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Mass problems and recursively bounded DNR functions
Mushfeq Khan
University of Hawai‘i at M¯anoa
ASL Meeting
Urbana, IL
March 28th, 2015
Based on:
Forcing with Bushy Trees, with Joseph S. Miller, in preparation.
(A preprint is available on my website.)
Connecting randomness and classical recursion theory
Diagonally non-recursive (DNR) functions play an important role both in classical
recursion theory and in algorithmic randomness.
Definition
A function f : ω → ω is DNR if for all e such that ϕe (e) ↓,
f (e) 6= ϕe (e).
For k ≥ 2, let DNRk denote the family of DNR functions that take values less than
k.
Theorem (Jockusch, Degrees of functions with no fixed points, 1989)
For each k > 2, every DNRk function computes a DNR2 function, and hence is of PA
degree. However, this is not uniform.
Bounded DNR functions and measure
From the point of view of the study of mass problems, PA degrees are quite
powerful. They are at the top of the lattice Ew of Muchnik (or weak) degrees of
nonempty Π01 sets of reals.
How much of this power generalizes to wider classes of DNR functions?
We now have much stronger results, but a simple “majority vote” argument shows
that the measure of reals that compute a DNR2 function (i.e., those of PA degree) is
0.
On the other hand, Kučera (Measure, Π01 classes, and complete extensions of PA,
1985) showed that there is a recursive function h such that every Martin-Löf
random real computes a h-bounded DNR function.
A minimal DNR
Theorem (Kumabe, A fixed-point-free minimal degree, unpublished; Kumabe and
Lewis, A fixed-point-free minimal degree, 2009)
There is a DNR function of minimal Turing degree.
This was the first appearance of “bushy tree forcing”. The construction can be
thought of as a combination of Spector forcing with some intricate combinatorics
involving trees that branch a lot at every level (hence “bushy”). Just as in the
Spector minimal degree construction, the generic object in the Kumabe
construction is hyperimmune-free.
Corollary
There is a recursively bounded DNR function that computes no Martin-Löf random
real.
Recursively bounded DNRs
Theorem (Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman, Comparing DNR and
WWKL, 2004)
For any recursive function h ∈ ω ω , there is a Turing ideal that contains no h-bounded
DNR function, but is a model of the principle DNR: for every degree in the ideal, there
is a DNR relative to it that is also in the ideal.
This showed that the principle DNR is weaker than WWKL0 , and answered a
question of Giusto and Simpson (Located sets and reverse mathematics, 2000).
They also showed:
Theorem (Ambos-Spies, et al, 2004)
There is a DNR function that computes no recursively bounded DNR function.
To summarize, in the Muchnik degrees:
0 < DNR < DNRrec < MLR < PA
DNRh
Interesting things happen when we fix a recursive bound h and consider what all
h-bounded DNR functions can compute, and how this power varies with h.
Definition
An order function is a function h : ω → ω \ {0, 1} that is recursive, nondecreasing
and unbounded. For an order function h, let
DNRh = {f ∈ DNR : (∀n) f (n) < h(n)}.
A note on numbering
Unlike the Martin-Löf random reals and the reals of PA degree, the definition of
the class of DNR functions is sensitive to the Gödel numbering of partial recursive
functions.
The Muchnik degrees of the DNR functions and the recursively-bounded DNR
functions are, however, well-defined. It isn’t clear that the same is true of the
DNRh classes, although Simpson (Mass problems and randomness, 2005) proposes
one way to define these classes more robustly.
From the purposes of this talk, we fix a numbering. When we say “... there is an
order function h such that ... ”, it should be understood that what h actually is
depends on the choice of numbering.
A basic bushy forcing argument
One might wonder if making h grow slowly enough ensures that all DNRh
functions have PA degree.
Proposition (Implicit in Greenberg and Miller, Diagonally nonrecursive functions and
effective Hausdorff dimension, 2011)
For every order function h, there is a DNRh function that is not of PA degree.
Even though this fact has been superseded by stronger (and more difficult) results,
the proof is a good introduction to bushy tree arguments.
Bushy trees
Definition
Let σ ∈ ω <ω . A tree T ⊆ ω <ω is n-bushy above σ if every element of T is
comparable with σ, and for each non-leaf τ of T that extends σ, τ has at least n
immediate extensions in T .
A tree that is 2-bushy above σ
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Big and small sets of strings
Definition
Given σ ∈ ω <ω , we say that a set B ⊆ ω <ω is n-big above σ if there is a finite
n-bushy tree T above σ such that all its leaves are in B. If B is not n-big above σ
then we say that B is n-small above σ.
