The torsion problem for the automorphism group of a full Z d -shift and its topological fullgroup.

Transcription

1 The torsion problem for the automorphism group of a full Z d -shift and its topological fullgroup. Sebastián Barbieri From a joint work with Jarkko Kari and Ville Salo LIP, ENS de Lyon CNRS INRIA UCBL Université de Lyon Wandering Seminar, Wrocław March, 216

2 Motivation Given a fullshift (A Z, σ) recall that its automorphism group is given by Aut(A Z ) = {φ : A Z A Z homeomorpism, [σ, φ] = id}

3 Motivation Given a fullshift (A Z, σ) recall that its automorphism group is given by Aut(A Z ) = {φ : A Z A Z homeomorpism, [σ, φ] = id} It is still unknown whether Aut({, 1} Z ) = Aut({, 1, 2} Z ), but we know that for any pair of alphabets A, B with at least two elements Aut(A Z ) Aut(B Z )

4 Motivation Given a fullshift (A Z, σ) recall that its automorphism group is given by Aut(A Z ) = {φ : A Z A Z homeomorpism, [σ, φ] = id} It is still unknown whether Aut({, 1} Z ) = Aut({, 1, 2} Z ), but we know that for any pair of alphabets A, B with at least two elements Aut(A Z ) Aut(B Z ) Nevertheless, we know that Aut({, 1} Z ) Aut({, 1, 2, 3} Z ). The proof comes from studying the roots of elements in the center.

5 Motivation It might be a good idea to understand torsion in these groups. Definition (Torsion problem) Let G = S R be a finitely generated group. The torsion problem of G is the language TP(G) where TP(G) = {w (S S 1 ) n N such that w n = G 1}

6 Motivation It might be a good idea to understand torsion in these groups. Definition (Torsion problem) Let G = S R be a finitely generated group. The torsion problem of G is the language TP(G) where Example TP(G) = {w (S S 1 ) n N such that w n = G 1} Let Z Z/3Z = a, b [a, b], b 3. Then TP(Z Z/3Z) = {w {a, b, a 1, b 1 } w a = w a 1}

7 Talk highlights Theorem (B, Kari, Salo) For any finite alphabet A 2, Aut(A Z ) contains a finitely generated subgroup with undecidable torsion problem. The same result also holds for any sofic subshift of positive entropy.

8 Talk highlights Theorem (B, Kari, Salo) For any finite alphabet A 2, Aut(A Z ) contains a finitely generated subgroup with undecidable torsion problem. The same result also holds for any sofic subshift of positive entropy. The topological fullgroup of a dynamical system (X, T ) where T : G X is the group [[T ]] = {φ Homeo(X) s : X G continuous, φ(x) = T s(x) (x)}. Theorem (B, Kari, Salo) Let (A Zd, σ) be a full shift and A 2. The topological fullgroup [[σ]] contains a finitely generated subgroup with undecidable torsion problem if and only if d 2.

9 Background Recall that a Turing machine is defined by a rule : δ T : Σ Q Σ Q { 1,, 1}

10 Background Recall that a Turing machine is defined by a rule : δ T : Σ Q Σ Q { 1,, 1} q δ T (, q) = (, r, 1)

11 Background Recall that a Turing machine is defined by a rule : δ T : Σ Q Σ Q { 1,, 1} r δ T (, q) = (, r, 1)

12 Background This defines a natural action T : Σ Z Q Σ Z Q q T r

13 Background The composition of two actions T T is not necessarily an action generated by a Turing machine. if the action T is a bijection then the inverse it not necessarily an action generated by a Turing machine.

14 Background The composition of two actions T T is not necessarily an action generated by a Turing machine. if the action T is a bijection then the inverse it not necessarily an action generated by a Turing machine. As in cellular automata, the class of CA with radius bounded by some k N is not closed under composition or inverses.

15 Definition Let s get rid of these constrains. Given F, F finite subsets of a group G, consider instead of δ T a function : f T : Σ F Q Σ F Q G,

16 Definition Let s get rid of these constrains. Given F, F finite subsets of a group G, consider instead of δ T a function : f T : Σ F Q Σ F Q G, Let F = F = {, 1, 2} 2, then f T (p, q) = (p, q, d) means : p p Turn state q into state q Move head by d.

