Uniform Martin's conjecture, locally

TL;DR

The paper links part I of uniform Martin's conjecture to a local Turing degree phenomenon, revealing new complexity results and raising questions about part II's implications.

Contribution

It shows that part I of the conjecture follows from a local property of Turing invariants and explores the complexity of computable reducibility on equivalence relations.

Findings

Part I of uniform Martin's conjecture is equivalent to Turing determinacy.

Computable reducibility on equivalence relations is highly complex, with Turing reducibility embedded.

The local phenomenon leads to new insights beyond the original conjecture.

Abstract

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part I of uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le_c$ on equivalence relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel reducible to it. We conclude raising the question "Is part II of uniform Martin's conjecture implied by local phenomena, too?" and briefly indicating a possible direction.

Equations

$$x{\equiv }_{T}y⟹f(x){\equiv }_{T}f(y),$$

$$x{\le }_{T}y⟹f(x){\le }_{T}f(y).$$

$$\left\{x\in 2^{\mathbb{N}}\right\}x{\ge }_{T}z,$$

$$\mu (A)=\{\begin{array}{ll}1& \text{if A contains a cone}\\ 0& \text{otherwise}\end{array}$$

$$x^{\prime }(n)=\{\begin{array}{ll}1& \text{if n∈dom(φnx),}\\ 0& \text{otherwise.}\end{array}$$

$$f{\le }_{M}g⟺f(x){\le }_{T}g(x)\text{ on a cone.}$$

$$x{\le }_{T}y\text{ via i}⟹f(x){\le }_{T}f(y)\text{ via u(i)}$$

$$x{\equiv }_{T}y\text{ via (i,j)}⟹f(x){\equiv }_{T}f(y)\text{ via u(i,j)}.$$

$$A=\left\{x\in 2^{\mathbb{N}}\right\}\text{f↾[x]≡T is constant}.$$

$$x{\equiv }_{T}y⟹f(x)=f(y).$$

$$\left\{x{\ge }_{T}z\right\}f(x)(i)=j$$

$$f(x)=\{\begin{array}{ll}\underline{0}=000\dots & \text{if x∈A,}\\ {\underline{0}}^{\prime }& \text{if x∈A.}\end{array}$$

$$(x\oplus y)(n)=\{\begin{array}{ll}x\left(\frac{n}{2}\right)& \text{if n is even,}\\ y\left(\frac{n-1}{2}\right)& \text{if n is odd.}\end{array}$$

$$\left(\underset{n}{⨁}{x}_n\right)(⟨i,j⟩)=x_j(i),$$

$$\underset{n}{⨁}{\phi }_{t(n)}^x$$

$$e{\odot }_{T}x=\{\begin{array}{ll}{\phi }_e^x& \text{if φex∈2N}\\ \text{undefined}& \text{otherwise.}\end{array}$$

$$e{\odot }_{T}x\simeq y⟹u(e){\odot }_{T}f(x)\simeq f(y),$$

$$e{\odot }_{T}x\text{ is defined and is in A}⟹u(e){\odot }_{T}f(x)\simeq f(e{\odot }_{T}x).$$

$$(i,j){}^s{\odot }_{T}x=\{\begin{array}{ll}{\phi }_i^x& \text{if φix∈2N and φjφix=x,}\\ \text{undefined}& \text{otherwise.}\end{array}$$

$$(i,j){}^s{\odot }_{T}x\simeq y⟺\{\begin{array}{l}i{\odot }_{T}x\simeq y\\ j{\odot }_{T}y\simeq x.\end{array}$$

$$(i,j){}^s{\odot }_{T}x\simeq y⟹u(i,j){}^s{\odot }_{T}f(x)\simeq f(y),$$

$$(i,j)\mathbin{{}^{s}\odot_{T}}x\text{ is defined and is in $A$}\implies u(i,j)\odot_{T}f(x)\simeq f\bigl{(}(i,j)\mathbin{{}^{s}\odot_{T}}x\bigr{)}.$$

$${\phi }_i^{{\phi }_j^x}={\phi }_{i{\ast }_{T}j}^x,$$

$$j{\odot }_{T}x\text{ is defined}⟹i{\odot }_T(j{\odot }_{T}x)\simeq (i{\ast }_{T}j){\odot }_{T}x.$$

