- •13.4 Shift complex sets
- •13.5 Partial randomness
- •13.9.1 Dimension in h-spaces
- •13.10 C-independence and Zimand’s Theorem
- •13.11 Other notions of dimension
- •13.11.1 Box counting dimension
- •13.11.3 Packing dimension
- •13.12 Packing dimension and complexity extraction
- •13.13 Clumpy trees and minimal degrees
- •13.14 Building sets of high packing dimension
- •13.15 Computable dimension and Schnorr dimension
- •13.15.1 Basics
- •13.15.2 Examples of Schnorr dimension
- •13.15.3 A machine characterization of Schnorr dimension
- •13.15.4 Schnorr dimension and computable enumerability
- •13.16 Kolmogorov complexity and the dimensions of individual strings
13.9. DNC functions and e ective Hausdor dimension |
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13.9.1 Dimension in h-spaces
Greenberg and Miller [170] generalized many notions we have seen in this chapter to h-spaces. The following is the natural generalization of the notions of weak and strong s-Martin-L¨of randomness to hω.
Definition 13.9.2. Let s [0, 1].
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(iii)A hω is weakly s-Martin-L¨of random if A / k Vk for all tests for weak s-Martin-L¨of randomness {Vk}k ω.
(iv)A hω is strongly s-Martin-L¨of random if A / k Vk for all tests for strong s-Martin-L¨of randomness {Vk}k ω.
Many results proved in Section 13.5 still hold for hω, such as Theorem 13.5.13, which says that if t < s and A is weakly s-random, then A is strongly t-random. Thus, for A hω, the supremum of all s for which A is weakly s-random equals the supremum of all s for which A is strongly s-random, and, by analogy with the 2ω case, can be called the e ective (Hausdor ) dimension of A, denoted by dimh(A).
We can also generalize the notion of a Solovay s-test.
Definition 13.9.3. Let s [0, 1]. A Solovay s-test is a c.e. set S h<ω
such that σ S μ( σ )s < ∞. A set A is covered by this test if A σ for infinitely many σ S.
Theorem 13.5.7 still holds: If A is weakly s-random then A is not covered by any Solovay s-test. If A is covered by some Solovay s-test then A is not weakly t-random for any t > s.
We can also define martingales for h-spaces.
Definition 13.9.4. A supermartingale (for h) is a function d : h<ω → R 0 such that for all σ h<ω,
d(σi) h(|σ|)d(σ).8
i<h(|σ|)
If s [0, 1], we say that a supermartingale d s-succeeds on A hω if lim supn d(A n)μ( A n )1−s = ∞.
8Note that this condition is equivalent to d(τ )μ( τ ) d(σ)μ( σ ), where the sum is taken over all immediate successors τ of σ.
620 13. Algorithmic Dimension
As in the 2ω case, A hω is strongly s-Martin-L¨of random iff there is no c.e. supermartingale that s-succeeds on A. The proofs are as before, using in particular the fact that Kolmogorov’s Inequality holds in
hω: For a supermartingale d and |
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It is also worth noting that there is an optimal c.e. supermartingale for h, constructed as usual.
Our goal is to use h-spaces to construct an element of 2ω of minimal degree with e ective dimension 1. To do so, we will need to be able to translate between hω and 2ω in a dimension-preserving way. If h is very
well-behaved then there is no problem. For example, if X 4ω and we let Y (2n) = #X(2n) $ and Y (2n+ 1) = X(n) mod 2, then Y ≡T X, and it is easy to check that dim(Y ) = dimh(X). To handle more complicated h-spaces, it is convenient to work through Euclidean space rather than going directly from hω to 2ω.
There is a natural measure-preserving surjection of hω onto the Euclidean interval [0, 1]. First map strings to closed intervals: Let πh(λ) = [0, 1]. Once πh(σ) = I is defined, divide I into h(|σ|) many intervals I0, I1, . . . , Ih(|σ)|−1
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not quite 1-1, but it is 1-1 if we ignore the (countably many) sequences that are eventually constant. Note that for all X hω, we have X ≡T πh(X).
The theory of e ective dimension can also be developed in the space [0, 1]. We do not have martingales, but we can still, for example, define Solovay tests, as we saw in Definition 13.5.8. Let dim[0,1](X) be the infimum of all s such that there is an interval Solovay s-test that covers X (that is, such that X is in infinitely many elements of the test).
Proposition 13.9.5 (Greenberg and Miller [170]). For all h and X hω, we have dim[0,1](πh(X)) dimh(X).
Proof. Let s > dimh(X) and let S be a Solovay s-test for h that covers X. Since πh is measure-preserving, the image of S under πh is an interval Solovay test covering πh(X).
