Two notes on subshifts

Transcription

1 Two notes on subshifts Joseph S. Miller Special Session on Logic and Dynamical Systems Joint Mathematics Meetings, Washington, DC January 6, 2009

2 First Note Every Π 0 1 Medvedev degree contains a Π0 1 subshift. 2 / 14

3 First Note Every Π 0 1 Medvedev degree contains a Π0 1 subshift. Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. 2 / 14

4 First Note Every Π 0 1 Medvedev degree contains a Π0 1 subshift. Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. This answers a question of Steve Simpson. 2 / 14

5 First Note Every Π 0 1 Medvedev degree contains a Π0 1 subshift. Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. This answers a question of Steve Simpson. First we should understand: What is a Π 0 1 class? What is a subshift? What does s mean (what are the Medvedev degrees)? 2 / 14

6 Subshifts Subshifts (or shift spaces) are the fundamental object of study in symbolic dynamics. 3 / 14

7 Subshifts Subshifts (or shift spaces) are the fundamental object of study in symbolic dynamics. We work in 2 N = {0, 1} N (but could work in 2 Z ). 3 / 14

8 Subshifts Subshifts (or shift spaces) are the fundamental object of study in symbolic dynamics. We work in 2 N = {0, 1} N (but could work in 2 Z ). The shift operator σ: 2 N 2 N removes the first letter in a sequence. 3 / 14

9 Subshifts Subshifts (or shift spaces) are the fundamental object of study in symbolic dynamics. We work in 2 N = {0, 1} N (but could work in 2 Z ). The shift operator σ: 2 N 2 N removes the first letter in a sequence. Example. σ( ) = / 14

10 Subshifts Subshifts (or shift spaces) are the fundamental object of study in symbolic dynamics. We work in 2 N = {0, 1} N (but could work in 2 Z ). The shift operator σ: 2 N 2 N removes the first letter in a sequence. Example. σ( ) = Definition Q 2 N is a subshift if it is closed (in the product topology) and σ(q) Q. 3 / 14

11 Another view of subshifts Let S 2 <N. 4 / 14

12 Another view of subshifts Let S 2 <N. Definition X 2 N avoids S if no σ S is a substring of X. 4 / 14

13 Another view of subshifts Let S 2 <N. Definition X 2 N avoids S if no σ S is a substring of X. Note. The class Q S 2 N of all sequences that avoid S is a subshift. 4 / 14

14 Another view of subshifts Let S 2 <N. Definition X 2 N avoids S if no σ S is a substring of X. Note. The class Q S 2 N of all sequences that avoid S is a subshift. Proposition Q 2 N is a subshift iff Q = Q S for some S 2 <N. 4 / 14

15 Another view of subshifts Let S 2 <N. Definition X 2 N avoids S if no σ S is a substring of X. Note. The class Q S 2 N of all sequences that avoid S is a subshift. Proposition Q 2 N is a subshift iff Q = Q S for some S 2 <N. If S is finite, Q S is said to have finite type. 4 / 14

16 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. 5 / 14

17 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. 5 / 14

18 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. If W is a computably enumerable set (i.e., there is an algorithm to list the elements of W) 5 / 14

19 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. If W is a computably enumerable set (i.e., there is an algorithm to list the elements of W) then we call [W] a Σ 0 1 class. 5 / 14

20 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. If W is a computably enumerable set (i.e., there is an algorithm to list the elements of W) then we call [W] a Σ 0 1 class. These are the effective open sets. 5 / 14

21 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. If W is a computably enumerable set (i.e., there is an algorithm to list the elements of W) then we call [W] a Σ 0 1 class. These are the effective open sets. Definition The complement of a Σ 0 1 class is a Π0 1 class. 5 / 14

22 Π 0 1 classes For W 2 <N, let [W] = {X 2 N : ( σ W) σ X} be the set of sequences with a prefix in W. Every open subset of 2 N is of the form [W]. If W is a computably enumerable set (i.e., there is an algorithm to list the elements of W) then we call [W] a Σ 0 1 class. These are the effective open sets. Definition The complement of a Σ 0 1 class is a Π0 1 class. What about Π 0 1 subshifts? 5 / 14

23 Π 0 1 subshifts If S is a computably enumerable set, then Q S is a Π 0 1 class. 6 / 14

24 Π 0 1 subshifts If S is a computably enumerable set, then Q S is a Π 0 1 class. Conversely, if Q is a Π 0 1 subshift, then the set S of all strings that appear in no element of Q is computably enumerable 6 / 14