17 Moving head model f T defines naturally an action T : Σ G Q Z d Σ G Q Z d T q 2 q 1 f (, q 1 ) = (, q 2, (1, 1)) F = {(, ), (1, ), (1, 1)}

18 Moving head model f T defines naturally an action T : Σ G Q Z d Σ G Q Z d T q 2 q 1 f (, q 1 ) = (, q 2, (1, 1)) F = {(, ), (1, ), (1, 1)} Let Σ = n and Q = k. (RTM(G, n, k), ) is the group of all such T which are bijective. It can be seen as a group of CA over a sofic shift.

19 Moving tape model f T defines naturally an action T : Σ G Q Σ G Q T f q 1 q 2 f (, q 1 ) = (, q 2, (1, 1)) F = {(, ), (1, ), (1, 1)}

20 Moving tape model f T defines naturally an action T : Σ G Q Σ G Q T f q 1 q 2 f (, q 1 ) = (, q 2, (1, 1)) F = {(, ), (1, ), (1, 1)} Let Σ = n and Q = k. (RTM fix (G, n, k), ) is the group of all such T which are bijective. It s like the topological fullgroup but admits local changes.

21 Equivalence of the models RTM fix (G, 1, k) = S k and G RTM(G, 1, k). Proposition If n 2 then : RTM fix (G, n, k) = RTM(G, n, k).

22 Properties of RTM Proposition If n 2 RTM(Z, n, k) is not finitely generated.

23 Properties of RTM Proposition If n 2 RTM(Z, n, k) is not finitely generated. Proof : We find an epimorphism from RTM to a non-finitely generated group. Let T RTM fix (Z, n, k), therefore, it has a cocycle s : Σ Z Q Z. Define α(t ) := E µ (s) = s(x, q)dµ, Σ Z Q One can check that α(t 1 T 2 ) = α(t 1 ) + α(t 2 ). Therefore α : RTM(Z, n, k) Q is an homomorphism

24 Properties of RTM Now consider the machine T SURF,m where for all a Σ and q Q : q a q T SURF,m a f ( m a, q) = (a m, q, 1). Otherwise f (u, q) = (u, q, ).

25 Properties of RTM Now consider the machine T SURF,m where for all a Σ and q Q : q a q T SURF,m a f ( m a, q) = (a m, q, 1). Otherwise f (u, q) = (u, q, ). T SURF,m RTM(Z, n, k) and α(t SURF,m ) = 1/n m

26 Properties of RTM Now consider the machine T SURF,m where for all a Σ and q Q : q a q T SURF,m a f ( m a, q) = (a m, q, 1). Otherwise f (u, q) = (u, q, ). T SURF,m RTM(Z, n, k) and α(t SURF,m ) = 1/n m (1/n m ) m N α(rtm(z, n, k)) which is thus a non-finitely generated subgroup of Q.

27 Computability properties Given a finite rules : f, f : It is decidable (in any model) whether T f = T f. We can effectively calculate a rule for T f T f. It is decidable whether T f is reversible. If it is, we can effectively compute a rule for T 1 f.

28 Computability properties Given a finite rules : f, f : It is decidable (in any model) whether T f = T f. We can effectively calculate a rule for T f T f. It is decidable whether T f is reversible. If it is, we can effectively compute a rule for T 1 f. RTM(Z d, n, k) is a recursively presented group with decidable word problem. (Unlike Aut(A Z ))

29 Interesting subgroups of RTM LP(G, n, k) Local permutations. q T π r

30 Interesting subgroups of RTM LP(G, n, k) Local permutations. RFA(G, n, k) Reversible finite-state automata. q T r Note that for ({,..., n 1} G, σ) then [[σ]] = RFA(G, n, 1).

31 Interesting subgroups of RTM LP(G, n, k) Local permutations. RFA(G, n, k) Reversible finite-state automata. OB(G, n, k) Oblivous machines LP, σ.

32 Interesting subgroups of RTM LP(G, n, k) Local permutations. RFA(G, n, k) Reversible finite-state automata. OB(G, n, k) Oblivous machines LP, σ. EL(G, n, k) Elementary machines LP, RFA.

33 The torsion problem for [[σ]]

34 The torsion problem for RFA RFA(Z, n, k) has decidable torsion problem. Proof idea : As Z is two-ended, any non-torsion machine must shift to the left or right in at least a periodic configuration.