A set of strings that is 2-big above σ
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The non-DNR strings
Example
The set that are non-DNR, i.e., there is an i < |σ|, σ(i) = ϕi (i), is 2-small above
any DNR string.
Any 2-big set of strings above a DNR string σ contains a DNR string
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Smallness preservation
Smallness preservation lemma (Kumabe and Lewis, 2009)
If A is n-small above σ and B is m-small above σ, then A ∪ B is (n + m − 1)-small
above σ.
Proof
Suppose A ∪ B is (n + m − 1)-big above σ and T is a tree with leaves in A ∪ B that
(n + m − 1)-bushy above σ. Let τ be the immediate predecessor of one of the
leaves of T of maximum length (note that T is finite).
...
Repeat this process until σ gets a label.
Smallness preservation
Smallness preservation lemma (Kumabe and Lewis, 2009)
If A is n-small above σ and B is m-small above σ, then A ∪ B is (n + m − 1)-small
above σ.
Proof
Suppose A ∪ B is (n + m − 1)-big above σ and T is a tree with leaves in A ∪ B that
(n + m − 1)-bushy above σ. Let τ be the immediate predecessor of one of the
leaves of T of maximum length (note that T is finite).
...
Repeat this process until σ gets a label.
Smallness preservation
Smallness preservation lemma (Kumabe and Lewis, 2009)
If A is n-small above σ and B is m-small above σ, then A ∪ B is (n + m − 1)-small
above σ.
Proof
Suppose A ∪ B is (n + m − 1)-big above σ and T is a tree with leaves in A ∪ B that
(n + m − 1)-bushy above σ. Let τ be the immediate predecessor of one of the
leaves of T of maximum length (note that T is finite).
...
Repeat this process until σ gets a label.
Smallness preservation
Smallness preservation lemma (Kumabe and Lewis, 2009)
If A is n-small above σ and B is m-small above σ, then A ∪ B is (n + m − 1)-small
above σ.
Proof
Suppose A ∪ B is (n + m − 1)-big above σ and T is a tree with leaves in A ∪ B that
(n + m − 1)-bushy above σ. Let τ be the immediate predecessor of one of the
leaves of T of maximum length (note that T is finite).
...
Repeat this process until σ gets a label.
A basic bushy forcing argument
Proposition (Greenberg and Miller, 2011)
For every order function h, there is a DNRh function that is not of PA degree.
Proof
We work entirely in
h<ω := {σ ∈ ω <ω : (∀i < |σ|) σ(i) < h(i)},
and force with conditions of the form (σ, B), where σ is a string, and B a set of
strings that is h(|σ|)-small above σ.
We may assume that B already contains all strings that it is h(|σ|)-big above. This
closure operation doesn’t change the smallness of B above σ.
Given a Turing reduction Γ, we would like to extend (σ, B) to another condition
(σ 0 , B0 ) that “defeats” Γ.
A basic bushy forcing argument
Proof
Extend σ to a τ ∈
/ B such that h(|τ |) ≥ 3h(|σ|).
A basic bushy forcing argument
Proof
For i ∈ {0, 1}, let Ai,x = {ρ τ : Γρ (x) ↓ = i}. Now let ϕe be the partial recursive
function that, on input x, searches for a set of strings D that is h(|σ|)-big above τ ,
and contained entirely in one of the Ai,x , and if it finds such a set, outputs i.
Suppose ϕe (e) ↓, say to 0. Then A0,e is h(|σ|)-big above τ .
0
Extend τ to a σ 0 ∈ A0,e \ B. Then Γσ (e) = ϕe (e), and (σ 0 , B) is our new condition.
A basic bushy forcing argument
Proof
For i ∈ {0, 1}, let Ai,x = {ρ τ : Γρ (x) ↓ = i}. Now let ϕe be the partial recursive
function that, on input x, searches for a set of strings D that is h(|σ|)-big above τ ,
and contained entirely in one of the Ai,x , and if it finds such a set, outputs i.
Suppose ϕe (e) ↓, say to 0. Then A0,e is h(|σ|)-big above τ .
0
Extend τ to a σ 0 ∈ A0,e \ B. Then Γσ (e) = ϕe (e), and (σ 0 , B) is our new condition.
A basic bushy forcing argument
Proof
For i ∈ {0, 1}, let Ai,x = {ρ τ : Γρ (x) ↓ = i}. Now let ϕe be the partial recursive
function that, on input x, searches for a set of strings D that is h(|σ|)-big above τ ,
and contained entirely in one of the Ai,x , and if it finds such a set, outputs i.
