Computability Theory and Asymptotic Density Denis R. Hirschfeldt - - PowerPoint PPT Presentation

Computability Theory and Asymptotic Density Denis R. Hirschfeldt — University of Chicago

Groups and Computation: In Honor of Paul Schupp’s 80th Birthday

Cantor space and relative computability We work in the space 2ω of infinite binary sequences, identifying A ∈ 2ω with {n : A(n) = 1}.

Cantor space and relative computability We work in the space 2ω of infinite binary sequences, identifying A ∈ 2ω with {n : A(n) = 1}. For σ ∈ 2<ω, let [σ] = {A ∈ 2ω : σ ≺ A}. These are the basic open sets of the standard topology on 2ω.

Cantor space and relative computability We work in the space 2ω of infinite binary sequences, identifying A ∈ 2ω with {n : A(n) = 1}. For σ ∈ 2<ω, let [σ] = {A ∈ 2ω : σ ≺ A}. These are the basic open sets of the standard topology on 2ω. Let µ the measure on 2ω defined by µ([σ]) = 2−|σ|.

Cantor space and relative computability We work in the space 2ω of infinite binary sequences, identifying A ∈ 2ω with {n : A(n) = 1}. For σ ∈ 2<ω, let [σ] = {A ∈ 2ω : σ ≺ A}. These are the basic open sets of the standard topology on 2ω. Let µ the measure on 2ω defined by µ([σ]) = 2−|σ|. We write A T B to mean that A is computable relative to B. A ≡T B if A T B and B T A. The ≡T-equivalence classes are the (Turing) degrees.

Describing a set We consider partial descriptions of elements of 2ω.

Describing a set We consider partial descriptions of elements of 2ω. For an input n, such a description might:

• give an answer (correctly or not);
• never give an answer;
• declare that it will not give an answer.

Describing a set We consider partial descriptions of elements of 2ω. For an input n, such a description might:

• give an answer (correctly or not);
• never give an answer;
• declare that it will not give an answer.

A description of A ∈ 2ω is a partial function ∆ : N → {0, 1, }. We write ∆(n)↓ if ∆(n) is defined, and ∆(n)↑ otherwise.

Describing a set We consider partial descriptions of elements of 2ω. For an input n, such a description might:

• give an answer (correctly or not);
• never give an answer;
• declare that it will not give an answer.

A description of A ∈ 2ω is a partial function ∆ : N → {0, 1, }. We write ∆(n)↓ if ∆(n) is defined, and ∆(n)↑ otherwise. When might we say that ∆ describes A “almost everywhere”?

Asymptotic density Let S ⊆ N. The upper (asymptotic) density of S is ρ(S) = lim supn |S ∩ [0, n)| n . The lower (asymptotic) density of S is ρ(S) = lim infn |S ∩ [0, n)| n . If ρ(S) = ρ(S) then S has (asymptotic) density ρ(S) = ρ(S).

Asymptotic density Let S ⊆ N. The upper (asymptotic) density of S is ρ(S) = lim supn |S ∩ [0, n)| n . The lower (asymptotic) density of S is ρ(S) = lim infn |S ∩ [0, n)| n . If ρ(S) = ρ(S) then S has (asymptotic) density ρ(S) = ρ(S). We think of sets of density 0 as negligible.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n:

• ∆(n)↑
• ∆(n)↓ = 1 − A(n)
• ∆(n)↓ =

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n:

• ∆(n)↑
• ∆(n)↓ = 1 − A(n)
• ∆(n)↓ =

Let D = {n : ∆(n)↑}, let M = {n : ∆(n)↓ = 1 − A(n)}, and let R = {n : ∆(n)↓ = }.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n:

• ∆(n)↑
• ∆(n)↓ = 1 − A(n)
• ∆(n)↓ =

Let D = {n : ∆(n)↑}, let M = {n : ∆(n)↓ = 1 − A(n)}, and let R = {n : ∆(n)↓ = }. ∆ is a dense description of A if D ∪ M ∪ R has density 0.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n:

• ∆(n)↑
• ∆(n)↓ = 1 − A(n)
• ∆(n)↓ =

Let D = {n : ∆(n)↑}, let M = {n : ∆(n)↓ = 1 − A(n)}, and let R = {n : ∆(n)↓ = }. ∆ is a dense description of A if D ∪ M ∪ R has density 0. If ∆ is a dense description then it is:

• a generic description of A if M ∪ R = ∅.
• a coarse description of A if D ∪ R = ∅.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n:

• ∆(n)↑
• ∆(n)↓ = 1 − A(n)
• ∆(n)↓ =

Let D = {n : ∆(n)↑}, let M = {n : ∆(n)↓ = 1 − A(n)}, and let R = {n : ∆(n)↓ = }. ∆ is a dense description of A if D ∪ M ∪ R has density 0. If ∆ is a dense description then it is:

• a generic description of A if M ∪ R = ∅.
• a coarse description of A if D ∪ R = ∅.
• an effective dense description of A if D ∪ M = ∅.

