Shift-complex Sequences

Transcription

1 University of Wisconsin Madison March 24th, ASL North American Annual Meeting Berkeley, CA

2 What are shift-complex sequences? K denotes prefix-free Kolmogorov complexity: For a string σ, K(σ) is the length of a shortest description of σ with respect to some prefix-free universal Turing machine. Initial segments of Martin-Löf randoms have high Kolmogorov complexity in the following sense: Theorem (Schnorr) A sequence X 2 ω is Martin-Löf random if and only if K(X n) n O(1). Contiguous substrings of Martin-Löf randoms fail to have this property.

3 What are shift-complex sequences? Question Are there sequences that are uniformly complex wherever we look? Definition Let δ (0, 1). A sequence X 2 ω is δ-shift-complex if for every substring σ of X, K(σ) δ σ O(1). Note: Every δ-shift-complex sequence with respect to K is also shift-complex with respect to C (plain complexity) for a slightly lower δ and vice versa.

4 Constructions Durand, Levin and Shen 2001 Builds a shift-complex sequence via extensions of constant length. Simple, but not easily effectivized. Rumyantsev and Ushakov 2006 Uses the Lovász Local Lemma and compactness. Miller 2010 Provides a condition on a set of forbidden strings that suffices (but is not necessary) to ensure the existence of a sequence which avoids it.

5 Constructions Theorem (Miller) Let S 2 <ω. If there is a c (1/2, 1) such that c τ 2c 1, τ S then there is an X 2 ω that avoids S. The condition is merely on the lengths of strings in S. For an appropriately chosen constant b, the set of strings σ such that K(σ) < δ σ b satisfies the condition. Further, this theorem is effective: Proposition Let S be as in the theorem above. There is an X T S that avoids S.

6 A few quick observations A 2 ω is δ-shift-complex with constant b if and only if ( n 1 n 2 > n 1 s)[k s (A n2 n 1 ) δ(n 2 n 1 ) b]. This a Π 0 1 condition. By any of the constructions just mentioned, for each δ, there is a nonempty Π 0 1 class of δ-shift-complex sequences. Thus, every PA degree computes a δ-shift-complex sequence. The basis theorems show that there are low and hyperimmune-free δ-shift-complex sequences. The class of shift-complex sequences has zero measure and is meager.

7 What computes shift-complex sequences? Theorem (Rumyantsev) The measure of oracles that compute shift-complex sequences is 0. Key observation: every shift-complex sequence either has a lot of strings of each length, or computes something of higher shift-complexity. More precisely: Definition We say that a shift-complex sequence Y is abundant if for some n, Y is δ-shift-complex for some δ > 1/n and further, for every m, Y contains at least 2 m(n 1)/n strings of length m. Lemma Every shift-complex sequence computes an abundant shift-complex sequence.

8 What computes shift-complex sequences? Proof sketch of Rumyantsev s theorem Assume a positive measure set of oracles computes shift-complex sequences, hence abundant shift-complex sequences. Fix a functional Γ such that a positive measure set of oracles computes shift-complex sequences that are abundant for some n. Each oracle computes a lot of strings of each length, so there must exist strings computed by a large measure of oracles. Compress such strings when they appear.

9 What computes shift-complex sequences? Question How random does an oracle have to be to ensure that it does not compute a shift-complex sequence? Using Rumyantsev s theorem as a black box, one can prove: Corollary No weak 2-random real computes a shift-complex sequence. With a little effort, one can do better.

10 What computes shift-complex sequences? Recall that a Martin-Löf test is a uniform sequence (U i ) i ω of Σ 0 1 open sets such that µ(u i ) 2 i. A real passes a Martin-Löf test (U i ) i ω if it is not contained in i ω U i. Franklin and Ng introduce a notion of randomness called difference randomness and show it to be strictly intermediate between weak 2-randomness and Martin-Löf randomness. Definition (Franklin and Ng 2011) A difference test is a uniform sequence of pairs (U i, V i ) of Σ 0 1 open sets such that µ(u i \ V i ) 2 i. A real passes a difference test ((U i, V i )) i ω if it is not contained in i ω (U i \ V i ). A real is difference random if it passes all difference tests.