$${\phi }_a^{0^e{1}^{⌢}x}={\phi }_e^x$$

$$e{\odot }_{T}x\simeq (a{b}^{e}c){\odot }_{T}x.$$

$$\displaystyle f(e\odot_{T}x)\simeq f\bigl{(}(ab^{e}c)\odot_{T}x\bigr{)}$$

$$⋮$$

$$\simeq u(a)u(b{)}^{e}u(c){\odot }_{T}f(x),$$

$$v(e)=u(a)u(b{)}^{e}u(c)$$

$$(i,j){}^s{\ast }_T(k,l)=(i{\ast }_{T}k,l{\ast }_{T}j).$$

Full text

1 Introduction to Martin’s conjecture

Providing evidence for the intricacy of the structure $(\mathcal{D},\leq_{T})$ of Turing degrees has been arguably one of the main concerns of computability theory since the mid ’50s, when the celebrated priority method was discovered. However, some have pointed out that if we restrict our attention to those Turing degrees that correspond to relevant problems occurring in mathematical practice, we see a much simpler structure: such “natural” Turing degrees appear to be well-ordered by $\leq_{T}$, and there seems to be no “natural” Turing degree strictly between a “natural” Turing degree and its Turing jump. Martin’s conjecture is a long-standing open problem whose aim was to provide a precise mathematical formalization of the previous insight. The leading idea is to formalize the notion of “natural” Turing degree as a suitable equivalence class of “definable” functions over Turing degrees.

A function $f:A\to{2^{\mathbb{N}}}$, where $A\subseteq{2^{\mathbb{N}}}$, is said to be Turing invariant (abbreviated TI) if, for all $x,y\in A$ one has

[TABLE]

whereas it is said to be order-preserving (abbreviated OP) if, for all $x,y\in A$,

[TABLE]

The intuition behind Martin’s conjecture is that “natural” Turing degrees are supposed to induce, by relativization, “definable” TI functions and, vice versa, “definable” TI functions are supposed to come from this process of relativizing some “natural” Turing degree. Precisely, ‘definable’ is formalized setting Martin’s conjecture under the Axiom of Determinacy ($\mathsf{AD}$).

Recall that, in this context, upward Turing cones, i.e. sets of the form

[TABLE]

are usually referred to just as cones. Also recall that $A\subseteq{2^{\mathbb{N}}}$ is said to be Turing-invariant if it is closed under $\equiv_{T}$. Turing Determinacy ($\mathsf{TD}$) denotes the statement that every Turing-invariant subset of ${2^{\mathbb{N}}}$ either contains a cone or is disjoint from a cone. Martin’s celebrated cone theorem [Mar68] states that $\mathsf{TD}$ follows from $\mathsf{AD}$. The importance of $\mathsf{TD}$ lies in the fact that it enables to define a natural notion of largeness of Turing-invariant sets: the map

[TABLE]

defines, under $\mathsf{TD}$, a measure on the $\sigma$-algebra of Turing-invariant subsets of ${2^{\mathbb{N}}}$.

Recall the definition of the Turing jump of $x\in{2^{\mathbb{N}}}$:

[TABLE]

Finally, given two TI functions $f,g:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$, one defines

[TABLE]

Conjecture 1.1** (Martin).**

\thlabel

mc In $\mathsf{ZF+DC+AD}$, the following are conjectured:

1. I.

if $f:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ is Turing invariant, then either $f(x)\geq_{T}x$ on a cone or there exists $y\in{2^{\mathbb{N}}}$ such that $f(x)\equiv_{T}y$ on a cone; 2. II.

the set of TI functions $f$ such that $f\geq_{M}\operatorname{id}_{{2^{\mathbb{N}}}}$ is pre-well-ordered by $\leq_{M}$; moreover, if such an $f$ has $\leq_{M}$-rank $\alpha$, then $f^{\prime}$ (defined by $f^{\prime}(x)=f(x)^{\prime}$) has $\leq_{M}$-rank $\alpha+1$.

On a side note, part I of Martin’s conjecture, in particular, has been in vogue since the discovery of its profound consequences in the theory of countable Borel equivalence relations (see [Tho09]).

Despite still being open, Martin’s conjecture was proved true when restricted to a particular class of functions — namely the UTI functions — by Slaman and Steel in [Ste82, SS88]. Let us recall what UTI functions are.

Let $(\varphi_{i}^{x})_{i\in\mathbb{N}}$ be the standard numbering of partial unary computable-in-$x$ functions, where the oracle $x$ is a function from $\mathbb{N}$ to $\mathbb{N}$. Given $x,y\in{2^{\mathbb{N}}}$ and $i,j\in\mathbb{N}$, we say that $x\leq_{T}y$ via $i$ if $x=\varphi_{i}^{y}$, and we say that $x\equiv_{T}y$ via $(i,j)$ if $x\geq_{T}y$ via $i$ and $x\leq_{T}y$ via $j$.