Asymptotic computability A is densely computable if it has a computable dense description. A is generically computable if it has a computable generic description. A is coarsely computable if it has a computable coarse description. A is effectively densely computable if it has a computable effective dense description.

Asymptotic computability A is densely computable if it has a computable dense description. A is generically computable if it has a computable generic description. A is coarsely computable if it has a computable coarse description. A is effectively densely computable if it has a computable effective dense description. These notions can be relativized to define dense computability relative to X, etc.

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability.

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability. Dense and effective dense computability were studied by Astor, Hirschfeldt, and Jockusch (ta).

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability. Dense and effective dense computability were studied by Astor, Hirschfeldt, and Jockusch (ta). Meyer (1973) defined effective dense computability, and asked a question answered by Lynch (1974). Terwijn (1998) studied coarse computability.

Relationships between notions of asymptotic computability computable ↓ effectively densely computable ւ ց generically coarsely computable computable ց ւ densely computable No implications other than the ones shown hold in general.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X, consider {2n : n ∈ X}.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X, consider {2n : n ∈ X}. Every nontrivial Turing degree contains a set that is not densely computable: Given X, consider

n∈X[2n, 2n+1).

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X, consider {2n : n ∈ X}. Every nontrivial Turing degree contains a set that is not densely computable: Given X, consider

n∈X[2n, 2n+1).

A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X, consider {2n : n ∈ X}. Every nontrivial Turing degree contains a set that is not densely computable: Given X, consider

n∈X[2n, 2n+1).

A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable. There is a c.e. set that is generically computable but not coarsely computable.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X, consider {2n : n ∈ X}. Every nontrivial Turing degree contains a set that is not densely computable: Given X, consider

n∈X[2n, 2n+1).

A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable. There is a c.e. set that is generically computable but not coarsely computable. {A : A is densely computable} has measure 0 and is meager.

Asymptotic reducibilities A is coarsely reducible to B, written A c B, if every coarse description of B computes a coarse description of A.

Asymptotic reducibilities A is coarsely reducible to B, written A c B, if every coarse description of B computes a coarse description of A. There are nonuniform and uniform versions of this reducibility, but we will ignore the distinction.

Asymptotic reducibilities A is coarsely reducible to B, written A c B, if every coarse description of B computes a coarse description of A. There are nonuniform and uniform versions of this reducibility, but we will ignore the distinction. A c ∅ iff A is coarsely computable.

Asymptotic reducibilities A is coarsely reducible to B, written A c B, if every coarse description of B computes a coarse description of A. There are nonuniform and uniform versions of this reducibility, but we will ignore the distinction. A c ∅ iff A is coarsely computable. A ≡c B if A c B and B c A. The ≡c-equivalence classes are the coarse degrees.

Asymptotic reducibilities A is coarsely reducible to B, written A c B, if every coarse description of B computes a coarse description of A. There are nonuniform and uniform versions of this reducibility, but we will ignore the distinction. A c ∅ iff A is coarsely computable. A ≡c B if A c B and B c A. The ≡c-equivalence classes are the coarse degrees. We can similarly define generic reducibility g, dense reducibility d, and effective dense reducibility ed.

Upper cones and minimal pairs in the Turing degrees Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

Upper cones and minimal pairs in the Turing degrees Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0. If Y is not computable then {A : A T Y} is countable, so µ  

• 0<TATY

{X : A T X}   = 0.

Upper cones and minimal pairs in the Turing degrees Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0. If Y is not computable then {A : A T Y} is countable, so µ  

• 0<TATY

{X : A T X}   = 0. Thus there is an X s.t. if A T X, Y then A is computable. We say that the Turing degrees of X and Y form a minimal pair.

Upper cones and minimal pairs in the Turing degrees Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0. If Y is not computable then {A : A T Y} is countable, so µ  

• 0<TATY

{X : A T X}   = 0. Thus there is an X s.t. if A T X, Y then A is computable. We say that the Turing degrees of X and Y form a minimal pair. By this argument, most pairs of Turing degrees are minimal pairs.

