Mikhail Volkov Completely Reachable Automata

April 11, 2019

From Trees to Automata: Example

Summary: The states of A (Γ) are the leaves of Γ. The input letters of A (Γ) are the non-root vertices of Γ. The image of a vertex v consists of the leaves in Γ \ Γv. 1 2 3 1 2 3 a1 a2, a3 a3 a1,2 a1,2 a1 a2, a3 a1, a2, a1,2

Mikhail Volkov Completely Reachable Automata

April 11, 2019

All Minimal Completely Reachable Automata

We say that two DFAs A = Q, Σ, δ and B = Q, ∆, ζ are syntactically equivalent if their transition monoids coincide.

• 1. For each respectful tree Γ, the automaton A (Γ) is a minimal

completely reachable automaton.

• 2. Non-isomorphic respectful trees give rise to non-isomorphic

automata.

• 3. Every minimal completely reachable automaton is

syntactically equivalent to an automaton of the form A (Γ) for a suitable respectful tree Γ.

• 4. Every minimal completely reachable automaton with n states

has at least 2n − 2 input letters.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

All Minimal Completely Reachable Automata

We say that two DFAs A = Q, Σ, δ and B = Q, ∆, ζ are syntactically equivalent if their transition monoids coincide.

• 1. For each respectful tree Γ, the automaton A (Γ) is a minimal

completely reachable automaton.

• 2. Non-isomorphic respectful trees give rise to non-isomorphic

automata.

• 3. Every minimal completely reachable automaton is

syntactically equivalent to an automaton of the form A (Γ) for a suitable respectful tree Γ.

• 4. Every minimal completely reachable automaton with n states

has at least 2n − 2 input letters.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

All Minimal Completely Reachable Automata

We say that two DFAs A = Q, Σ, δ and B = Q, ∆, ζ are syntactically equivalent if their transition monoids coincide.

• 1. For each respectful tree Γ, the automaton A (Γ) is a minimal

completely reachable automaton.

• 2. Non-isomorphic respectful trees give rise to non-isomorphic

automata.

• 3. Every minimal completely reachable automaton is

syntactically equivalent to an automaton of the form A (Γ) for a suitable respectful tree Γ.

• 4. Every minimal completely reachable automaton with n states

has at least 2n − 2 input letters.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

All Minimal Completely Reachable Automata

We say that two DFAs A = Q, Σ, δ and B = Q, ∆, ζ are syntactically equivalent if their transition monoids coincide.

• 1. For each respectful tree Γ, the automaton A (Γ) is a minimal

completely reachable automaton.

• 2. Non-isomorphic respectful trees give rise to non-isomorphic

automata.

• 3. Every minimal completely reachable automaton is

syntactically equivalent to an automaton of the form A (Γ) for a suitable respectful tree Γ.

• 4. Every minimal completely reachable automaton with n states

has at least 2n − 2 input letters.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

All Minimal Completely Reachable Automata

We say that two DFAs A = Q, Σ, δ and B = Q, ∆, ζ are syntactically equivalent if their transition monoids coincide.

• 1. For each respectful tree Γ, the automaton A (Γ) is a minimal

completely reachable automaton.

• 2. Non-isomorphic respectful trees give rise to non-isomorphic

automata.

• 3. Every minimal completely reachable automaton is

syntactically equivalent to an automaton of the form A (Γ) for a suitable respectful tree Γ.

• 4. Every minimal completely reachable automaton with n states

has at least 2n − 2 input letters.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. n−1 1 n−2 2 a, b b b b a a a a . . .

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. n−1 1 n−2 2 a, b b b b a a a a . . . n−1 1 n−2 2 b b, c b, c b, c c . . . c := ab

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton? In other words, is it true that an automaton of the form A (Γ) ‘hides’ within every completely reachable automaton?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton? In other words, is it true that an automaton of the form A (Γ) ‘hides’ within every completely reachable automaton? Here we have recently found a negative answer.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton? In other words, is it true that an automaton of the form A (Γ) ‘hides’ within every completely reachable automaton? Here we have recently found a negative answer. 3) What is the computational complexity of recognizing complete reachability?

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton? In other words, is it true that an automaton of the form A (Γ) ‘hides’ within every completely reachable automaton? Here we have recently found a negative answer. 3) What is the computational complexity of recognizing complete reachability? We know that the property of a DFA to be completely reachable of maximal syntactic complexity is in P.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Three Further Questions

1) What are lower bounds for the syntactic complexity

• f completely reachable automata with restricted alphabet?

A DFA A = Q, Σ, δ induces a DFA B = Q, ∆, ζ on the same state set if the transition monoid of A contains that of B. This means that for every letter b ∈ ∆, there exists a word w ∈ Σ∗ such that ζ(q, b) = δ(q, w) for every q ∈ Q. 2) Is it true that every completely reachable automaton induces a minimal completely reachable automaton? In other words, is it true that an automaton of the form A (Γ) ‘hides’ within every completely reachable automaton? Here we have recently found a negative answer. 3) What is the computational complexity of recognizing complete reachability? We know that the property of a DFA to be completely reachable of maximal syntactic complexity is in P. We do not know the answer for the property of a DFA to be minimal completely reachable.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Synchronization

The reset threshold of a minimal completely reachable automaton is equal to the minimum length of a path from the root to a leaf in the corresponding tree. In particular, the reset threshold of minimal completely reachable automaton with n states does not exceed log2 n, and this bound is attained for the minimal completely reachable automata corresponding to perfect binary trees.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Synchronization

The reset threshold of a minimal completely reachable automaton is equal to the minimum length of a path from the root to a leaf in the corresponding tree. In particular, the reset threshold of minimal completely reachable automaton with n states does not exceed log2 n, and this bound is attained for the minimal completely reachable automata corresponding to perfect binary trees.

Mikhail Volkov Completely Reachable Automata

April 11, 2019

Synchronization

The reset threshold of a minimal completely reachable automaton is equal to the minimum length of a path from the root to a leaf in the corresponding tree. In particular, the reset threshold of minimal completely reachable automaton with n states does not exceed log2 n, and this bound is attained for the minimal completely reachable automata corresponding to perfect binary trees.

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Mikhail Volkov Completely Reachable Automata