Equality of these dimensions does not hold in general, but we will see that we do get equality if h does not grow too quickly or irregularly. Suppose that S is an interval Solovay s-test covering some πh(X). We would like to cover X by something like (πh)−1(S). The problem, of course, is that the basic sets (closed intervals with rational endpoints) in [0, 1] are finer than the basic sets in hω; not every closed interval is in the range of πh. We thus need to refine S by replacing every I S by finitely many intervals in the range of πh. While we can control the Lebesgue measure of such a
13.9. DNC functions and e ective Hausdor dimension |
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collection, if the exponent s is smaller than 1 then the process of replacing large intervals by a collection of smaller ones may increase the s-weighted sum of the lengths of the intervals significantly. We show that if h does not grow too irregularly and we increase the exponent s slightly, then this sum remains finite.
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Lemma 13.9.6 (Greenberg and Miller [170]). Suppose that (13.1) holds
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Proof. Let I be any closed interval in [0, > k and k h(n),
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1]. Let n = nI and k = kI . Since
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which by assumption is finite.
Thus for all X hω, if (13.1) holds for t, s, and h, and dim[0,1](πh(X)) < s, then dimh(X) t. So if (13.1) holds for all t > s 0, then dim[0,1](πh(X)) = dimh(X) for all X hω.
622 13. Algorithmic Dimension
For example, (13.1) holds for all t > s 0 for the constant function h(n) = 2. Thus, as discussed above, dimension in [0, 1] is the same as dimension in 2ω. However, this condition holds for some unbounded functions
has well (for example h(n) = 2n).
The following is a su cient condition for (13.1) to hold for all t > s 0.
Lemma 13.9.7 (Greenberg and Miller [170]). If |
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Proof. Let f (n) = log h(n). Let t > s |
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which is finite because 2δ < 1 and log g(n) n (since h(n) 2).
The regularity condition of Lemma 13.9.7 is not, strictly speaking, a slowness condition, because, for example, h(n) = 2n2 satisfies this condition, yet there is a monotone function that is dominated by h but does not satisfy the condition. However, the condition does hold for all su ciently slow monotone functions.
Lemma 13.9.8 (Greenberg and Miller [170]). If h is nondecreasing and dominated by 2kn for some k, then
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Proof. If h is bounded then it is eventually constant, and the condition is easily verified, so assume that h is unbounded. Fix c > 0. There is an Nc such that log h(n) > c for all n Nc. For n > Nc,
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13.9. DNC functions and e ective Hausdor dimension |
623 |
Finally, we get the result we will need to translate between hω and 2ω.
Corollary 13.9.9 (Greenberg and Miller [170]).
dominated by 2kn for some k, then every X hω that dimh(X) = dim(Y ).
If h is nondecreasing and computes a Y 2ω such
13.9.2Slow-growing DNC functions and sets of high e ective dimension
For natural numbers a > b > 0, let Qba be the collection of functions f such that for all n,
1.f (n) {0, . . . , a − 1},
2.|f (n)| = b, and
3.if Φn(n) ↓ then Φn(n) / f (n).
By standard coding, Qba can be seen as a computably bounded Π01 subclass of ωω. Note that Q1a is essentially the same as DNCa.
To connect these sets for di erent values of a and b, we will use the concept of strong reducibility of mass problems from Definition 8.9.1. Recall that P ωω is strongly reducible to R ωω if there is a Turing functional Ψ such that Ψf P for all f R. If P is a Π01 class, then we may assume that the Ψ in the above definition is total, and hence the reduction is a truth table reduction, since we can always define ΨX (n) = 0 if there is an s such that X / P [s] and ΨX (n)[s] ↑.
Lemma 13.9.10 (Greenberg and Miller [170]). If a > b > 0 then Qba+1+1 s
Qba, uniformly in a and b. (Uniformity here means that an index for the reduction functional Ψ can be obtained e ectively from a and b.)
Proof. From n and y < a we can compute an mn,y such that
1.for all x < a such that x =y, we have Φmn,y (mn,y ) ↓= x iff Φn(n) ↓= x; and
2.Φmn,y (mn,y ) ↓= y iff either Φn(n) ↓= y or Φn(n) ↓= a.
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624 13. Algorithmic Dimension
Corollary 13.9.11 (Greenberg and Miller [170]). If a 2 then Qba+1+b s
DNCa, uniformly in a and b.
For a 2 and c > 0, let Pac be the collection of functions f aω such
that for all n and all x < c, if Φ n,x ( n, x ) ↓ then Φ n,x ( n, x ) =f (n). Note that Pa1 ≡s DNCa.
Lemma 13.9.12 (Greenberg and Miller [170]). For all a > b > 0 and c > 0, if c(a − b) < a then Pac s Qba, uniformly in a, b, and c.
Proof. Fix f Qba and n. For all x < c, if Φ n,x ( n, x ) ↓ then
Φ n,x ( n, x ) [0, a) \ |
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The set on the right has size at most c(a − b), so if c(a − b) < a then we can choose some x < a not in that set and define Ψf (n) = x.
Corollary 13.9.13 (Greenberg and Miller [170]). If a 2 and c > 0 then
Pcac s DNCa, uniformly in a and c.