Upper cones and minimal pairs in the coarse degrees Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0.

Upper cones and minimal pairs in the coarse degrees Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0. Cor (Hirschfeldt, Jockusch, Kuyper, and Schupp). There are X, Y c ∅ s.t. any A that is coarsely computable relative both to X and to Y is coarsely computable. In particular, if A c X, Y then A c ∅, so there are minimal pairs in the coarse degrees.

Upper cones and minimal pairs in the coarse degrees Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0. Cor (Hirschfeldt, Jockusch, Kuyper, and Schupp). There are X, Y c ∅ s.t. any A that is coarsely computable relative both to X and to Y is coarsely computable. In particular, if A c X, Y then A c ∅, so there are minimal pairs in the coarse degrees. Astor, Hirschfeldt, and Jockusch showed that these facts also hold for dense computability.

Upper cones and minimal pairs in the generic degrees Thm (Astor, Hirschfeldt, and Jockusch). If A g ∅ then µ({X : A is generically computable relative to X}) = 0. So proper upper cones in the generic degrees have measure 0.

Upper cones and minimal pairs in the generic degrees Thm (Astor, Hirschfeldt, and Jockusch). If A g ∅ then µ({X : A is generically computable relative to X}) = 0. So proper upper cones in the generic degrees have measure 0. Thm (Igusa). If X, Y g ∅ then there is an A g ∅ s.t. A is generically computable relative both to X and to Y.

Upper cones and minimal pairs in the generic degrees Thm (Astor, Hirschfeldt, and Jockusch). If A g ∅ then µ({X : A is generically computable relative to X}) = 0. So proper upper cones in the generic degrees have measure 0. Thm (Igusa). If X, Y g ∅ then there is an A g ∅ s.t. A is generically computable relative both to X and to Y. Open Question. Are there minimal pairs in the generic degrees?

Upper cones and minimal pairs in the generic degrees Thm (Astor, Hirschfeldt, and Jockusch). If A g ∅ then µ({X : A is generically computable relative to X}) = 0. So proper upper cones in the generic degrees have measure 0. Thm (Igusa). If X, Y g ∅ then there is an A g ∅ s.t. A is generically computable relative both to X and to Y. Open Question. Are there minimal pairs in the generic degrees? Astor, Hirschfeldt, and Jockusch showed that the situation is the same for effective dense computability.

Algorithmic randomness We can use computability theory to define notions of randomness for individual elements of 2ω.

Algorithmic randomness We can use computability theory to define notions of randomness for individual elements of 2ω. Early attempts focused on specific statistical tests.

Algorithmic randomness We can use computability theory to define notions of randomness for individual elements of 2ω. Early attempts focused on specific statistical tests. Martin-L¨

• f gave the first robust definition, based on the idea of

considering all (theoretically) performable tests.

Algorithmic randomness We can use computability theory to define notions of randomness for individual elements of 2ω. Early attempts focused on specific statistical tests. Martin-L¨

• f gave the first robust definition, based on the idea of

considering all (theoretically) performable tests. A Martin-L¨

• f test is an open set U ⊂ 2ω of measure 0, given as an

intersection of rapidly shrinking effectively open sets.

Algorithmic randomness We can use computability theory to define notions of randomness for individual elements of 2ω. Early attempts focused on specific statistical tests. Martin-L¨

• f gave the first robust definition, based on the idea of

considering all (theoretically) performable tests. A Martin-L¨

• f test is an open set U ⊂ 2ω of measure 0, given as an

intersection of rapidly shrinking effectively open sets. X ∈ 2ω passes this test if X / ∈ U. X is 1-random (or Martin-L¨

• f random) if it passes every ML-test.

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

We can also relativize these notions.

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

We can also relativize these notions. Write X = X0 ⊕ X1 ⊕ · · · ⊕ Xn−1 to mean that Xk = {i : ni + k ∈ X}.

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

We can also relativize these notions. Write X = X0 ⊕ X1 ⊕ · · · ⊕ Xn−1 to mean that Xk = {i : ni + k ∈ X}. Thm (van Lambalgen). X = X0 ⊕ X1 is 1-random iff X0 and X1 are 1-random relative to each other.

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

We can also relativize these notions. Write X = X0 ⊕ X1 ⊕ · · · ⊕ Xn−1 to mean that Xk = {i : ni + k ∈ X}. Thm (van Lambalgen). X = X0 ⊕ X1 is 1-random iff X0 and X1 are 1-random relative to each other. In this case, X0 and X1 are “a bit more random” than X.

