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

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. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Example a a 0 1 b b b a a 3 2 b 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 C n found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton C n has n states, 2 input letters and reset threshold ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Example a a 0 1 b b b a a 3 2 b 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 C n found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton C n has n states, 2 input letters and reset threshold ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Example a a 0 1 b b b a a 3 2 b 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 C n found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton C n has n states, 2 input letters and reset threshold ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Example a a 0 1 b b b a a 3 2 b 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 C n found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton C n has n states, 2 input letters and reset threshold ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Example a a 0 1 b b b a a 3 2 b 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 C n found by Jan ˇ Cern´ y in 1964. For each n > 1, the automaton C n has n states, 2 input letters and reset threshold ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Series The states of C n 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 C 4 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Series The states of C n 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 C 4 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Series The states of C n 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 C 4 . Here is a generic automaton from the ˇ Cern´ y series: a a a b 0 b n − 1 1 b b a a n − 2 2 . . . . . . April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Series The states of C n 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 C 4 . Here is a generic automaton from the ˇ Cern´ y series: a a a b 0 b n − 1 1 b b a a n − 2 2 . . . . . . ˇ Cern´ y has proved that the shortest reset word for C n is ( ab n − 1 ) n − 2 a of length n ( n − 2) + 1 = ( n − 1) 2 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

The ˇ Cern´ y Conjecture Define the ˇ Cern´ y function C ( n ) as the maximum reset threshold of all synchronizing automata with n states. The above property of the series { C n } , 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 ) ≤ n 3 − 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 6 at n 3 by in n but improves the coefficient 1 125 511104 ≈ 0 . 000245. The new bound is (85059 n 3 + 90024 n 2 + 196504 n − 10648) / 511104. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Observation Observation (Marina Maslennikova, arXiv:1404.2816; Henk Don, Electronic J. Combinatorics 23 (2016) #P3.12) The ˇ Cern´ y automata C n are completely reachable. April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Observation Observation (Marina Maslennikova, arXiv:1404.2816; Henk Don, Electronic J. Combinatorics 23 (2016) #P3.12) The ˇ Cern´ y automata C n are completely reachable. For an illustration, consider the power-set automaton of the ˇ Cern´ y automaton C 4 . April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Observation a a , b 0 1 b b 3 2 b a a April 11, 2019 Mikhail Volkov Completely Reachable Automata

An Observation b a a a 0123 123 023 b b b b 012 013 a a a a a , b b b a 0 1 01 12 02 13 a a b b b b b b a 3 2 03 23 b a a a April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of completely reachable automata. Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of completely reachable automata. Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of completely reachable automata. Therefore completely reachable automata form quite a natural class to study from the viewpoint of the ˇ Cern´ y conjecture. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of its ‘software library’ appears to be fairly natural. Define the syntactic complexity of a DFA A as the size of its transition monoid M ( A ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of its ‘software library’ appears to be fairly natural. Define the syntactic complexity of a DFA A as the size of its transition monoid M ( A ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of its ‘software library’ appears to be fairly natural. Define the syntactic complexity of a DFA A as the size of its transition monoid M ( A ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of transformations of the state set. Observe that the ˇ Cern´ y automata C n with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid T n requires at least 3 generators. We constructed a series of n -state automata in this class with the reset threshold n ( n − 1) , thus establishing a lower bound 2 for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2 n 2 − 6 n + 5. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of transformations of the state set. Observe that the ˇ Cern´ y automata C n with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid T n requires at least 3 generators. We constructed a series of n -state automata in this class with the reset threshold n ( n − 1) , thus establishing a lower bound 2 for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2 n 2 − 6 n + 5. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of transformations of the state set. Observe that the ˇ Cern´ y automata C n with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid T n requires at least 3 generators. We constructed a series of n -state automata in this class with the reset threshold n ( n − 1) , thus establishing a lower bound 2 for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2 n 2 − 6 n + 5. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 of transformations of the state set. Observe that the ˇ Cern´ y automata C n with n ≥ 3 are not in this family of completely reachable automata since the full transformation monoid T n requires at least 3 generators. We constructed a series of n -state automata in this class with the reset threshold n ( n − 1) , thus establishing a lower bound 2 for the ˇ Cern´ y function restricted to maximal completely reachable automata, and found an upper bound with the same growth rate, namely, 2 n 2 − 6 n + 5. April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

Minimal Completely Reachable Automata Here we focus on automata being in a sense the extreme opposites of 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 2 n − 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 2 n − 1 transformations of 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? April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 T n , 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 T n . Here one wants to find B n (where B n stands for the Bell number) transformations of 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). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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, Γ v 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 of 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 Γ u → Γ v . April 11, 2019 Mikhail Volkov Completely Reachable Automata

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. April 11, 2019 Mikhail Volkov Completely Reachable Automata

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): April 11, 2019 Mikhail Volkov Completely Reachable Automata

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): A N April 11, 2019 Mikhail Volkov Completely Reachable Automata

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: April 11, 2019 Mikhail Volkov Completely Reachable Automata

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. April 11, 2019 Mikhail Volkov Completely Reachable Automata

Enumeration? It is easy to show that there exist respectful trees with any number of leaves. April 11, 2019 Mikhail Volkov Completely Reachable Automata

Enumeration? It is easy to show that there exist respectful trees with any number of 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. April 11, 2019 Mikhail Volkov Completely Reachable Automata

Enumeration? It is easy to show that there exist respectful trees with any number of 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 April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

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 2 n − 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(Γ s ) and Λ(Γ d ). If v � = s and v � = d , then v is a non-root vertex in one of the subtrees Γ s or Γ d ; WLOG assume that v belongs to Γ s . By the induction assumption applied to Γ s , the action of v on the set Λ(Γ s ) is already defined; we extend this action to the set Λ(Γ d ) by setting y . v := y for each y ∈ Λ(Γ d ). April 11, 2019 Mikhail Volkov Completely Reachable Automata

From Trees to Automata, Illustration For an illustration consider the respectful tree shown before. r s d April 11, 2019 Mikhail Volkov Completely Reachable Automata

From Trees to Automata, Illustration For an illustration consider the respectful tree shown before. r s d v April 11, 2019 Mikhail Volkov Completely Reachable Automata