11 What computes shift-complex sequences? We already know the difference randoms in another guise: Theorem (Franklin and Ng 2011) The difference random reals are precisely the incomplete Martin-Löf random reals. Further, no difference random is of PA degree: Theorem (Stephan 2002) A Martin-Löf random has PA degree if and only if it is complete.

12 What computes shift-complex sequences? If Y computes a shift-complex sequence then we can trap it with a difference test: Theorem (K.) No difference random real computes a shift-complex sequence. Clearly, every complete Martin-Löf random real computes a shift-complex sequence, so we have: Corollary A Martin-Löf random real computes a shift-complex sequence if and only if it is complete.

13 Computing Martin-Löf randoms Question Does every shift-complex sequence compute a Martin-Löf random? We want to bound a shift-complex sequence by something weaker than a PA degree. Idea: Use a slow-growing diagonally noncomputable (DNC) function. Recall that a function f : ω ω is diagonally noncomputable if f(e) Φ e (e) for any e such that Φ e (e).

14 Computing Martin-Löf randoms Definition For any non-decreasing function h : ω ω \ {0, 1}, DNC h = {f DNC ( n)f(n) < h(n)}. Note: If h is a constant function, then any f DNC h is of PA degree. If we allow h to grow unboundedly, we can obtain something weaker: Theorem (Greenberg and Miller 2011) For any non-decreasing, unbounded, computable function h : ω ω \ {0, 1}, there is an f DNC h that does not compute a Martin-Löf random real.

15 Computing Martin-Löf randoms Theorem (K.) For every δ (0, 1) there is a non-decreasing, unbounded, computable function h : ω ω \ {0, 1} such that every f DNC h computes a δ-shift-complex sequence. Proof sketch Let S = {σ 2 <ω K(σ) < δ σ } and let S m = S 2 m. Uniformly in m, we can cover S m with a set S m 2 m, using any function f of sufficient DNC strength. Further, as m increases, the required DNC strength decreases. Using the uniformity above, produce an h such that every f DNC h computes an S 2 <ω that 1 covers S except for finitely many strings 2 satisfies the hypothesis of Miller s theorem

16 Computing Martin-Löf randoms Corollary For every δ (0, 1) there is a δ-shift-complex sequence that does not compute any Martin-Löf random.

17 Open questions Definition The effective Hausdorff dimension of a real X, denoted by dim(x), is lim inf n K(X n) n Clearly, a δ-shift-complex sequence has effective Hausdorff dimension at least δ. Question Do there exist δ-shift-complex sequences that have effective Hausdorff dimension exactly δ? We know now that a shift-complex sequence need not compute a Martin-Löf random. However, the following is still open: Question Does every shift-complex sequence compute a real of effective Hausdorff dimension 1?

18 Bibliography Bruno Durand, Leonid A. Levin, and Alexander Shen. Complex tilings. J. Symbolic Logic, 73(2): , Johanna N. Y. Franklin and Keng Meng Ng. Difference randomness. Proc. Amer. Math. Soc., 139(1): , Noam Greenberg and Joseph S. Miller. Diagonally non-recursive functions and effective Hausdorff dimension. Bulletin of the London Mathematical Society, to appear. Joseph S. Miller. Two notes on subshifts. Proceedings of the American Mathematical Society. To appear. A. Yu. Rumyantsev and M. A. Ushakov. Forbidden substrings, Kolmogorov complexity and almost periodic sequences. In STACS 2006, volume 3884 of Lecture Notes in Comput. Sci., pages Springer, Berlin, Frank Stephan. Martin-Löf random and PA-complete sets. In Logic Colloquium 02, volume 27 of Lect. Notes Log., pages Assoc. Symbol. Logic, La Jolla, CA, 2006.