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Mealy machines, automaton (semi)groups, decision problems, and random generation Thibault Godin S eminaire CALIN Paris 13, October 3, 2017 ANR JCJC 12 JS02 012 01 1 / 35 finite Analogue to Dixon theorem [ANALCO16] groups S k S

  1. finite Analogue to Dixon theorem [ANALCO’16] groups  S k × S k  infinite σ  � � = ( A k × A k ) ⋊ � ( π, π ) � groups τ  A k × A k  random generation finiteness Invertible reversible non-coreversible automata generate infinite non Burn- 1 | 1 side groups [LATA’15 w. Klimann and Picantin] b d 1 | 1 1 | 1 Bireversible automata of 0 | 0 c 0 | 0 automaton prime size cannot generate 0 | 0 infinite Burnside groups infinite patterns [MFCS’16 w. Klimann] and group Burnside a e properties 0 | 1 1 | 0 0 | 0 1 | 1 x 3 Mealy automata A ℓ − 1 i | . 3 x � = y z i | . A ℓ dynamics 1 2 of y growth the action A ℓ +1 The set of singular points singular of a contracting automaton is described by a B¨ uchi points automaton [DGKPR’16] Schreier graphs ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ξ singular q � � � � � Wang A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] tillings N 0 | 1 0 x , 0 x , 1 y x 1 | 0 x y 1 | 1 0 1 0 | 0 y y y , 0 y , 1 1 9 / 35

  2. finiteness Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of automaton prime size cannot generate infinite Burnside groups patterns infinite [MFCS’16 w. Klimann] and group Burnside properties x 3 A ℓ − 1 i | . 3 x � = y z i | . A ℓ 1 2 y growth A ℓ +1 The set of of a contracting is described automaton 9 / 35

  3. 0 | 1 0 | 0 1 | 0 1 | 1 Mealy automata dynamics of the action The set of singular points singular of a contracting automaton is described by a B¨ uchi points automaton [DGKPR’16] Schreier graphs ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ξ singular q � � � � � Wang A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] tillings N 0 | 1 0 x , 0 x , 1 y x 1 | 0 x y 1 | 1 0 1 0 | 0 y y y , 0 y , 1 1 9 / 35

  4. finite Analogue to Dixon theorem [ANALCO’16] groups  S k × S k  infinite σ  � � = ( A k × A k ) ⋊ � ( π, π ) � groups τ  A k × A k  random generation 9 / 35

  5. Finite random groups Theorem Any finite group G is a subgroup of S | G | . 10 / 35

  6. Finite random groups Theorem Any finite group G is a subgroup of S | G | . First idea Pick up some permutations σ 1 , . . . , σ n of { 1 , . . . , k } , look at � σ 1 , . . . , σ n � . 10 / 35

  7. Finite random groups Theorem Any finite group G is a subgroup of S | G | . First idea Pick up some permutations σ 1 , . . . , σ n of { 1 , . . . , k } , look at � σ 1 , . . . , σ n � . 1 2 3 k ! # permutations 10 / 35

  8. Finite random groups Theorem Any finite group G is a subgroup of S | G | . First idea Pick up some permutations σ 1 , . . . , σ n of { 1 , . . . , k } , look at � σ 1 , . . . , σ n � . 1 2 3 k ! # permutations cyclic groups 10 / 35

  9. Finite random groups Theorem Any finite group G is a subgroup of S | G | . First idea Pick up some permutations σ 1 , . . . , σ n of { 1 , . . . , k } , look at � σ 1 , . . . , σ n � . 1 2 3 k ! # permutations cyclic S k groups 10 / 35

  10. Finite random groups Theorem Any finite group G is a subgroup of S | G | . First idea Pick up some permutations σ 1 , . . . , σ n of { 1 , . . . , k } , look at � σ 1 , . . . , σ n � . 1 2 3 k ! # permutations cyclic ? S k groups 10 / 35

  11. Finite random groups Theorem (Dixon, 1969) � S k w.g.p. � σ, τ � = A k 1 2 3 k ! # permutations cyclic groups 10 / 35

  12. Finite random groups Theorem (Dixon, 1969) � S k S k w.g.p. � σ, τ � = A k A k 1 2 3 k ! # permutations cyclic groups 10 / 35

