The domino problem for structures between Z and Z 2.

Transcription

1 1/29 The domino problem for structures between Z and Z 2. Sebastián Barbieri LIP, ENS de Lyon Turku October, 215

2 2/29 G-subshifts Consider a group G. I A is a finite alphabet. Ex : A = {, 1}. I A G is the set of functions x : G æa. I : G A G æa G is the shift action given by : g (x) h = x g 1 h.

3 2/29 G-subshifts Consider a group G. I A is a finite alphabet. Ex : A = {, 1}. I A G is the set of functions x : G æa. I : G A G æa G is the shift action given by : Definition : G-subshift g (x) h = x g 1 h. X µa G is a G-subshift if it is invariant under the action of and closed for the product topology on A G.

4 2/29 G-subshifts Consider a group G. I A is a finite alphabet. Ex : A = {, 1}. I A G is the set of functions x : G æa. I : G A G æa G is the shift action given by : Definition : G-subshift g (x) h = x g 1 h. X µa G is a G-subshift if it is invariant under the action of and closed for the product topology on A G. Alternative definition : G-subshift X is a G-subshift if it can be defined as the set of configurations which avoid a set forbidden patterns : F µ t F µg, F <Œ AF such that : X = X F := {x œa G p œf: x}.

5 3/29 Example in Z 2 :Fibonaccishift Example : Fibonacci shift. X Fib is the set of assignments of Z 2 to {, 1} such that there are no two adjacent ones.

6 Example in Z 2 :Fibonaccishift Example : Fibonacci shift. X Fib is the set of assignments of Z 2 to {, 1} such that there are no two adjacent ones F = /29

7 Example : one-or-less subshift Example : one-or-less subshift. X Æ1 := {x œ{, 1} Zd {z œ Z d : x z = 1} Æ 1}. 1 3/29

8 4/29 Fibonacci in F s 2 F = 1 s

9 5/29 Subshifts of finite type. I What about if we only consider local rules? Definition : subshift of finite type. A G-subshift is of finite type (SFT) if it can be defined by a finite set F of forbidden patterns. Example : Both Fibonacci subshifts shown before are of finite type. X Æ1 isn t.

10 5/29 Subshifts of finite type. I What about if we only consider local rules? Definition : subshift of finite type. A G-subshift is of finite type (SFT) if it can be defined by a finite set F of forbidden patterns. Example : Both Fibonacci subshifts shown before are of finite type. X Æ1 isn t. I Given a finite set of forbidden patterns, can we decide if the G-subshift produced by them is non-empty?

11 6/29 The domino problem. I Every finite alphabet can be identified as a finite subset of N. Domino problem. DP(G) ={F µ N ú G F < Œ, X F = ÿ}.

12 6/29 The domino problem. I Every finite alphabet can be identified as a finite subset of N. Domino problem. DP(G) ={F µ N ú G F < Œ, X F = ÿ}. I if G is finitely generated by the set S, wecancodifyeach pattern as a function from a finite set of words in (S fi S 1 ) ú to N. I Therefore, DP(G) can be written as a formal language. We say G has decidable domino problem if DP(G) is Turing-decidable.

13 6/29 The domino problem. I Every finite alphabet can be identified as a finite subset of N. Domino problem. DP(G) ={F µ N ú G F < Œ, X F = ÿ}. I if G is finitely generated by the set S, wecancodifyeach pattern as a function from a finite set of words in (S fi S 1 ) ú to N. I Therefore, DP(G) can be written as a formal language. We say G has decidable domino problem if DP(G) is Turing-decidable. Question : Which groups have decidable domino problem?

14 7/29 The easy case G = Z. Theorem : The set of configurations of a Z-SFT can be characterized as the set of bi-infinite walks in a finite graph.

15 7/29 The easy case G = Z. Theorem : The set of configurations of a Z-SFT can be characterized as the set of bi-infinite walks in a finite graph. Example : Consider the Fibonacci shift given by F = {11}

16 7/29 The easy case G = Z. Theorem : The set of configurations of a Z-SFT can be characterized as the set of bi-infinite walks in a finite graph. Example : Consider the Fibonacci shift given by F = {11}. 1 1 As the graph is finite, a Z-SFT is non-empty if and only if its Rauzy graph contains a cycle, thus DP(Z) is decidable. 1

17 8/29 The not so easy case : G = Z 2 The name "Domino problem" comes from the G = Z 2 case.