A function $f:A\to{2^{\mathbb{N}}}$, with $A\subseteq{2^{\mathbb{N}}}$, is said to be uniformly order-preserving (abbreviated UOP) if every time we have $x,y\in A$ such that $x\leq_{T}y$ via $i$, we can choose uniformly in $x$ and $y$ an index $j$ such that $f(x)\leq_{T}f(y)$ via $j$. In other words, $f$ is UOP if there is a function $u:\mathbb{N}\to\mathbb{N}$ such that

[TABLE]

for all $x,y\in A$. Similarly, $f$ is said to be uniformly Turing invariant (abbreviated UTI) if there is a function $u:\mathbb{N}^{2}\to\mathbb{N}^{2}$ such that, for all $x,y\in A$,

[TABLE]

Uniform Martin’s conjecture refers to Martin’s conjecture for UTI functions only, projective Martin’s conjecture refers to Martin’s conjecture for projective TI functions only, and so on.

2 Part I of uniform Martin’s conjecture, from local to global

In Steel’s paper [Ste82] and in Slaman and Steel’s paper [SS88], it was proved respectively that part II and part I of uniform Martin’s conjecture hold. We shall present a soft proof of the following slight improvement of the latter result. Recall that Turing Determinacy ($\mathsf{TD}$) denotes the statement that every $A\subseteq{2^{\mathbb{N}}}$ which is closed under $\equiv_{T}$ either contains a cone or is disjoint from a cone, and also recall this follows from $\mathsf{AD}$ by Martin’s cone theorem.

Theorem 2.1**.**

\thlabel

umc1 Assume $\mathsf{TD}$ and let $f:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ be UTI on a cone. Then, either $f(x)\geq_{T}x$ on a cone, or there exists $y\in{2^{\mathbb{N}}}$ such that $f(x)=y$ on a cone.

Let us stress the differences between the results: Slaman and Steel showed that, under $\mathsf{AD}$, UTI functions are either increasing or constant up to Turing equivalence on a cone. By contrast, \threfumc1 tells us that, under the sole assumption of $\mathsf{TD}$, UTI functions are either increasing or literally constant on a cone.

The interesting thing about our proof of \threfumc1 is that it shows us how the global dichotomy in such theorem actually arises from an analogous local dichotomy, i.e. a dichotomy that UTI functions exhibit on each single Turing degree.

Theorem 2.2**.**

\thlabel

lmc1 Let $x\in{2^{\mathbb{N}}}$ and $f:[x]_{\equiv_{T}}\to{2^{\mathbb{N}}}$ be UTI. Then, either $f(x)\geq_{T}x$ or $f$ is constant.

This \thnamereflmc1, which could be called a local version of Slaman and Steel’s theorem, is the main result of this paper. Before we prove it, let us show how easily \threfumc1 descends from it.

Proof of \threfumc1.

Suppose $f$ is UTI in the cone above $z$. Consider

[TABLE]

$A$ is Turing invariant, so by $\mathsf{TD}$ either ${2^{\mathbb{N}}}\setminus A$ or $A$ contains a cone. In the former case — say ${2^{\mathbb{N}}}\setminus A$ contains the cone above $w$ — given any $x\geq_{T}z\oplus w$, we can apply \threflmc1 and deduce $f(x)\geq_{T}x$. Otherwise, if $A$ contains a cone, next folklore \thnamereflemcost applies. ∎

Fact 2.3**.**

\thlabel

lemcost Suppose $f:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ is such that the following holds for all $x,y$ in a cone:

[TABLE]

Then, assuming $\mathsf{TD}$, $f$ is literally constant on a cone.

The easy yet classic argument for \threflemcost is probably found for the first time in [SS88], in the form of a remark that $\mathsf{AD}$ implies there is no choice function on Turing degrees. We present it here for the reader’s convenience.

Proof of \threflemcost.

Suppose that the hypothesis holds in the cone based in $z$. Then, the sets of the form

[TABLE]

are Turing invariant, and so $\mathsf{TD}$ implies that the $i$-th digit of $f(x)$ is constant for all $x$ in a cone $C_{i}$. Hence, $f$ is constant on the intersection of the $C_{i}$’s (which trivially contains a cone). ∎

\thref

lmc1 enables us to calibrate the strength of the statement of part I of uniform Martin’s conjecture over $\mathsf{ZF+DC}$. We actually have two different statements for uniform Martin’s conjecture part I, namely the original one and the statement of \threfumc1. However, our calibration holds for both.