Suppose ϕe (e) ↓, say to 0. Then A0,e is h(|σ|)-big above τ .
0
Extend τ to a σ 0 ∈ A0,e \ B. Then Γσ (e) = ϕe (e), and (σ 0 , B) is our new condition.
A basic bushy forcing argument
Proof
If ϕe (e) ↑, then both A0,e and A1,e are h(|σ|)-small above τ .
By smallness preservation, B0 = A0,e ∪ A1,e ∪ B is 3h(|σ|)-small above τ . Since
h(|τ |) ≥ 3h(|σ|), (τ, B0 ) is our new condition.
Finally, note that (hi, B0 ), where B0 is the set of non-DNR strings, is a condition.
A basic bushy forcing argument
Proof
If ϕe (e) ↑, then both A0,e and A1,e are h(|σ|)-small above τ .
By smallness preservation, B0 = A0,e ∪ A1,e ∪ B is 3h(|σ|)-small above τ . Since
h(|τ |) ≥ 3h(|σ|), (τ, B0 ) is our new condition.
Finally, note that (hi, B0 ), where B0 is the set of non-DNR strings, is a condition.
A basic bushy forcing argument
Proof
If ϕe (e) ↑, then both A0,e and A1,e are h(|σ|)-small above τ .
By smallness preservation, B0 = A0,e ∪ A1,e ∪ B is 3h(|σ|)-small above τ . Since
h(|τ |) ≥ 3h(|σ|), (τ, B0 ) is our new condition.
Finally, note that (hi, B0 ), where B0 is the set of non-DNR strings, is a condition.
Elaborations
An interesting feature of this argument is that it partially relativizes. We make no
use of the effectivity of the set B (it is r.e.).
Proposition
For every oracle X , for every order function h, there is a DNRhX that is not of PA degree.
This method can be elaborated upon to prove:
Theorem (Ambos-Spies, et al, 2004; K., Miller)
Fix an order function h. Suppose X computes no DNRh function. Then there is an f
that is DNR relative to X such that f ⊕ X computes no DNRh function.
Note: This is an iterative version of the main result in Ambos-Spies, et al, 2004,
which was proved via a simultaneous construction of a sequence of functions, each
DNR relative to the ones preceding it.
The “DNR degrees”
In addition, one can obtain the following structural results about the classes of
DNRh functions:
Theorem (Ambos-Spies, et al, 2004)
Given any order function g, there is an order function g + and an f ∈ DNRg+ such that
f computes no DNRg function.
Theorem (K., Miller)
Given any order function g, there is an order function g − and an f ∈ DNRg− such
that f computes no DNRg− function.
Theorem (K., Miller)
Given any order function g0 , there is another order function g1 and functions
f0 ∈ DNRg0 and f1 ∈ DNRg1 such that f0 computes no DNRg1 function and f1 computes
no DNRg0 function.
Effective Hausdorff dimension and shift-complexity
These results paint a reasonable picture of the relationships between various DNRh
classes.
What about other mass problems that are below PA?
Theorem (Greenberg and Miller, 2011)
There is an order function h such that every DNRh function computes a real of
effective Hausdorff dimension 1.
Shift-complex sequences (also called everywhere-complex sequences) are reals
such that all contiguous substrings have uniformly high Kolmogorov complexity, as
measured by a coefficient 0 < δ < 1.
Theorem (K., Shift-complex sequences, 2013)
For every δ such that 0 < δ < 1, there is an order function h such that every DNRh
function computes a δ-shift-complex sequence.
Martin-Löf randomness
With respect to randomness, the behavior is quite different:
Theorem (Greenberg and Miller, 2011)
For every order function h, there is a function in DNRh that computes no Martin-Löf
random real.
This is another example of a forcing argument involving bushy tree combinatorics.
It also partially relativizes: for any oracle X , the result remains true if we replace
DNRh , with DNRXh .
Kurtz randomness
If we sacrifice partial relativization, we can prove something stronger:
Theorem (K., Miller)
For every order function h, there is a function in DNRh that computes no Kurtz
random real.
The approach in this proof is quite different from the Greenberg-Miller result
above. We force with recursive trees and make strong use of the fact that the set of
bad strings is r.e., adapting techniques from the following result:
Theorem (Downey, Greenberg, Jockusch, and Milans, Binary subtrees with few
labeled paths, 2011)
There is no single reduction Γ such that Γf is Kurtz random for all f ∈ DNR3 .
We know that the most general partial relativization of the theorem above is false:
Theorem (Miller)
0
Every DNR∅ is hyperimmune.
Hence such DNRs compute Kurtz random reals.
