Shift-complex Sequences Mushfeq Khan University of Wisconsin—Madison March 24th, 2011 2011 ASL North American Annual Meeting Berkeley, CA Mushfeq Khan Shift-complex Sequences
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. Mushfeq Khan Shift-complex Sequences
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. Mushfeq Khan Shift-complex Sequences
Constructions Durand, Levin and Shen 2001 Builds a shift-complex sequence via extensions of constant length. Simple, but not easily e ff ectivized. Rumyantsev and Ushakov 2006 Uses the Lovász Local Lemma and compactness. Miller 2010 Provides a condition on a set of forbidden strings that su ffi ces (but is not necessary) to ensure the existence of a sequence which avoids it. Mushfeq Khan Shift-complex Sequences
Constructions Theorem (Miller) Let S ⊆ 2 <ω . If there is a c ∈ ( 1 / 2 , 1 ) such that c | τ | ≤ 2 c − 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 satis fi es the condition. Further, this theorem is e ff ective: Proposition Let S be as in the theorem above. There is an X ≤ T S that avoids S . Mushfeq Khan Shift-complex Sequences
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 ↾ n 2 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. Mushfeq Khan Shift-complex Sequences
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: De fi nition 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. Mushfeq Khan Shift-complex Sequences
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. Mushfeq Khan Shift-complex Sequences
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 e ff ort, one can do better. Mushfeq Khan Shift-complex Sequences
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 di ff erence randomness and show it to be strictly intermediate between weak 2-randomness and Martin-Löf randomness. De fi nition (Franklin and Ng 2011) A di ff erence 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 di ff erence test (( U i , V i )) i ∈ ω if it is not contained in � i ∈ ω ( U i \ V i ) . A real is di ff erence random if it passes all di ff erence tests. Mushfeq Khan Shift-complex Sequences
What computes shift-complex sequences? We already know the di ff erence randoms in another guise: Theorem (Franklin and Ng 2011) The di ff erence random reals are precisely the incomplete Martin-Löf random reals. Further, no di ff erence random is of PA degree: Theorem (Stephan 2002) A Martin-Löf random has PA degree if and only if it is complete. Mushfeq Khan Shift-complex Sequences
What computes shift-complex sequences? If Y computes a shift-complex sequence then we can trap it with a di ff erence test: Theorem (K.) No di ff erence 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. Mushfeq Khan Shift-complex Sequences
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 ) ↓ . Mushfeq Khan Shift-complex Sequences
Computing Martin-Löf randoms De fi nition 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. Mushfeq Khan Shift-complex Sequences
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 su ffi cient 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 covers S except for fi nitely many strings 1 satis fi es the hypothesis of Miller’s theorem 2 Mushfeq Khan Shift-complex Sequences
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. Mushfeq Khan Shift-complex Sequences
Open questions De fi nition The e ff ective Hausdor ff dimension of a real X , denoted by dim ( X ) , is K ( X ↾ n ) lim inf n n →∞ Clearly, a δ -shift-complex sequence has e ff ective Hausdor ff dimension at least δ . Question Do there exist δ -shift-complex sequences that have e ff ective Hausdor ff 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 e ff ective Hausdor ff dimension 1? Mushfeq Khan Shift-complex Sequences
Bibliography Bruno Durand, Leonid A. Levin, and Alexander Shen. Complex tilings. J. Symbolic Logic , 73(2):593–613, 2008. Johanna N. Y. Franklin and Keng Meng Ng. Di ff erence randomness. Proc. Amer. Math. Soc. , 139(1):345–360, 2011. Noam Greenberg and Joseph S. Miller. Diagonally non-recursive functions and e ff ective Hausdor ff 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 396–407. Springer, Berlin, 2006. Frank Stephan. Martin-Löf random and PA-complete sets. In Logic Colloquium ’02 , volume 27 of Lect. Notes Log. , pages 342–348. Assoc. Symbol. Logic, La Jolla, CA, 2006. Mushfeq Khan Shift-complex Sequences
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