Corollary 2.4**.**

\thlabel

metac The following statements are equivalent over $\mathsf{ZF+DC}$:

1. $(a)$

$\mathsf{TD}$; 2. $(b)$

for all $f:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ which is either UTI on a cone, either $f(x)\geq_{T}x$ on a cone, or $f$ is literally constant on a cone; 3. $(c)$

for all $f:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ which is either UTI on a cone, either $f(x)\geq_{T}x$ on a cone, or $f$ is constant up to $\equiv_{T}$ on a cone.

Proof.

$(a)\implies(b)$ is precisely \threfumc1, whereas $(b)\implies(c)$ is trivial. Let us prove $(a)$ from $(c)$. Fix $A\subseteq{2^{\mathbb{N}}}$ which is Turing invariant (i.e. closed under $\equiv_{T}$), and define

[TABLE]

Of course, $f$ is UTI, so by $(c)$ we get that $f$ is constant on a cone up to $\equiv_{T}$. Then, either $f(x)\equiv_{T}\underline{0}$ on a cone, or $f(x)\equiv_{T}\underline{0}^{\prime}$ on a cone. In the former case, $A$ contains a cone, in the latter one, the complement of $A$ does. ∎

In [CWY10], the authors calibrated the strength of part II of uniform Martin’s conjecture for projective functions.

Theorem 2.5** (Chong, Wang and Yu, [CWY10]).**

\thlabel

thmchong Over $\mathsf{ZFC}$, part II of projective uniform Martin’s conjecture is equivalent to Projective Determinacy (which, by unpublished work by Woodin, is equivalent over $\mathsf{ZFC}$ to Projective Turing Determinacy).

Putting together \threfmetac and \threfthmchong, and observing that our assumption of $\mathsf{TD}$ in the proof of \threfmetac “localizes”, we get:

Theorem 2.6**.**

The following are equivalent over $\mathsf{ZFC}$:

• •

Projective Determinacy;

• •

Projective Turing Determinacy;

• •

part I of projective uniform Martin’s conjecture;

• •

part II of projective uniform Martin’s conjecture.

3 The proof

We now address the proof of \threflmc1. The argument itself is very short and easy, but we need a few preliminaries and notation first. Recall that the join (or merge) of $x,y\in{2^{\mathbb{N}}}$, is the element of ${2^{\mathbb{N}}}$ denoted by $x\oplus y$ and defined by

[TABLE]

Moreover, the join (or merge) of a sequence $(x_{n})_{n\in\mathbb{N}}$ of elements of ${2^{\mathbb{N}}}$ is the element of ${2^{\mathbb{N}}}$ denoted by $\bigoplus_{n}x_{n}$ defined by

[TABLE]

where $\braket{\cdot,\cdot}$ is a computable bijection between $\mathbb{N}^{2}$ and $\mathbb{N}$ chosen once for all. We shall say that $x_{j}$ is the $j$-th column of $\bigoplus_{n}x_{n}$, while $n\mapsto x_{n}(i)$ is its $i$-th row. The following fact easily descends from the existence of a universal oracle Turing machine.

Fact 3.1**.**

\thlabel

complem Fix $x\in{2^{\mathbb{N}}}$ and a computable $t:\mathbb{N}\to\mathbb{N}$. If

[TABLE]

is in ${2^{\mathbb{N}}}$, then it is Turing reducible to $x$.

Definition 3.2**.**

For $e\in\mathbb{N}$ and $x\in{2^{\mathbb{N}}}$, set

[TABLE]

The graph of $\odot_{T}$ is the set of $(e,x,y)$ in $\mathbb{N}\times{2^{\mathbb{N}}}\times{2^{\mathbb{N}}}$ such that $y\leq_{T}x$ via $e$; for this reason, we call $\odot_{T}$ “Turing reducibility via”.

Notation 3.3**.**

Let $\simeq$ denote Kleene’s equality: $\varphi\simeq\psi$ holds exactly when, if either $\varphi$ or $\psi$ is defined, then the other is defined as well and the two are equal.

Rephrasing the definition of UOP, given $A\subseteq{2^{\mathbb{N}}}$, we have that $f:A\to{2^{\mathbb{N}}}$ is UOP when there is $u:\mathbb{N}\to\mathbb{N}$ such that, for all $e\in\mathbb{N}$ and $x,y\in A$,

[TABLE]

or equivalently,

[TABLE]

A function $u$ as above is called uniformity function for $f$.