Algorithmic randomness By varying the complexity of tests, we obtain a hierarchy:

1-random ⇐ weakly 2-random ⇐ 2-random ⇐ weakly 3-random ⇐ · · ·

We can also relativize these notions. Write X = X0 ⊕ X1 ⊕ · · · ⊕ Xn−1 to mean that Xk = {i : ni + k ∈ X}. Thm (van Lambalgen). X = X0 ⊕ X1 is 1-random iff X0 and X1 are 1-random relative to each other. In this case, X0 and X1 are “a bit more random” than X. This theorem generalizes to X = X0 ⊕ X1 ⊕ · · · ⊕ Xn−1.

Upper cones and minimal pairs in the Turing degrees revisited Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

Upper cones and minimal pairs in the Turing degrees revisited Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

• Cor. If A T ∅ and X is weakly 2-random relative to A then A T X.

Upper cones and minimal pairs in the Turing degrees revisited Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

• Cor. If A T ∅ and X is weakly 2-random relative to A then A T X.

Cor (Kautz). If X and Y are weakly 2-random relative to each

• ther then the Turing degrees of X and Y form a minimal pair.

Upper cones and minimal pairs in the Turing degrees revisited Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

• Cor. If A T ∅ and X is weakly 2-random relative to A then A T X.

Cor (Kautz). If X and Y are weakly 2-random relative to each

• ther then the Turing degrees of X and Y form a minimal pair.

Let ∅′ be the Halting Problem. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair.

Upper cones and minimal pairs in the Turing degrees revisited Thm (de Leeuw, Moore, Shannon, and Shapiro / Sacks). If A is not computable then µ({X : A T X}) = 0.

• Cor. If A T ∅ and X is weakly 2-random relative to A then A T X.

Cor (Kautz). If X and Y are weakly 2-random relative to each

• ther then the Turing degrees of X and Y form a minimal pair.

Let ∅′ be the Halting Problem. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair. A is a base for 1-randomness if A T X for some X that is 1-random relative to A. Cor (Kuˇ cera). There is a noncomputable base for 1-randomness.

Upper cones and minimal pairs in the coarse degrees revisited Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0.

• Cor. If A is not coarsely computable and X is weakly 3-random

relative to A then X does not compute a coarse description of A.

Upper cones and minimal pairs in the coarse degrees revisited Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0.

• Cor. If A is not coarsely computable and X is weakly 3-random

relative to A then X does not compute a coarse description of A. Cor (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X and Y are weakly 3-random relative to each other then the coarse degrees of X and Y form a minimal pair.

Upper cones and minimal pairs in the coarse degrees revisited Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If A c ∅ then µ({X : A is coarsely computable relative to X}) = 0. So proper upper cones in the coarse degrees have measure 0.

• Cor. If A is not coarsely computable and X is weakly 3-random

relative to A then X does not compute a coarse description of A. Cor (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X and Y are weakly 3-random relative to each other then the coarse degrees of X and Y form a minimal pair. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). There exist relatively 2-random X, Y whose (nonuniform) coarse degrees do not form a minimal pair.

Kolmogorov complexity Kolmogorov complexity measures the complexity of a finite

• bject in terms of the length of its shortest description.

Kolmogorov complexity Kolmogorov complexity measures the complexity of a finite

• bject in terms of the length of its shortest description.

We can think of σ ∈ 2<ω as random if it has no descriptions shorter than itself, and as far from random if it has a very short description.

Kolmogorov complexity Kolmogorov complexity measures the complexity of a finite

• bject in terms of the length of its shortest description.

We can think of σ ∈ 2<ω as random if it has no descriptions shorter than itself, and as far from random if it has a very short description. Let K(σ) be the prefix-free Kolmogorov complexity of σ ∈ 2<ω.

Kolmogorov complexity Kolmogorov complexity measures the complexity of a finite

• bject in terms of the length of its shortest description.

We can think of σ ∈ 2<ω as random if it has no descriptions shorter than itself, and as far from random if it has a very short description. Let K(σ) be the prefix-free Kolmogorov complexity of σ ∈ 2<ω. Let X ↾ n be the first n bits of X ∈ 2ω. Schnorr showed that X is 1-random iff K(X ↾ n) n − O(1).

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial.

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial. Solovay showed that noncomputable K-trivials exist.

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial. Solovay showed that noncomputable K-trivials exist. The K-trivials form a Turing ideal: they are closed downwards under T and are closed under ⊕.