18 8/29 The not so easy case : G = Z 2 The name "Domino problem" comes from the G = Z 2 case. Wang tiles are unit squares with colored edges, the forbidden patterns are implicit in the alphabet.

19 9/29 Wang s conjecture Wang s conjecture (1961) If a set of Wang tiles can tile the plane, then they can always be arranged to do so periodically.

20 9/29 Wang s conjecture Wang s conjecture (1961) If a set of Wang tiles can tile the plane, then they can always be arranged to do so periodically. If Wang s conjecture is true, we can decide if a set of Wang tiles can tile the plane! Semi-algorithm 1 : 1 Accept if there is a periodic configuration. 2 loops otherwise Semi-algorithm 2 : 1 Accept if a block [, n] 2 cannot be tiled without breaking local rules. 2 loops otherwise

21 1/29 Wang s conjecture Theorem[Berger 1966] Wang s conjecture is FALSE

22 1/29 Wang s conjecture Theorem[Berger 1966] Wang s conjecture is FALSE His construction encodes a Turing machine using an alphabet of size His proof was later simplified by Robinson[1971]. A proof with a di erent approach was also presented by Kari[1996].

23 11/29 Robinson tileset The Robinson tileset, where tiles can be rotated.

24 12/29 General structure of the Robinson tiling Macro-tiles of level 1.

25 12/29 General structure of the Robinson tiling Macro-tiles of level 1. They behave like large.

26 From macro-tiles of level 1 to macro-tiles of level 2 13/29

27 From macro-tiles of level 1 to macro-tiles of level 2 13/29

28 From macro-tiles of level 1 to macro-tiles of level 2 13/29

29 From macro-tiles of level 1 to macro-tiles of level 2 13/29

30 From macro-tiles of level 1 to macro-tiles of level 2 13/29

31 From macro-tiles of level 1 to macro-tiles of level 2 13/29

32 From macro-tiles of level 1 to macro-tiles of level 2 13/29

33 From macro-tiles of level 1 to macro-tiles of level 2 13/29

34 From macro-tiles of level 1 to macro-tiles of level 2 13/29

35 From macro-tiles of level 1 to macro-tiles of level 2 13/29

36 From macro-tiles of level 1 to macro-tiles of level 2 13/29

37 14/29 From macro-tiles of level n to macro-tiles of level n + 1

38 15/29 Some recent results and facts in f.g. groups I If a group G has undecidable word problem DP(G) is undecidable. I Virtually free groups have decidable domino problem. I For virtually nilpotent groups : DP(G) is decidable if and only if it has two or more ends (213 Ballier, Stein). I Every virtually polycyclic group which is not virtually Z has undecidable domino problem (work in progress by Jeandel). I The domino problem is a quasi-isometry invariant for finitely presented groups (215 Cohen).

39 16/29 Going between Z and Z 2 (Joint work with M. Sablik) So far we have : I DP(Z) is decidable. I DP(Z 2 ) is undecidable. And if H Æ Z 2,theneitherH = 1, H = Z or H = Z 2.

40 16/29 Going between Z and Z 2 (Joint work with M. Sablik) So far we have : I DP(Z) is decidable. I DP(Z 2 ) is undecidable. And if H Æ Z 2,theneitherH = 1, H = Z or H = Z 2. We need to lose the group structure if we want to study intermediate structures.

41 Toy case : SierpiÒski triangle 17/29

42 18/29 Coding subsets of Z 2 as configurations. Let F µ Z 2 and define the configuration x F œ{, 1} Z2 : (x F ) z = And let Y = t zœz 2 { z(x F )}. I 1ifz œ F if not.

43 18/29 Coding subsets of Z 2 as configurations. Let F µ Z 2 and define the configuration x F œ{, 1} Z2 : (x F ) z = I 1ifz œ F if not. And let Y = t zœz 2 { z(x F )}. Given a set of forbidden patterns F we can define colorings of F as the configurations of X F over an alphabet A suchthatthe application fi : A Z2 æ{, 1} Z2 : yields an element of Y. fi(x) z = I 1ifxz = ifx z =

44 19/29 Formally... I Let Y Z2 be a Z 2 -subshift over the alphabet {, 1}. I Let F be a set of forbidden patterns over an alphabet A which does not forbid any pattern consisting only of.

45 19/29 Formally... I Let Y Z2 be a Z 2 -subshift over the alphabet {, 1}. I Let F be a set of forbidden patterns over an alphabet A which does not forbid any pattern consisting only of. Definition : Y -based subshift The Y -based subshift defined by F is the set : X Y,F := fi 1 (Y ) fl X F.