Proof. Let b
so P c −
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Pca, and hence Pca s Pc(a−1)+1. All these reductions are
Greenberg and Miller [170] used the classes Pcac to construct sequences of positive e ective dimension.
Theorem 13.9.14 (Greenberg and Miller [170]). Let a 2 and ε > 0. Every f DNCa computes a set of e ective Hausdor dimension greater than 1 − ε, via a reduction that is uniform in a and ε.9
Proof. Fix c > 1. We work in the space (ca)ω . Let d be the universal c.e. supermartingale for this space. By scaling we may assume that d(λ) < 1.
For σ (ca)<ω , let Sσ be the set of k < c such that d(σk) a|σ|+1. Note that these sets are uniformly c.e. From σ, we can compute an mσ such that
for each x < c, we have Φ mσ ,x ( mσ , x ) ↓= k if k is the xth element to be enumerated into Sσ , and Φ mσ ,x ( mσ , x ) ↑ if |Sσ | < x.
The idea here is that if d(σ) a|σ| then |Sσ| c, by the supermartingale condition, so all elements of Sσ are “captured” by the map e →Φe(e). We can thus use a function g Pcac to avoid all such extensions: given such g, inductively define X (ca)<ω by letting the (n + 1)st digit of X be g(mX n). Then, by induction, d(X n) an for all n.
9That each f DNCa computes such a set is of course not a new fact, since the Turing degree of a bounded DNC function is PA and so computes a 1-random set. The extra information is the uniformity.
13.9. DNC functions and e ective Hausdor dimension |
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Now let s 0 and suppose that d s-succeeds on X, that is, that d(X n)μca(X n)1−s is unbounded. Since μca(X n) = (ca)−n,
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Theorem 13.9.15 (Greenberg and Miller [170]). Let a 2. Every f
DNCa computes a set of e ective Hausdor dimension 1, via a reduction that is uniform in a.
Proof. We combine the constructions of sets of e ective dimensions closer and closer to 1 into one construction. Let h(n) = (n + 1)a. Let d be the universal c.e. supermartingale for hω. Given f DNCa, obtain gn Pnan for all n > 0 uniformly from f .
For σ hn, let Sσ be the set of k n such that d(σk) a|σ|+1 and
compute an mσ such that for each x n, we have Φ mσ ,x ( mσ , x ) ↓= k if k is the xth element to be enumerated into Sσ, and Φ mσ ,x ( mσ , x ) ↑ if
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Let s < 1. Let k be such that ks−1a2s−1 < 1. For almost all n we have n! > (ka)n, so for almost all n we have
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and hence d cannot s-succeed on X. Thus dimh(X) = 1, so by Corollary 13.9.9, X computes a Y 2ω of e ective dimension 1.
Finally, we can paste together these constructions for all a 2 to get the desired result.
Theorem 13.9.16 (Greenberg and Miller [170]). There is a computable
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DNCh˜ computes a set of |
order h : N → N \ {0, 1} such that every f |
e ective Hausdor dimension 1.
Proof. Let h(n) = (n+1)2n, and let d be the universal c.e. supermartingale for h. For σ hn, let Sσ be the set of k < 2n such that d(σk) (n + 1)! and
compute an mσ such that for each x < 2n, we have Φ mσ ,x ( mσ , x ) ↓= k if k is the xth element to be enumerated into Sσ, and Φ mσ ,x ( mσ , x ) ↑ if
|Sσ| < x.
626 13. Algorithmic Dimension
n
For n > 0, we have Ph2(n) DNCn+1 uniformly in n, so there is an e ective list of truth table functionals Ψn such that Ψfn Ph2(nn) for all f DNCn+1. Let ψn be a computable bound on the use function of Ψn. Let
mn = 1 + sup{mσ, x : σ hn x < 2n}
and let un = ψn(mn). Let u0 = 0.
For all n > 0, if ρ is a sequence of length un that is a DNCn+1-string (that is, ρ (n+1)un and for all y < un such that Φy (y) ↓, we have Φy(y) =ρ(y); or equivalently, ρ is an initial segment of a sequence in DNCn+1) then Ψn(ρ)
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for all x < 2n, if Φ mσ ,x ( mσ , x ) ↓ then g(mσ) = Φmσ ,x ( mσ , x ). |
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We can now use g to define X hω as in the last two constructions, by letting X(n) = g(mX n). By induction on n, we can show that d(X n) n!. As before, we can do so because if σ hn and d(σ) n!, then there are at most 2n many immediate successors τ of σ such that d(τ ) (n + 1)!, and so they are all captured by the function e →Φe(e) and avoided by g.
Finally, we show that dimh(X) = 1, which by Corollary 13.9.9 implies that X computes a Y 2ω of e ective dimension 1.
Let s < 1. For any σ hn,
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which is bounded (and indeed tends to 0). Thus d does not s-succeed on
X.
Thus, by Theorem 13.9.1, we have the following.
Theorem 13.9.17 (Greenberg and Miller [170]). There is a minimal degree of e ective Hausdor dimension 1.