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial. Solovay showed that noncomputable K-trivials exist. The K-trivials form a Turing ideal: they are closed downwards under T and are closed under ⊕. Thm (Nies). A is K-trivial iff every 1-random set is 1-random relative to A.

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial. Solovay showed that noncomputable K-trivials exist. The K-trivials form a Turing ideal: they are closed downwards under T and are closed under ⊕. Thm (Nies). A is K-trivial iff every 1-random set is 1-random relative to A. Thm (Nies and Hirschfeldt). A is K-trivial iff K(σ) K A(σ) + O(1).

K-triviality K(n) K(A ↾ n) + O(1) for any A ∈ 2ω. A is K-trivial if K(A ↾ n) K(n) + O(1). Every computable set is K-trivial. Solovay showed that noncomputable K-trivials exist. The K-trivials form a Turing ideal: they are closed downwards under T and are closed under ⊕. Thm (Nies). A is K-trivial iff every 1-random set is 1-random relative to A. Thm (Nies and Hirschfeldt). A is K-trivial iff K(σ) K A(σ) + O(1). Thm (Hirschfeldt, Nies, and Stephan). A is K-trivial iff A is a base for 1-randomness.

Algorithmically random sets as oracles Thm (Kuˇ cera / G´ acs). Every set is computable from some 1-random set.

Algorithmically random sets as oracles Thm (Kuˇ cera / G´ acs). Every set is computable from some 1-random set. Thm (Downey, Nies, Weber, and Yu). Every weakly 2-random set forms a minimal pair with ∅′, and in particular does not compute any noncomputable c.e. set.

Algorithmically random sets as oracles Thm (Kuˇ cera / G´ acs). Every set is computable from some 1-random set. Thm (Downey, Nies, Weber, and Yu). Every weakly 2-random set forms a minimal pair with ∅′, and in particular does not compute any noncomputable c.e. set. 1-randoms that do not compute ∅′ are “more random” than those that do, as quantified by Franklin and Ng.

Algorithmically random sets as oracles Thm (Kuˇ cera / G´ acs). Every set is computable from some 1-random set. Thm (Downey, Nies, Weber, and Yu). Every weakly 2-random set forms a minimal pair with ∅′, and in particular does not compute any noncomputable c.e. set. 1-randoms that do not compute ∅′ are “more random” than those that do, as quantified by Franklin and Ng. Thm (Hirschfeldt, Nies, and Stephan). If X T ∅′ is 1-random and A T X is c.e. then A is K-trivial.

Algorithmically random sets as oracles Thm (Kuˇ cera / G´ acs). Every set is computable from some 1-random set. Thm (Downey, Nies, Weber, and Yu). Every weakly 2-random set forms a minimal pair with ∅′, and in particular does not compute any noncomputable c.e. set. 1-randoms that do not compute ∅′ are “more random” than those that do, as quantified by Franklin and Ng. Thm (Hirschfeldt, Nies, and Stephan). If X T ∅′ is 1-random and A T X is c.e. then A is K-trivial. Thm (Bienvenu, Day, Greenberg, Kuˇ cera, Miller, Nies, and Turetsky). There is a 1-random X T ∅′ that computes every K-trivial set.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality. Recently, more subclasses of the K-trivials have been uncovered.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality. Recently, more subclasses of the K-trivials have been uncovered. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair. Let X ⊕ Y T ∅′ be 1-random and let A T X, Y.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality. Recently, more subclasses of the K-trivials have been uncovered. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair. Let X ⊕ Y T ∅′ be 1-random and let A T X, Y. By van Lambalgen’s Theorem, X is 1-random relative to Y, and hence relative to A.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality. Recently, more subclasses of the K-trivials have been uncovered. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair. Let X ⊕ Y T ∅′ be 1-random and let A T X, Y. By van Lambalgen’s Theorem, X is 1-random relative to Y, and hence relative to A. Thus A is a base for 1-randomness, and hence is K-trivial.

Subclasses of the K-trivials Many notions of randomness-theoretic weakness have been shown to be equivalent to K-triviality. Recently, more subclasses of the K-trivials have been uncovered. Thm (Kuˇ cera). If X, Y T ∅′ are 1-random, then their degrees do not form a minimal pair. Let X ⊕ Y T ∅′ be 1-random and let A T X, Y. By van Lambalgen’s Theorem, X is 1-random relative to Y, and hence relative to A. Thus A is a base for 1-randomness, and hence is K-trivial. Thm(Bienvenu, Greenberg, Kuˇ cera, Nies, and Turetsky). There is a K-trivial that is not computable from both halves of a 1-random.