46 19/29 Formally... I Let Y Z2 be a Z 2 -subshift over the alphabet {, 1}. I Let F be a set of forbidden patterns over an alphabet A which does not forbid any pattern consisting only of. Definition : Y -based subshift The Y -based subshift defined by F is the set : X Y,F := fi 1 (Y ) fl X F. Definition : Y -based domino problem DP(Y ):={F µ N ú Z 2 F < Œ and X Y,F = { Z2 }}.

47 2/29 Back to the fractal structures... We focus on subshifts Y with a self-similar structure generated by substitutions.

48 2/29 Back to the fractal structures... We focus on subshifts Y with a self-similar structure generated by substitutions. I If Y contains a strongly periodic point which is not Z2 then DP(Y ) is undecidable. I It is easy to calculate a Hausdor dimension (in this case box-counting dimension). Is there a threshold in the dimension which enforces undecidability? I These subshifts can be defined by local rules (sofic subshifts) according to Mozes Theorem.

49 2/29 Back to the fractal structures... We focus on subshifts Y with a self-similar structure generated by substitutions. I If Y contains a strongly periodic point which is not Z2 then DP(Y ) is undecidable. I It is easy to calculate a Hausdor dimension (in this case box-counting dimension). Is there a threshold in the dimension which enforces undecidability? I These subshifts can be defined by local rules (sofic subshifts) according to Mozes Theorem. In particular we consider : substitutions over {, 1} such that the image of is a rectangle of zeros.

50 21/29 Example 1 : SierpiÒski triangle Consider the alphabet A = {, } and the self-similar substitution s such that : and

51 Example 2 : SierpiÒski carpet 22/29

52 Example 3 : The Bridge. 23/29

53 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle.

54 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy : I Consider a rectangle containing the union of the support of all forbidden patterns.

55 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy : I Consider a rectangle containing the union of the support of all forbidden patterns. I Suppose we can tile locally an iteration n of the substitution without producing forbidden patterns.

56 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy : I Consider a rectangle containing the union of the support of all forbidden patterns. I Suppose we can tile locally an iteration n of the substitution without producing forbidden patterns. I To construct a tiling of the next level, it su ces to "paste" three tilings of the iteration n without producing forbidden patterns.

57 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy : I Consider a rectangle containing the union of the support of all forbidden patterns. I Suppose we can tile locally an iteration n of the substitution without producing forbidden patterns. I To construct a tiling of the next level, it su ces to "paste" three tilings of the iteration n without producing forbidden patterns.

58 Toy case 1 : SierpiÒski triangle. 24/29

59 Toy case 1 : SierpiÒski triangle. 24/29

60 Toy case 1 : SierpiÒski triangle. 24/29

61 Toy case 1 : SierpiÒski triangle. 24/29

62 Toy case 1 : SierpiÒski triangle. 24/29

63 Toy case 1 : SierpiÒski triangle. 24/29

64 Toy case 1 : SierpiÒski triangle. 24/29

65 Toy case 1 : SierpiÒski triangle. 24/29

66 Toy case 1 : SierpiÒski triangle. 24/29

67 Toy case 1 : SierpiÒski triangle. 24/29

68 Toy case 1 : SierpiÒski triangle. 24/29

69 Toy case 1 : SierpiÒski triangle. 24/29

70 Toy case 1 : SierpiÒski triangle. 24/29

71 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy (continued) : I Keep the information about the pasting places (finite tuples) and build pasting rules (T 1, T 2, T 3 ) æ T 4.

72 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy (continued) : I Keep the information about the pasting places (finite tuples) and build pasting rules (T 1, T 2, T 3 ) æ T 4. I For each iteration n, construct the set of tuples observed in the pasting places. Construct the next set using this one.

73 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy (continued) : I Keep the information about the pasting places (finite tuples) and build pasting rules (T 1, T 2, T 3 ) æ T 4. I For each iteration n, construct the set of tuples observed in the pasting places. Construct the next set using this one. I This process either cycles (arbitrary iterations can be tiled) or ends up producing the empty set (the only valid tiling is Z2 ).