Also define, for $(i,j)\in\mathbb{N}^{2}$ and $x\in{2^{\mathbb{N}}}$,

[TABLE]

The symbol s stands for ‘symmetrization’: $\mathbin{{}^{s}\odot_{T}}$ can be viewed as some kind of symmetrization of $\odot_{T}$, as we have

[TABLE]

We call $\mathbin{{}^{s}\odot_{T}}$ Turing equivalence via, since its graph is the set of \bigl{(}(i,j),x,y\bigr{)} in $\mathbb{N}^{2}\times{2^{\mathbb{N}}}\times{2^{\mathbb{N}}}$ such that $x\equiv_{T}y$ via $(i,j)$.

Similarly as above, note that $f:A\to{2^{\mathbb{N}}}$ is UTI when there is $u:\mathbb{N}^{2}\to\mathbb{N}^{2}$ such that, for all $(i,j)\in\mathbb{N}^{2}$ and $x,y\in A$,

[TABLE]

or equivalently,

[TABLE]

Also in this case, $u$ is called a uniformity function for $f$.

Lemma 3.4**.**

\thlabel

comvar Let $A\subseteq{2^{\mathbb{N}}}$ be such that for all $x\in A$, the concatenations $0^{\frown}x$ and $1^{\frown}x$ are in $A$, too. Let $f:A\to{2^{\mathbb{N}}}$ be either UOP or UTI. In either case, there is a computable uniformity function for $f$.

Proof.

First, suppose $f$ is UOP and $u$ is a uniformity function for it. Consider the obvious binary operation $*_{T}$ on $\mathbb{N}$ that leads

[TABLE]

so that we have

[TABLE]

Observe that $*_{T}$ is defined, at least implicitly, when showing that $\leq_{T}$ is transitive. The crucial thing to note here is that $*_{T}$ is computable.

Let $a,b,c\in\mathbb{N}$ be such that $\varphi_{c}^{x}=1^{\frown}x$, $\varphi_{b}^{x}=0^{\frown}x$ and

[TABLE]

for all $x\in{2^{\mathbb{N}}}$ ($0^{e}1$ is shorthand for $\underbrace{0\dots 0}_{e}1$). Also, let $ij$ be shorthand for $i*_{T}j$, $ijk$ for $i*_{T}(j*_{T}k)$, and so on.111In fact, $*_{T}$ is associative and (2) tells us that $\odot_{T}$ resembles an action of the semi-group $(\mathbb{N},*_{T})$. Now, fix $x\in A$ and $e\in\mathbb{N}$ such that $e\odot_{T}x$ is defined and is in $A$ and notice that we have

[TABLE]

Therefore:

[TABLE]

where we used (2), (1) and the hypothesis that $(b^{k}c)\odot_{T}x$ is defined and is in $A$ for all $k\leq e$.222The author wishes to thank Kirill Gura for pointing out the necessity of the hypothesis that $A$ is closed under initial appending of [math]’s and $1$’s in order to carry on this argument. Thus, setting

[TABLE]

we get that $v$ is a uniformity function for $f$, and since $u(a)$, $u(b)$, $u(c)$ are three fixed natural numbers and $*_{T}$ is computable, $v$ is computable, too.

When $f$ is UTI, the argument analogous. This time, define

[TABLE]

Abbreviate $(i,j)\mathbin{{}^{s}*_{T}}(k,l)$ as $(i,j)(k,l)$ and (i,j)\bigl{(}(k,l)(m,n)\bigr{)} as $(i,j)(k,l)(m,n)$.333Again, ${}^{s}*_{T}$ is associative, but this is unnecessary for the scope of this proof. Observe that we have

[TABLE]

Pick $m\in\mathbb{N}$ such that $\varphi_{m}^{x}(n)=x(n+1)$ for all $n$, and notice that, with $b,c\in\mathbb{N}$ as before, we have, for all $x\in{2^{\mathbb{N}}}$:

[TABLE]

Also let $d\in\mathbb{N}$ be such that, for all $x\in{2^{\mathbb{N}}}$,

[TABLE]

Now observe that, for $x,y\in{2^{\mathbb{N}}}$:

[TABLE]

Thus, if $u$ is a uniformity function for $f$, we can set

[TABLE]

and the same argument as before gives us that $v$ is a computable uniformity function for $f$. ∎

Proof of \threflmc1.