Algorithmic randomness, K-triviality, and information coding X c = {A : A is computable from each coarse description of X}. If X is random, we expect X not to code information robustly, and hence expect X c to be trivial.

Algorithmic randomness, K-triviality, and information coding X c = {A : A is computable from each coarse description of X}. If X is random, we expect X not to code information robustly, and hence expect X c to be trivial. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X is 1-random then X c ⊆ K-trivials.

Algorithmic randomness, K-triviality, and information coding X c = {A : A is computable from each coarse description of X}. If X is random, we expect X not to code information robustly, and hence expect X c to be trivial. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X is 1-random then X c ⊆ K-trivials.

• Cor. If X is 1-random then X c = computable.

Algorithmic randomness, K-triviality, and information coding X c = {A : A is computable from each coarse description of X}. If X is random, we expect X not to code information robustly, and hence expect X c to be trivial. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X is 1-random then X c ⊆ K-trivials.

• Cor. If X is 1-random then X c = computable.

Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X T ∅′ is 1-random then X c has a noncomputable element.

Algorithmic randomness, K-triviality, and information coding X c = {A : A is computable from each coarse description of X}. If X is random, we expect X not to code information robustly, and hence expect X c to be trivial. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X is 1-random then X c ⊆ K-trivials.

• Cor. If X is 1-random then X c = computable.

Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). If X T ∅′ is 1-random then X c has a noncomputable element. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). There is a K-trivial that is not in X c for any 1-random X.

Robust codability and K-triviality Let C = {X c : X is 1-random}. As mentioned above, if A ∈ C then A is K-trivial.

Robust codability and K-triviality Let C = {X c : X is 1-random}. As mentioned above, if A ∈ C then A is K-trivial. The proof shows that there is a splitting X = X1 ⊕ · · · ⊕ Xn s.t. A is computable from every join of n − 1 many Xi.

Robust codability and K-triviality Let C = {X c : X is 1-random}. As mentioned above, if A ∈ C then A is K-trivial. The proof shows that there is a splitting X = X1 ⊕ · · · ⊕ Xn s.t. A is computable from every join of n − 1 many Xi. By van Lambalgen’s Theorem, such a join is 1-random, so A is a base for 1-randomness, and hence is K-trivial.

k n-bases

A is a k

n-base if there is a 1-random X1 ⊕ · · · ⊕ Xn s.t. A is

computable from every join of k many Xi.

k n-bases

A is a k

n-base if there is a 1-random X1 ⊕ · · · ⊕ Xn s.t. A is

computable from every join of k many Xi. As mentioned above, if A ∈ C then A is an n−1

n -base for some n.

k n-bases

A is a k

n-base if there is a 1-random X1 ⊕ · · · ⊕ Xn s.t. A is

computable from every join of k many Xi. As mentioned above, if A ∈ C then A is an n−1

n -base for some n.

Thm (Greenberg, Miller, and Nies). A ∈ C iff A is an n−1

n -base for

some n.

k n-bases

A is a k

n-base if there is a 1-random X1 ⊕ · · · ⊕ Xn s.t. A is

computable from every join of k many Xi. As mentioned above, if A ∈ C then A is an n−1

n -base for some n.

Thm (Greenberg, Miller, and Nies). A ∈ C iff A is an n−1

n -base for

some n. Thm (Greenberg, Miller, and Nies). The class of k

n-bases depends

• nly on the value of k

n as a rational.

k n-bases

A is a k

n-base if there is a 1-random X1 ⊕ · · · ⊕ Xn s.t. A is

computable from every join of k many Xi. As mentioned above, if A ∈ C then A is an n−1

n -base for some n.

Thm (Greenberg, Miller, and Nies). A ∈ C iff A is an n−1

n -base for

some n. Thm (Greenberg, Miller, and Nies). The class of k

n-bases depends

• nly on the value of k

n as a rational.

Thm (Greenberg, Miller, and Nies). The p-bases for p ∈ Q ∩ (0, 1) form a proper hierarchy of subideals of the K-trivials. Thm (Hirschfeldt, Jockusch, Kuyper, and Schupp). The union of this hierarchy is also a proper subset of the K-trivials.

Computability Theory and Asymptotic Density Denis R. Hirschfeldt — University of Chicago

Groups and Computation: In Honor of Paul Schupp’s 80th Birthday