74 24/29 Toy case 1 : SierpiÒski triangle. Theorem : The domino problem is decidable in the SierpiÒski triangle. Proof strategy (continued) : I Keep the information about the pasting places (finite tuples) and build pasting rules (T 1, T 2, T 3 ) æ T 4. I For each iteration n, construct the set of tuples observed in the pasting places. Construct the next set using this one. I This process either cycles (arbitrary iterations can be tiled) or ends up producing the empty set (the only valid tiling is Z2 ). This technique can be extended to a big class of self-similar substitutions!

75 25/29 Toy case 2 : SierpiÒski carpet. Theorem : The domino problem is undecidable in the SierpiÒski carpet. Proof strategy : I Suppose we can simulate substitutions over the SierpiÒski carpet (using a bigger alphabet).

76 25/29 Toy case 2 : SierpiÒski carpet. Theorem : The domino problem is undecidable in the SierpiÒski carpet. Proof strategy : I Suppose we can simulate substitutions over the SierpiÒski carpet (using a bigger alphabet).

77 25/29 Toy case 2 : SierpiÒski carpet. æ sõ Ï Ï, Ï æ sõ Ï Ï Ï Ï, æ sõ Ï Ï.

78 25/29 Toy case 2 : SierpiÒski carpet. Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï

79 25/29 Toy case 2 : SierpiÒski carpet. a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 Ωæ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï a 1 a 2 a 2 a 3 a 3 a 3 a 3 a 4 a 4 b 1 b 2 ı ı b 3 b 4 b 1 b 2 b 2 b 3 b 3 b 3 b 3 b 4 b 4 c 1 ı c 2 c 3 ı c 4 c 1 c 2 c 3 c 4 c 1 ı c 2 c 3 ı c 4 c 1 c 2 c 2 c 3 c 3 c 3 c 3 c 4 c 4 d 1 d 2 ı ı d 3 d 4 d 1 d 2 d 2 d 3 d 3 d 3 d 3 d 4 d 4

80 25/29 Toy case 2 : SierpiÒski carpet. Theorem : The domino problem is undecidable in the SierpiÒski carpet. Proof strategy : I Suppose we can simulate substitutions over the SierpiÒski carpet (using a bigger alphabet). Use the substitution shown above to simulate arbitrarily big patterns of a Z 2 -subshift

81 25/29 Toy case 2 : SierpiÒski carpet. Theorem : The domino problem is undecidable in the SierpiÒski carpet. Proof strategy : I Suppose we can simulate substitutions over the SierpiÒski carpet (using a bigger alphabet). Use the substitution shown above to simulate arbitrarily big patterns of a Z 2 -subshift DP(Z 2 ) is reduced to the domino problem in the carpet.

82 25/29 Toy case 2 : SierpiÒski carpet. Theorem : The domino problem is undecidable in the SierpiÒski carpet. Proof strategy : I Suppose we can simulate substitutions over the SierpiÒski carpet (using a bigger alphabet). Use the substitution shown above to simulate arbitrarily big patterns of a Z 2 -subshift DP(Z 2 ) is reduced to the domino problem in the carpet. It only remains to show that we can simulate substitutions with local rules.

83 26/29 Toy case 2 : SierpiÒski carpet and Mozes We need to prove a modified version of Mozes theorem : Theorem : Mozes. The subshifts generated by Z 2 -substitutions are sofic (are the image of an SFT under a cellular automaton) We can prove a similar version for some Y -based subshifts. Among them the SierpiÒski carpet.

84 26/29 Toy case 2 : SierpiÒski carpet and Mozes (3, 1) (3, 2) (3, 3) (2, 1) (2, 3) (1, 1) (1, 1) (1, 2) (1, 3)

85 27/29 Conclusion We can generalize the ideas in the previous toy problems to attack classes of substitutions :

86 27/29 Conclusion We can generalize the ideas in the previous toy problems to attack classes of substitutions : Bounded Connectivity Decidable domino problem

87 27/29 Conclusion We can generalize the ideas in the previous toy problems to attack classes of substitutions : Strong grid Undecidable domino problem

88 27/29 Conclusion And separate the substitutions which we cannot classify into two groups :

89 27/29 Conclusion And separate the substitutions which we cannot classify into two groups : Isthmus Unknown

90 27/29 Conclusion And separate the substitutions which we cannot classify into two groups : Weak grid Unknown

91 28/29 Weak grid We got some ideas of how it might be...

92 29/29 Isthmus We don t know anything about this one.

93 29/29 Conclusion And about the Hausdor dimension?...

94 29/29 Conclusion And about the Hausdor dimension?... s n : n s Õ n : n There is no threshold.

95 Thank you for your attention! 29/29