35 The torsion problem for RFA RFA(Z, n, k) has decidable torsion problem. Proof idea : As Z is two-ended, any non-torsion machine must shift to the left or right in at least a periodic configuration. Theorem RFA(Z d, n, k) has a finitely generated subgroup with undecidable torsion problem for d, n 2.

36 The torsion problem for RFA RFA(Z, n, k) has decidable torsion problem. Proof idea : As Z is two-ended, any non-torsion machine must shift to the left or right in at least a periodic configuration. Theorem RFA(Z d, n, k) has a finitely generated subgroup with undecidable torsion problem for d, n 2.

37 The snake problem ɛ Can we tile the plane in a way which produces a bi-infinite path?

38 The snake problem Theorem (Kari) The snake tiling problem is undecidable. The proof uses a plane filling curve generated by a substitution.

39 The snake problem Theorem (Kari) The snake tiling problem is undecidable. The proof uses a plane filling curve generated by a substitution. For every instance of the snake tiling problem, one can construct T RFA which walks the path of the snake, and turns back if it encounters a problem.

40 The torsion problem for RFA : Cheating version We ll first do it by cheating : Arbitrary alphabet τ as an instance of the snake tiling problem and at least two states L, R. Let t be the tile at (, ). If t = ɛ, do nothing. Otherwise : If the state is L. Check the tile in the direction left(t). If it matches correctly with t move the head to that position, otherwise switch the state to R. If the state is R. Check the tile in the direction right(t). If it matches correctly with t move the head to that position, otherwise switch the state to L

41 The torsion problem for RFA : The real thing We are going to code everything in a binary alphabet and use no states l 1 l 2 r 1 r 2 b 1 b 2 b 3 b 4 b 5 Let C be the set of all patterns of this form.

42 The torsion problem for RFA : The real deal Consider the group spanned by the following machines : 1 {T v } v D that shifts by v 2 T walk that walks along the direction codified by l 1, l 2 or r 1, r 2 depending on the direction bit. 3 {g c } c C that flips the direction bit if the current pattern is c C, 4 {h c } c C that flips the auxiliary bit if the current pattern is c C, 5 {g +,c } c C that adds the auxiliary bit to the direction bit if the current pattern is c C, and 6 {h +,c } c C that adds the direction bit to the auxiliary bit if the current pattern is c C,

43 The torsion problem for RFA : The real deal The previous group spans the machines g p and h p for patterns p composed of fragments of c in compatible positions.

44 The torsion problem for RFA : The real deal The previous group spans the machines g p and h p for patterns p composed of fragments of c in compatible positions. g p = (T 7 v g +,c T 7 v h p F \{ v} )2. h p = (T 7 v h +,c T 7 v g p F \{ v} )2. Finally, we use these machines to code the first ones.

45 The torsion problem for RFA : The real deal M(t) =

46 The torsion problem for RFA : The real deal T = (T walk ) M g p p M c C Acts as the first machine, but using these coded macrotiles. g c

47 The torsion problem for RFA : The real deal T = (T walk ) M p M g p c C Acts as the first machine, but using these coded macrotiles. Corollary Let d 2 and σ be the shift action of Z d over a full shift A Zd where A 2. Then the full group [[σ]] contains a finitely generated subgroup with undecidable torsion problem. g c

48 The torsion problem for Aut(A Z )

49 TP(Aut(A Z )) is undecidable. We want to prove that the torsion problem is undecidable for a f.g. subgroup of Aut(A Z ). The sketch is as follows : 1 The torsion problem for reversible classical Turing machines is undecidable [Kari, Ollinger 28].

50 TP(Aut(A Z )) is undecidable. We want to prove that the torsion problem is undecidable for a f.g. subgroup of Aut(A Z ). The sketch is as follows : 1 The torsion problem for reversible classical Turing machines is undecidable [Kari, Ollinger 28]. 2 Classical Turing machines embed into EL(Z, n, k).

51 TP(Aut(A Z )) is undecidable. We want to prove that the torsion problem is undecidable for a f.g. subgroup of Aut(A Z ). The sketch is as follows : 1 The torsion problem for reversible classical Turing machines is undecidable [Kari, Ollinger 28]. 2 Classical Turing machines embed into EL(Z, n, k). 3 EL(Z, n, k) is finitely generated.