25 Π 0 1 subshifts If S is a computably enumerable set, then Q S is a Π 0 1 class. Conversely, if Q is a Π 0 1 subshift, then the set S of all strings that appear in no element of Q is computably enumerable and Q = Q S. 6 / 14

26 Π 0 1 subshifts If S is a computably enumerable set, then Q S is a Π 0 1 class. Conversely, if Q is a Π 0 1 subshift, then the set S of all strings that appear in no element of Q is computably enumerable and Q = Q S. We will show that, from a computability theoretic perspective, Π 0 1 subshifts can exhibit all of the behavior possible from arbitrary Π 0 1 subclasses of 2N. 6 / 14

27 Medvedev (strong) degrees Definition (Medvedev reducibility) P s Q if there is an algorithm that, given any element of Q, computes an element of P. 7 / 14

28 Medvedev (strong) degrees Definition (Medvedev reducibility) P s Q if there is an algorithm that, given any element of Q, computes an element of P. We define P s Q and the Medvedev degrees as usual. 7 / 14

29 Medvedev (strong) degrees Definition (Medvedev reducibility) P s Q if there is an algorithm that, given any element of Q, computes an element of P. We define P s Q and the Medvedev degrees as usual. Theorem (Simpson) Every Π 0 1 Medvedev degree contains a 2-dimensional subshift of finite type. 7 / 14

30 Medvedev (strong) degrees Definition (Medvedev reducibility) P s Q if there is an algorithm that, given any element of Q, computes an element of P. We define P s Q and the Medvedev degrees as usual. Theorem (Simpson) Every Π 0 1 Medvedev degree contains a 2-dimensional subshift of finite type. Note. Nonempty 1-dimensional subshifts of finite type contain periodic sequences, so they are all Medvedev equivalent. 7 / 14

31 First Note Simpson asked about and Cenzer, Dashti and King studied Π 0 1 subshifts (in dimension 1). 8 / 14

32 First Note Simpson asked about and Cenzer, Dashti and King studied Π 0 1 subshifts (in dimension 1). We prove: Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. 8 / 14

33 First Note Simpson asked about and Cenzer, Dashti and King studied Π 0 1 subshifts (in dimension 1). We prove: Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. Proof Idea A sequence Y P is coded by another sequence X in such a way that every tail of X also codes Y. 8 / 14

34 First Note Simpson asked about and Cenzer, Dashti and King studied Π 0 1 subshifts (in dimension 1). We prove: Theorem 1 If P is a Π 0 1 class, then there is a Π0 1 subshift Q such that P s Q. Proof Idea A sequence Y P is coded by another sequence X in such a way that every tail of X also codes Y. We code into the texture of X, not into specific positions. 8 / 14

35 Proof sketch Let a λ = 0 and b λ = 1, where λ represents the empty string. 9 / 14

36 Proof sketch Let a λ = 0 and b λ = 1, where λ represents the empty string. For σ 2 <N, let a σ0 = b σ a σ a σ, b σ0 = b σ a σ a σ a σ, a σ1 = a σ b σ b σ and b σ1 = a σ b σ b σ b σ. 9 / 14

37 Proof sketch Let a λ = 0 and b λ = 1, where λ represents the empty string. For σ 2 <N, let a σ0 = b σ a σ a σ, b σ0 = b σ a σ a σ a σ, a σ1 = a σ b σ b σ and b σ1 = a σ b σ b σ b σ. Let Q 0 be the set of all X 2 N such that: 9 / 14

38 Proof sketch Let a λ = 0 and b λ = 1, where λ represents the empty string. For σ 2 <N, let a σ0 = b σ a σ a σ, b σ0 = b σ a σ a σ a σ, a σ1 = a σ b σ b σ and b σ1 = a σ b σ b σ b σ. Let Q 0 be the set of all X 2 N such that: For each n N there is a unique σ 2 n such that X is formed from a σ and b σ (disregarding an initial segment shorter than a σ ). 9 / 14

39 Proof sketch Let a λ = 0 and b λ = 1, where λ represents the empty string. For σ 2 <N, let a σ0 = b σ a σ a σ, b σ0 = b σ a σ a σ a σ, a σ1 = a σ b σ b σ and b σ1 = a σ b σ b σ b σ. Let Q 0 be the set of all X 2 N such that: For each n N there is a unique σ 2 n such that X is formed from a σ and b σ (disregarding an initial segment shorter than a σ ). It is not hard to see that Q 0 is a Π 0 1 subshift. 9 / 14