  13. Finite random groups Theorem (Dixon, 1969) � S k S k w.g.p. � σ, τ � = A k A k 1 2 3 k ! # permutations cyclic S k or A k groups 10 / 35

  14. Random automata 1 | 2 1 | 3 2 | 3 3 | 1 2 | 2 3 | 1 3 | 1 1 | 3 a c b 2 | 2 Is the generated group finite? . 11 / 35

  15. Random automata 1 | 2 1 | 3 2 | 3 3 | 1 2 | 2 3 | 1 3 | 1 1 | 3 a c b 2 | 2 Is the generated group finite? Yes, size 2 64 · 3 4 . 11 / 35

  16. Random automata 1 | 2 1 | 3 2 | 3 3 | 1 2 | 2 3 | 1 3 | 1 1 | 3 a c b 2 | 2 Is the generated group finite? Yes, size 2 64 · 3 4 . Difficult problem + unefficient rejection sampling. 11 / 35

  17. Random automata 0 Antonenko + Russeiev 2 | 3 3 | 2 1 3 2 | 2 2 | 1 1 | 1 1 | 3 4 3 | 1 1 | 3 1 | 3 2 | 1 3 | 2 3 | 2 1 | 1 8 2 | 3 2 1 | 1 3 | 2 2 | 3 1 | 2 3 | 2 1 | 2 2 | 3 7 5 6 2 | 1 3 | 1 3 | 3 1 | 1 9 2 | 2 1 | 1 3 | 3 2 | 2 3 | 3 11 / 35

  18. Random automata 0 Antonenko + Russeiev 2 | 3 3 | 2 1 3 2 | 2 2 | 1 1 | 1 1 | 3 4 3 | 1 1 | 3 1 | 3 2 | 1 3 | 2 3 | 2 1 | 1 8 2 | 3 2 1 | 1 3 | 2 2 | 3 1 | 2 3 | 2 1 | 2 2 | 3 7 5 6 2 | 1 3 | 1 3 | 3 1 | 1 9 2 | 2 1 | 1 3 | 3 2 | 2 3 | 3 cyclic automata 11 / 35

  19. Random 2-state cyclic automata σ � � = � ( σ, τ ) , ( τ, σ ) � τ 12 / 35

  20. Random 2-state cyclic automata σ � � = � ( σ, τ ) , ( τ, σ ) � τ Contribution  S k × S k  σ  � � = ( A k × A k ) ⋊ � ( π, π ) � τ  A k × A k  12 / 35

  21. Random 2-state cyclic automata σ � � = � ( σ, τ ) , ( τ, σ ) � τ Contribution  S k × S k  σ  � � = ( A k × A k ) ⋊ � ( π, π ) � τ  A k × A k  S k × S k A k × A k ( A k × A k ) ⋊ � ( π, π ) � 12 / 35

  22. Random 2-state cyclic automata σ � � = � ( σ, τ ) , ( τ, σ ) � τ Contribution  S k × S k  σ  � � = ( A k × A k ) ⋊ � ( π, π ) � τ  A k × A k  S k × S k S k × S k A k × A k ( A k × A k ) ⋊ � ( π, π ) � S k × A k A k × A k ( A k × A k ) ⋊ � ( π, π ) � 12 / 35

  23. Random automata 0 Antonenko + Russeiev 2 | 3 3 | 2 1 3 2 | 2 2 | 1 1 | 1 1 | 3 4 3 | 1 1 | 3 1 | 3 2 | 1 3 | 2 3 | 2 1 | 1 8 2 | 3 2 1 | 1 3 | 2 2 | 3 1 | 2 3 | 2 1 | 2 2 | 3 7 5 6 2 | 1 3 | 1 3 | 3 1 | 1 9 2 | 2 1 | 1 3 | 3 2 | 2 3 | 3 cyclic automata 13 / 35

  24. 1 | 1 b d 0 | 0 0 | 0 0 | 0 σ 1 σ n 1 | 1 1 | 1 0 | 1 . . . a e n f 1 0 | 1 1 | 2 0 | 1 1 | 0 2 | 0 1 | 0 σ a b 3 | 3 2 | 0 0 | 2 3 | 3 τ 1 | 1 ? ”complexity” structurally finite structurally infinite S k × S k S k finite by construction A k S k × A k A k × A k ⋊ � ( π, π ) � md reduction Dixon like decidable finiteness Dixon like (conj.) 2-state bireversible automata 14 / 35