52 TP(Aut(A Z )) is undecidable. We want to prove that the torsion problem is undecidable for a f.g. subgroup of Aut(A Z ). The sketch is as follows : 1 The torsion problem for reversible classical Turing machines is undecidable [Kari, Ollinger 28]. 2 Classical Turing machines embed into EL(Z, n, k). 3 EL(Z, n, k) is finitely generated. 4 There exists a "torsion preserving function" from EL(Z, n, k) to Aut(A Z ) Classical EL Aut(A Z )

53 OB(Z d, n, k) is finitely generated. This proof is inspired both on the existence of strongly universal reversible gates for permutations of Σ m and the Juschenko Monod proof for the fullgroup of minimal actions.

54 OB(Z d, n, k) is finitely generated. This proof is inspired both on the existence of strongly universal reversible gates for permutations of Σ m and the Juschenko Monod proof for the fullgroup of minimal actions. The set A 1 A 2 A 3 generate OB(Z d, n, k) = LP(Z d, n, k), σ. A 1 = Shifts T ei for {e i } i d a base of Z d. A 2 =All T π LP(Z d, n, k) of fixed support E Z d of size 4. A 3 = The swaps of symbols in positions (, e i ).

55 EL(Z, n, k) is finitely generated. EL(Z, n, k) = OB(Z, n, k), RFA(Z, n, k)

56 EL(Z, n, k) is finitely generated. EL(Z, n, k) = OB(Z, n, k), RFA(Z, n, k) We can show that RFA(Z, n, k) is generated by shifts and controlled position swaps. f is controlled position swap if for some u, v Σ, f (xu.avy) = xua.vy and f (xua.vy) = xu.avy.

57 EL(Z, n, k) is finitely generated. EL(Z, n, k) = OB(Z, n, k), RFA(Z, n, k) We can show that RFA(Z, n, k) is generated by shifts and controlled position swaps. f is controlled position swap if for some u, v Σ, f (xu.avy) = xua.vy and f (xua.vy) = xu.avy. We only need to implement controlled position swaps [technical].

58 From EL(Z, n, k) to Aut(A Z ) Definition Let G and H be groups. We say a function φ : G H is finiteness preserving (FP) if the following holds : If F G is finite, then the group w w F G is infinite if and only if the group φ(w 1 )φ(w 2 ) φ(w w ) w F is infinite.

59 From EL(Z, n, k) to Aut(A Z ) Definition Let G and H be groups. We say a function φ : G H is finiteness preserving (FP) if the following holds : If F G is finite, then the group w w F G is infinite if and only if the group φ(w 1 )φ(w 2 ) φ(w w ) w F is infinite. Lemma An FP function from a f.g. H to G forces the torsion problem of G to be harder than the one of H.

60 Construction of the FP function Let A = {Σ 2 ({, } (Q {, }))}.

61 Construction of the FP function Let A = {Σ 2 ({, } (Q {, }))}. Parse the third layer into zones ( (q, a) ) q q q q q

62 Construction of the FP function Let A = {Σ 2 ({, } (Q {, }))}. Parse the third layer into zones ( (q, a) ). Define φ to act as a conveyor belt over each zone u u 1 u 2 u 3 u 4 u 5 u 6 u u 1 u 2 u 3 u 4 u 5 T w w 1 u 1 w 2 u w 3 u u 1 w w 1 w 2 w 3 w 4 v v 1 v 2 v 3 v 4 v 5 v 6 q u 6 r w 4 v v 1 v 2 v 3 v 4 v 5 w 5 q v v 1 v 2 v3 v 4 v 5 v 6 v v 1 v 2 v3 v 4 v 5 w 5 r f T (u 2u 3.u 4u 5u 6v 6, q) = (w w 1.w 2w 3w 4w 5, r, 4)

63 Construction of the FP function Let A = {Σ 2 ({, } (Q {, }))}. Parse the third layer into zones ( (q, a) ). Define φ to act as a conveyor belt over each zone φ is a computable FP function.

64 Construction of the FP function Let A = {Σ 2 ({, } (Q {, }))}. Parse the third layer into zones ( (q, a) ). Define φ to act as a conveyor belt over each zone φ is a computable FP function. There is a finitely generated subgroup of Aut(A Z ) with undecidable torsion problem. As Aut(A Z ) Aut({, 1} Z ) the same is valid for any full shift, mixing SFTs, sofic shift of positive entropy, etc.

65 Thank you for your attention!