Suppose $f$ is not constant, so that there is $z\equiv_{T}x$ such that $f(x)\neq f(z)$. Obviously, there is a computable function $r$ such that

[TABLE]

Also obviously, there is $e\in\mathbb{N}$ such that

[TABLE]

Setting $t:\mathbb{N}\to\mathbb{N}^{2},n\mapsto(r(n),e)$, we get that $t$ is computable and

[TABLE]

We thus have

[TABLE]

This means the columns of $\bigoplus_{n}f(t(n)\mathbin{{}^{s}\odot_{T}}x)$ are either $f(x)$ or $f(z)$, and they alternate exactly as the bits of $x$ do. So, supposing that $f(x)$ and $f(z)$ differ on the $k$-th digit, the $k$-th row of \bigoplus_{n}f\bigl{(}t(n)\mathbin{{}^{s}\odot_{T}}x\bigr{)} is either $x$ or $i\mapsto 1-x(i)$, and hence

[TABLE]

But also, if we let $u$ be a computable uniformity function for $f$ (which exists by \threfcomvar) and $\pi:\mathbb{N}^{2}\to\mathbb{N}$ be the projection on the first coordinate, we get

[TABLE]

so \threfcomplem tells us that

[TABLE]

4 Applications

4.1 Comparing Turing degrees as structures

Although Turing degrees are usually viewed as the “atoms” of the main structure investigated in computability theory, namely $(\mathcal{D},\leq_{T})$, Turing reductions provide each Turing degree with a structure, so we might study Turing degrees as structures themselves.

Given $A\subseteq{2^{\mathbb{N}}}$, we call Turing reducibility on $A$ the following two-sorted relation:

[TABLE]

or, with an abuse of language, the underlying two-sorted structure $(A,\mathbb{N};\odot_{T,A})$.444Note that $\mathbb{N}$ does not carry any structure with it in $(A,\mathbb{N};\odot_{T,A})$. Turing equivalence via on $A$ is defined analogously.

Even though, single Turing degrees are trivial structures when equipped with Turing reducibility or equivalence, they are not trivial when endowed with Turing reducibility via or Turing equivalence via. So, for instance, we might want to understand, if the complexity of $\bm{x}$ as a structure depends on the computational complexity of $\bm{x}$ as a Turing degree, or how the structure on $\bm{x}$ relates to the structure on a different $\bm{y}$.

An embedding of $(\bm{x},\mathbb{N};\odot_{T,\bm{x}})$ into $(\bm{y},\mathbb{N};\odot_{T,\bm{y}})$ (or, more shortly, of $\odot_{T,\bm{x}}$ to $\odot_{T,\bm{y}}$) is a pair of functions $(f,u)$, with $f:\bm{x}\to\bm{y}$ and $u:\mathbb{N}\to\mathbb{N}$ preserving the truth of atomic formulas in both directions, which means

[TABLE]

for all $i,j\in\mathbb{N}$ and $x,z\in\bm{x}$. In other words, $f$ and $u$ are injective and preserve Turing reducibility via in both directions.

Theorem 4.1**.**

\thlabel

thm2 For all Turing degrees $\bm{x}$ and $\bm{y}$, the following are equivalent:

the structure on $\bm{x}$ is embeddable in the structure on $\bm{y}$, when the structure is given by Turing reducibility via; 2. 2.

the structure on $\bm{x}$ is embeddable in the structure on $\bm{y}$, when the structure is given by Turing equivalence via; 3. 3.

$\bm{x}\leq_{T}\bm{y}$.

Proof.

$\mathit{\ref{emb}}\implies\mathit{\ref{emb2}}$: if $(f,u)$ is an embedding of $\odot_{T,\bm{x}}$ into $\odot_{T,\bm{y}}$, then $(f,u\times u)$ is an embedding of $\mathbin{{}^{s}\odot_{T,\bm{x}}}$ into $\mathbin{{}^{s}\odot_{T,\bm{y}}}$, where $u\times u$ is the map $(i,j)\mapsto(u(i),u(j))$.

$\mathit{\ref{emb2}}\implies\mathit{\ref{red}}$: if $(f,u)$ is an embedding of $\mathbin{{}^{s}\odot_{T,\bm{x}}}$ into $\mathbin{{}^{s}\odot_{T,\bm{y}}}$, then $f$ is an injective (hence non-constant) and UTI function from $\bm{x}$ to $\bm{y}$, so we get $\bm{x}\leq_{T}\bm{y}$ from \threflmc1.