  25. Asymptotics Theorem (Dixon 1969,2005) P ( � σ, τ � = S or A ) ∼ 1 − 1 / k − 1 / k 2 − 4 / k 3 − 23 / k 4 − 171 / k 5 − · · · 15 / 35

  26. Asymptotics Theorem (Dixon 1969,2005) P ( � σ, τ � = S or A ) ∼ 1 − 1 / k − 1 / k 2 − 4 / k 3 − 23 / k 4 − 171 / k 5 − · · · SNC: ∃ w ( σ, τ ) , | w ( σ, τ ) | � = | w ( τ, σ ) | Lemma P ( | σ | = | τ | ) → 0 15 / 35

  27. Asymptotics Theorem (Dixon 1969,2005) P ( � σ, τ � = S or A ) ∼ 1 − 1 / k − 1 / k 2 − 4 / k 3 − 23 / k 4 − 171 / k 5 − · · · SNC: ∃ w ( σ, τ ) , | w ( σ, τ ) | � = | w ( τ, σ ) | Lemma P ( | σ | = | τ | ) → 0 proof: [Erd˝ os, Tur´ an 1967] log | σ | − 1 / 2 log 2 k → N (0 , 1) √ 3 log 3 / 2 k 1 / 15 / 35

  28. Asymptotics Theorem (Dixon 1969,2005) P ( � σ, τ � = S or A ) ∼ 1 − 1 / k − 1 / k 2 − 4 / k 3 − 23 / k 4 − 171 / k 5 − · · · SNC: ∃ w ( σ, τ ) , | w ( σ, τ ) | � = | w ( τ, σ ) | Lemma P ( | σ | = | τ | ) → 0 Conjecture P ( | σ | � = | τ | ) ∼ C / k 2 15 / 35

  29. Asymptotics Theorem (Dixon 1969,2005) P ( � σ, τ � = S or A ) ∼ 1 − 1 / k − 1 / k 2 − 4 / k 3 − 23 / k 4 − 171 / k 5 − · · · SNC: ∃ w ( σ, τ ) , | w ( σ, τ ) | � = | w ( τ, σ ) | Lemma P ( | σ | = | τ | ) → 0 Conjecture P ( | σ | � = | τ | ) ∼ C / k 2 rmq: 1 √ pk k (1 − 1 / p ) exp ( − k (1 − 1 / p ) + k 1 / p ) P ( | σ | = p ) ∼ 15 / 35

  30. 0 | 1 0 | 0 1 | 0 1 | 1 Mealy automata dynamics of the action The set of singular points singular of a contracting automaton is described by a B¨ uchi points automaton [DGKPR’16] Schreier graphs ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ξ singular q � � � � � Wang A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] tillings N 0 | 1 0 x , 0 x , 1 y x 1 | 0 x y 1 | 1 0 1 0 | 0 y y y , 0 y , 1 1 16 / 35

  31. Stabilisers and singular points The stabilisers of an infinite point ξ is Stab �A� ( ξ ) = { g ∈ �A� | g ( ξ ) = ξ } 17 / 35

  32. Stabilisers and singular points The stabilisers of an infinite point ξ is Stab �A� ( ξ ) = { g ∈ �A� | g ( ξ ) = ξ } ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] . . . q � � � � ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] 17 / 35

  33. Stabilisers and singular points The stabilisers of an infinite point ξ is Stab �A� ( ξ ) = { g ∈ �A� | g ( ξ ) = ξ } ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] . . . q � � � � ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] Example 1 | 1 b d 1 | 1 1 | 1 0 | 0 c 0 | 0 0 | 0 a e 0 | 1 1 | 0 0 | 0 1 | 1 ρ e , ρ b , ρ c , ρ d ∈ Stab �G� (1 ω ) studied by Y. Vorobets 17 / 35