40 Proof sketch (part 2) Let W 2 <N be computably enumerable and P = 2 N [W]. 10 / 14

41 Proof sketch (part 2) Let W 2 <N be computably enumerable and P = 2 N [W]. Let T = {a σ : σ W}. 10 / 14

42 Proof sketch (part 2) Let W 2 <N be computably enumerable and P = 2 N [W]. Let T = {a σ : σ W}. This is also computably enumerable. 10 / 14

43 Proof sketch (part 2) Let W 2 <N be computably enumerable and P = 2 N [W]. Let T = {a σ : σ W}. This is also computably enumerable. Thus Q = Q 0 Q T is a Π 0 1 subshift. 10 / 14

44 Proof sketch (part 2) Let W 2 <N be computably enumerable and P = 2 N [W]. Let T = {a σ : σ W}. This is also computably enumerable. Thus Q = Q 0 Q T is a Π 0 1 subshift. It is not hard to prove that P s Q. End of Sketch. 10 / 14

45 Second Note A sufficient condition for a subshift to be nonempty. 11 / 14

46 Second Note A sufficient condition for a subshift to be nonempty. Now we work in n N = {0,..., n 1} N for some n. 11 / 14

47 Second Note A sufficient condition for a subshift to be nonempty. Now we work in n N = {0,..., n 1} N for some n. Theorem 2 Let S n <N. If there is a c (1/n, 1) such that c τ nc 1, then Q S is nonempty. τ S 11 / 14

48 Second Note A sufficient condition for a subshift to be nonempty. Now we work in n N = {0,..., n 1} N for some n. Theorem 2 Let S n <N. If there is a c (1/n, 1) such that c τ nc 1, then Q S is nonempty. τ S Cenzer, Dashti and King gave a sequence of lengths such that any sequence of words with those lengths is avoidable. 11 / 14

49 Second Note A sufficient condition for a subshift to be nonempty. Now we work in n N = {0,..., n 1} N for some n. Theorem 2 Let S n <N. If there is a c (1/n, 1) such that c τ nc 1, then Q S is nonempty. τ S Cenzer, Dashti and King gave a sequence of lengths such that any sequence of words with those lengths is avoidable. The theorem gives us nice examples. 11 / 14

50 Nonempty subshifts Corollary Assume that S n <N contains at most one string of each length and let L = { σ : σ S}. If 1 n = 2 and L {5, 6, 7,... }, or 2 n = 2 and L {4, 6, 8,... }, or 3 n = 3 and L {2, 3, 4,... }, or 4 n = 4 and L {1, 2, 3,... }, then Q S is nonempty. 12 / 14

51 Nonempty subshifts Corollary Assume that S n <N contains at most one string of each length and let L = { σ : σ S}. If 1 n = 2 and L {5, 6, 7,... }, or 2 n = 2 and L {4, 6, 8,... }, or 3 n = 3 and L {2, 3, 4,... }, or 4 n = 4 and L {1, 2, 3,... }, then Q S is nonempty. Proof. In (a) and (b) we can apply the theorem with c = 5 1 2, the inverse of the golden ratio. For (c) and (d) we can use c = 1/2. 12 / 14

52 An application to effective randomness 13 / 14

53 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. 13 / 14

54 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). 13 / 14

55 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). So a sequence s initial segments can have high complexity; 13 / 14

56 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). So a sequence s initial segments can have high complexity; what about the complexity of substrings? 13 / 14

57 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). So a sequence s initial segments can have high complexity; what about the complexity of substrings? Corollary Let d < 1. There is an X 2 N such that if τ 2 <N is a substring of X, then K(τ) > d τ O(1). 13 / 14

58 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). So a sequence s initial segments can have high complexity; what about the complexity of substrings? Corollary Let d < 1. There is an X 2 N such that if τ 2 <N is a substring of X, then K(τ) > d τ O(1). Note. If X avoids any τ 2 <N, then lim sup K(X n)/n < / 14

59 An application to effective randomness Prefix(-free Kolmogorv) complexity, K, measures the complexity of finite binary strings. X 2 N is Martin-Löf random if and only if K(X n) n O(1). So a sequence s initial segments can have high complexity; what about the complexity of substrings? Corollary Let d < 1. There is an X 2 N such that if τ 2 <N is a substring of X, then K(τ) > d τ O(1). Note. If X avoids any τ 2 <N, then lim sup K(X n)/n < 1. So the result fails for d = / 14

60 Thank You 14 / 14