$\mathit{\ref{red}}\implies\mathit{\ref{emb}}$: choose $y\in\bm{y}$ and define $f:\bm{x}\to\bm{y},z\mapsto z\oplus y$. Observe that $f$ is injective and its range is indeed included in $\bm{y}$ because $\bm{x}\leq_{T}\bm{y}$. It is easy to see that there is an injective $u:\mathbb{N}\to\mathbb{N}$ such that, for all $z_{1},z_{2},z_{3}\in{2^{\mathbb{N}}}$ and all $i\in\mathbb{N}$, we have

[TABLE]

Thus, $(f,u)$ is an embedding of $\odot_{T,\bm{x}}$ into $\odot_{T,\bm{y}}$. ∎

Remark 4.2**.**

The clearness of \threfthm2 seems rather peculiar of the Turing case: for example, we can formulate a via version for arithmetic reducibility and equivalence, too, but the analog of \threfthm2 would fail in the arithmetic case. Indeed, taking $g:{2^{\mathbb{N}}}\to{2^{\mathbb{N}}}$ to be a counter-example of the arithmetic uniform Martin’s conjecture, part I (proved to exist by Slaman and Steel, see [MSS16]), we get that there is $z\in{2^{\mathbb{N}}}$ such that, for all $x\geq_{A}z$, arithmetic reducibility via on $[x]_{\equiv_{A}}$ is embeddable into arithmetic reducibility via on $[g(x)]_{\equiv_{A}}$ even though $x\not\leq_{A}g(x)$.

Indeed, although almost every part of the proof remains valid in the arithmetic case, \threfcomplem does not, as there is no universal arithmetic reduction. Hence, the best that we can get from the analog of \threflmc1 is that every non-constant uniformly arithmetically invariant $f:[x]_{\equiv_{A}}\to{2^{\mathbb{N}}}$ satisfies $x\leq_{T}{f(x)}^{(\omega)}$. However, this is not enough to characterize those pairs of arithmetic degrees $(\bm{x},\bm{y})$ such that there is an embedding (or a homomorphism) from $\bm{x}$ to $\bm{y}$.

4.2 Reducing $\leq_{T}$ to computable reducibility

Recall that, given two binary relations $R$ and $S$ on sets $X$ and $Y$ respectively, a homomorphism from $R$ to $S$ is a function $f:X\to Y$ such that

[TABLE]

Furthermore, such $f$ is a reduction if

[TABLE]

When $X$ and $Y$ are standard Borel spaces and there is a Borel reduction from $R$ to $S$, one says that $R$ is Borel reducible to $S$, and writes $R\leq_{B}S$. On the other hand, when $X=Y=\mathbb{N}$ and there is a computable reduction from $R$ to $S$, one says that $R$ is computably reducible to $S$, and writes $R\leq_{c}S$. When $R\leq_{c}S$ and $S\leq_{c}R$, one says that $R$ and $S$ are computably bi-reducible, and writes $R\sim_{c}S$. Analogously, one defines Borel bi-reducibility $\sim_{B}$ as the symmetrization of $\leq_{B}$.

Such reducibility notions are well-established tools to compare the complexity of equivalence relations, and thus measure the difficulty of the classification problems that equivalence relations embody (see, for example, [Gao08] for $\leq_{B}$ and [CHM11] for $\leq_{c}$). Borel reducibility is frequently used to compare quasi-orders, too.

Computable reducibility and bi-reducibility are themselves a Borel quasi-order and a Borel equivalence relation respectively, whether they considered on the Polish space $2^{\mathbb{N}\times\mathbb{N}}$ of all binary relations, or they are considered on the closed subset (hence, Polish space itself) $\mathrm{ER}\subseteq 2^{\mathbb{N}\times\mathbb{N}}$ of all equivalence relations on $\mathbb{N}$. For the rest of the paper, we refer to $\leq_{c}$ and $\sim_{c}$ as being defined on $\mathrm{ER}$.

Definition 4.3**.**

When a Borel quasi-order (resp. equivalence relation) has countable downward cones (resp. equivalence classes) it is called countable Borel quasi-order (resp. countable Borel equivalence relation).

These are well-studied classes of Borel relations (see, for instance, [Wil14] and [JKL02]). For instance, $\leq_{T}$ and $\leq_{c}$ are countable Borel quasi-orders and $\equiv_{T}$ and $\sim_{c}$ are countable Borel equivalence relations. As we are going to show, essentially the same argument that led to \threflmc1 entails that $\leq_{T}$ is Borel reducible to ${\leq_{c}}$, and hence $\equiv_{T}$ is Borel reducible to ${\sim_{c}}$.

Definition 4.4**.**

For $x\in{2^{\mathbb{N}}}$, define $\approx_{T}^{x}$ to be the equivalence relation on $\mathbb{N}$ given by

[TABLE]

Theorem 4.5**.**

\thlabel

redu The map $x\mapsto{\approx_{T}^{x}}$ is a Borel reduction from $\leq_{T}$ to $\leq_{c}$ (and hence from $\equiv_{T}$ to $\sim_{c}$).