A minimal DNRX
It is possible to partially relativize Kumabe’s theorem:
Theorem (K.)
For every oracle X there is a DNRX function of minimal Turing degree.
The motivation for this line of investigation is the following question:
Question
What is the classical Hausdorff dimension of the set of reals of minimal Turing
degree?
Earlier we saw:
Theorem (Greenberg and Miller, 2011)
There is an order function h such that every DNRh function computes a real of
effective Hausdorff dimension 1.
A minimal DNRX
It is possible to partially relativize Kumabe’s theorem:
Theorem (K.)
For every oracle X there is a DNRX function of minimal Turing degree.
The motivation for this line of investigation is the following question:
Question
What is the classical Hausdorff dimension of the set of reals of minimal Turing
degree?
Earlier we saw:
Theorem (Greenberg and Miller, 2011)
There is an order function h such that, for every X , every DNRhX function computes a
real of effective Hausdorff dimension 1 relative to X .
If we could show that minimal DNRX functions can be DNRXh (more on this later),
then we would have answered the question.
A minimal DNRX
It is possible to partially relativize Kumabe’s theorem:
Theorem (K.)
For every oracle X there is a DNRX function of minimal Turing degree.
The motivation for this line of investigation is the following question:
Question
What is the classical Hausdorff dimension of the set of reals of minimal Turing
degree?
Earlier we saw:
Theorem (Greenberg and Miller, 2011)
There is an order function h such that, for every X , every DNRhX function computes a
real of effective Hausdorff dimension 1 relative to X .
If we could show that minimal DNRX functions can be DNRXh (more on this later),
then we would have answered the question.
The Kumabe-Lewis forcing partial order
The Kumabe-Lewis construction of a minimal DNR can be formulated as a forcing
argument using recursive subtrees of g <ω (for some fixed, but very fast growing
order function g), and r.e. bad sets. The effectivity of the components of the forcing
conditions is used throughout the proof.
For example, when searching for bushy splittings above a bushy set A of strings
extending τ , it is guaranteed that either they will be found above a subset A0 of A
and the strings A \ A0 will enter the bad set B, or τ itself will enter B.
The Kumabe-Lewis forcing partial order
The Kumabe-Lewis construction of a minimal DNR can be formulated as a forcing
argument using recursive subtrees of g <ω (for some fixed, but very fast growing
order function g), and r.e. bad sets. The effectivity of the components of the forcing
conditions is used throughout the proof.
For example, when searching for bushy splittings above a bushy set A of strings
extending τ , it is guaranteed that either they will be found above a subset A0 of A
and the strings A \ A0 will enter the bad set B, or τ itself will enter B.
Our partial order
We have no access to the bad set (the non-DNR strings relative to X are r.e. in X ),
and our trees are partial recursive, with a co-r.e. set of terminal nodes.
We search for splittings above a sufficiently big subset A0 of A. If we find them, we
add A \ A0 to the bad set B, and must ensure that B does not get too big as a result.
Some elements of A0 may belong to B, and τ itself might be in B.
Our partial order
We have no access to the bad set (the non-DNR strings relative to X are r.e. in X ),
and our trees are partial recursive, with a co-r.e. set of terminal nodes.
We search for splittings above a sufficiently big subset A0 of A. If we find them, we
add A \ A0 to the bad set B, and must ensure that B does not get too big as a result.
Some elements of A0 may belong to B, and τ itself might be in B.
The cost of bushy splitting
The splitting operation is very costly in terms of bushiness. The current
combinatorial arguments require us to start with g <ω , where g is a very
fast-growing function.
Question
Can this construction be carried out below any preimposed order function? Or
even just one slow enough to ensure being able to compute a real of effective
dimension 1?
The cost of bushy splitting
The splitting operation is very costly in terms of bushiness. The current
combinatorial arguments require us to start with g <ω , where g is a very
fast-growing function.
Question
Can this construction be carried out below any preimposed order function? Or
even just one slow enough to ensure being able to compute a real of effective
dimension 1?
The cupping property
Definition
A Turing degree x has the cupping property if for every y such that y > x, there is
a z < y such that x ∨ z ≥ y.
The cupping property should be thought of as a form of computational strength: x
possesses some nontrivial information about every degree that is properly Turing
above it.
Theorem (Kučera)
Every PA degree has the cupping property.
Question
Do sufficiently slow-growing DNR functions have the cupping property?
Summary
PA degrees
Sufficiently slow-growing DNRs
bound Kurtz randoms
No
bound shift-complex sequences
Yes
bound effective Hausdorff
dimension 1 sequences
Yes
have the cupping property
?
are not minimal
?
Thanks