  34. Stabilisers and singular points The stabilisers of an infinite point ξ is Stab �A� ( ξ ) = { g ∈ �A� | g ( ξ ) = ξ } ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] . . . q � � � � ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] Interesting elements Example 0 | 0 0 | 1 1 | 1 1 | 1 1 | 0 b d 2 | 2 2 | 2 1 | 1 e a 1 | 1 0 | 0 c 0 | 0 0 | 0 2 ω is stabilised by ρ a a e 0 | 1 1 | 0 0 | 0 1 | 1 ρ e , ρ b , ρ c , ρ d ∈ Stab �G� (1 ω ) studied by Y. Vorobets 17 / 35

  35. Stabilisers and singular points The stabilisers of an infinite point ξ is Stab �A� ( ξ ) = { g ∈ �A� | g ( ξ ) = ξ } ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] . . . q � � � � ξ [1] ξ [2] ξ [3] ξ [4] ξ [5] Interesting elements Example 0 | 0 0 | 1 1 | 1 1 | 1 1 | 0 b d 2 | 2 2 | 2 1 | 1 e a 1 | 1 0 | 0 c 0 | 0 0 | 0 2 ω is stabilised by ρ a a e 0 | 1 Singular points 1 | 0 0 | 0 1 | 1 ρ e , ρ b , ρ c , ρ d ∈ Stab �G� (1 ω ) ξ singular if ∃ g stabilizing ξ and studied by Y. Vorobets avoiding ending in e 17 / 35

  36. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N 18 / 35

  37. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N A ℓ 18 / 35

  38. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N A ℓ q 18 / 35

  39. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N q δ ξ [: n ] ( q ) 18 / 35

  40. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N q δ ξ [: n ] ( q ) δ ζ [: n ′ ] ( q ) ζ 18 / 35

  41. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N q δ ξ [: n ] ( q ) p δ ζ [: n ′ ] ( q ) ζ 18 / 35

  42. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N A m q δ ξ [: n ] ( q ) u p v δ ζ [: n ′ ] ( q ) ζ 18 / 35

  43. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N A m q δ ξ [: n ] ( q ) u p v δ ζ [: n ′ ] ( q ) ζ 1 | 1 0 | 1 0 | 0 e t 1 | 1 18 / 35

  44. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N A m q δ ξ [: n ] ( q ) u p v δ ζ [: n ′ ] ( q ) ζ 1 | 1 1 | 1 0 | 1 0 | 1 0 | 0 t 2 ee et 1 | 1 18 / 35

  45. Contracting automata A contracting ⇐ ⇒ ∃ finite N , ∀ q , ∀ ξ , ∃ n , δ ξ [: n ] ( q ) ∈ N ξ A ℓ N A m q δ ξ [: n ] ( q ) u p v δ ζ [: n ′ ] ( q ) ζ 1 | 1 1 | 1 1 | 1 0 | 1 0 | 1 0 | 1 0 | 0 t 2 e t ℓ t 1 | 1 18 / 35

  46. Contracting automata and singular points 0 | 1 ba − 1 ab − 1 Basilica N automaton 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  47. Contracting automata and singular points 0 | 1 ba − 1 ab − 1 0 0 Basilica N automaton a 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 a 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  48. Contracting automata and singular points 0 | 1 ba − 1 ab − 1 0 0 Basilica N automaton a b 1 | 0 1 | 0 0 | 1 0 1 | 0 1 | 1 a − 1 b a b 0 | 0 0 | 1 e 1 | 0 0 | 0 1 a b − 1 a e 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  49. Contracting automata and singular points 0 | 1 ba − 1 ab − 1 0 0 Basilica N automaton a a ∈ N b 1 | 0 1 | 0 0 | 1 0 0 1 | 0 1 | 1 a − 1 b a e b 0 | 0 0 | 1 e 1 | 0 0 | 0 1 a b − 1 a e e 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  50. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ξ singular q � � � � � ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 0 | 1 ba − 1 ab − 1 Basilica N automaton 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  51. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 0 | 1 ba − 1 ab − 1 Basilica N automaton 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  52. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 0 | 1 ba − 1 ab − 1 Basilica N automaton 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  53. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 0 | 1 ba − 1 ab − 1 Basilica N automaton 1 | 0 1 | 0 0 | 1 1 | 0 1 | 1 a − 1 b 0 | 0 0 | 1 e 1 | 0 0 | 0 a b − 1 1 | 1 0 | 1 1 | 1 , 0 | 0 19 / 35

  54. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ba − 1 ab − 1 Basilica N automaton 1 | 1 a − 1 b 0 | 0 e 0 | 0 a b − 1 1 | 1 1 | 1 , 0 | 0 19 / 35

  55. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ba − 1 ab − 1 B¨ uchi automaton 1 a − 1 b 1 e 0 0 a b − 1 1 , 0 19 / 35

  56. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] ba − 1 ab − 1 B¨ uchi automaton 1 a − 1 b 1 e 0 0 a b − 1 1 , 0 Contribution Sing( B ) = ∅ . 19 / 35

  57. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 1 | 1 b d 1 | 1 1 | 1 0 | 0 c 0 | 0 0 | 0 a e 0 | 1 1 | 0 0 | 0 1 | 1 19 / 35

  58. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 1 | 1 1 b d b d 1 1 | 1 1 | 1 1 0 | 0 c 0 | 0 c 0 0 0 | 0 0 a e a e 0 | 1 1 | 0 0 | 0 0 1 | 1 1 19 / 35

  59. Contracting automata and singular points ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] N ξ singular q � � � � � A m ξ [0] ξ [1] ξ [ ℓ ] ξ [ ℓ + 1] 1 | 1 1 b d b d 1 1 | 1 1 | 1 1 0 | 0 c 0 | 0 c 0 0 0 | 0 0 a e a e 0 | 1 1 | 0 0 | 0 0 1 | 1 1 Proposition [Vorobets, DGKPR] Sing( G ) = (0 + 1) ∗ 1 ω . 19 / 35

  60. finiteness Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of automaton prime size cannot generate infinite Burnside groups patterns infinite [MFCS’16 w. Klimann] and group Burnside properties x 3 A ℓ − 1 i | . 3 x � = y z i | . A ℓ 1 2 y growth A ℓ +1 The set of of a contracting is described automaton 20 / 35

  61. About the Grigorchuk automaton 1 | 1 b d 1 | 1 1 | 1 0 | 0 c 0 | 0 0 | 0 a e 0 | 1 0 | 0 1 | 0 1 | 1 21 / 35

  62. About the Grigorchuk automaton Actions of the states on the letters: 1 | 1 ρ a : 0 �→ 1 �→ 0 b d 1 | 1 ρ b , ρ c , ρ d , ρ e : 0 �→ 0; 1 �→ 1 1 | 1 0 | 0 c 0 | 0 → permutations 0 | 0 a e 0 | 1 0 | 0 1 | 0 1 | 1 21 / 35

  63. About the Grigorchuk automaton Actions of the states on the letters: 1 | 1 ρ a : 0 �→ 1 �→ 0 b d 1 | 1 ρ b , ρ c , ρ d , ρ e : 0 �→ 0; 1 �→ 1 1 | 1 0 | 0 c 0 | 0 → permutations 0 | 0 → invertible a e 0 | 1 0 | 0 1 | 0 1 | 1 21 / 35

  64. About the Grigorchuk automaton Actions of the states on the letters: 1 | 1 ρ a : 0 �→ 1 �→ 0 b d 1 | 1 ρ b , ρ c , ρ d , ρ e : 0 �→ 0; 1 �→ 1 1 | 1 0 | 0 c 0 | 0 → permutations 0 | 0 → invertible a e Action of a letter on the states: 0 | 1 0 | 0 1 | 0 δ 0 : a , d , e �→ e ; b , c �→ a 1 | 1 → not a permutation 21 / 35

  65. About the Grigorchuk automaton Actions of the states on the letters: 1 | 1 ρ a : 0 �→ 1 �→ 0 b d 1 | 1 ρ b , ρ c , ρ d , ρ e : 0 �→ 0; 1 �→ 1 1 | 1 0 | 0 c 0 | 0 → permutations 0 | 0 → invertible a e Action of a letter on the states: 0 | 1 0 | 0 1 | 0 δ 0 : a , d , e �→ e ; b , c �→ a 1 | 1 → not a permutation → non-reversible 21 / 35

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