Proof.

If $x\leq_{T}y$, say $x=\varphi_{k}^{y}$, then recall the definition of $*_{T}$ from the proof of \threfcomvar and note that

[TABLE]

so the map $i\mapsto i*_{T}k$ is a computable reduction from $\approx_{T}^{x}$ to $\approx_{T}^{y}$.

Vice versa, suppose $(\approx_{T}^{x})\leq_{c}(\approx_{T}^{y})$ as witnessed by the computable reduction $v$. We exploit the same idea as in \threflmc1. Choose any two $a,b\in\mathbb{N}$ such that $a\not\approx_{T}^{x}b$ and hence, since $v$ is a reduction, $v(a)\not\approx_{T}^{y}v(b)$. This means there is some $k$ such that $\varphi_{v(a)}^{y}(k)\not\simeq\varphi_{v(b)}^{y}(k)$. Thus, $\varphi_{v(a)}^{y}(k)$ and $\varphi_{v(b)}^{y}(k)$ cannot be both undefined, so suppose, for example, that $\varphi_{v(a)}^{y}(k)$ is defined and equals, say, $m$; then, whether is defined or not, $\varphi_{v(b)}^{y}(k)$ does not equal $m$. Take now a computable function $r$ such that, for all $n$,

[TABLE]

Using the fact that $v$ is a reduction (in particular, a homomorphism), we get

[TABLE]

Now, to know if $x(n)$ equals $1$ or [math], it suffices for $y$ to parallel compute $\varphi_{v(r(2n))}^{y}(k)$ and $\varphi_{v(r(2n+1))}^{y}(k)$ and wait for $m$ to come out as the output of either computation. If $m$ comes from $\varphi_{v(r(2n))}^{y}(k)$, then $x(n)=1$, otherwise, if it comes from $\varphi_{v(r(2n+1))}^{y}(k)$, then $x(n)=0$.

Since $v$ and $r$ are computable and there exists a universal oracle Turing machine, the function $n\mapsto\varphi_{v(r(n))}^{y}$ is computable in $y$, and hence the procedure above describes a program that computes $x$ from $y$.

Thus, we have

[TABLE]

and the Borelness of the map is clear. ∎

Remark 4.6**.**

The connection between \threflmc1 and \threfredu is the following. If we examine the proof of the former, we can observe that it holds not only when $f:[x]_{\equiv_{T}}\to{2^{\mathbb{N}}}$ is UTI, but it suffices that $f$ admits a computable function $u$ such that

[TABLE]

Note that such $u$ need not be a uniformity function for $f$, as the previous formula need not hold for all elements of $[x]_{\equiv_{T}}$, but just for $x$.

On the other hand, if we define $\mathop{\downarrow_{T}}x=\set{\varphi_{e}^{x}}{e\in\mathbb{N}}$, then a homomorphism $v$ from $\approx_{T}^{x}$ to $\approx_{T}^{y}$ defines a function $f:(\mathop{\downarrow_{T}}x)\to(\mathop{\downarrow_{T}}y)$ by

[TABLE]

and vice versa. When, $v$ is not just a homomorphism, but a reduction, then $f$ is injective, in particular non-constant. Thus, we can view the proof that $(\approx_{T}^{x})\leq_{c}(\approx_{T}^{y})$ implies $x\leq_{T}y$ as an argument in the style of \threflmc1 applied to such $f$.

Remark 4.7**.**

In [CHM11], the authors indicated a way to turn an equivalence relation $E$ on ${2^{\mathbb{N}}}$ to an equivalence relation $E^{\mathit{ce}}$ on $\mathbb{N}$, defined by

[TABLE]

where $W_{i}$ denotes the $i$-th computably enumerable set, i.e. $\operatorname{dom}(\varphi_{i})$. In particular, they studied $=^{\mathit{ce}}$. Of course, the same process can be done relative to any oracle $x\in{2^{\mathbb{N}}}$: we could define

[TABLE]

Then, it is easy to see that $(=^{\mathit{ce},x})\sim_{c}(\approx_{T}^{x})$ for all $x$, so the map $x\mapsto(=^{\mathit{ce},x})$ is another Borel reduction from $\leq_{T}$ to $\leq_{c}$.

It would be interesting to understand the behavior of the map $T$ that takes a countable Borel equivalence relation $E$ to the Borel equivalence relation $T(E)$ that makes the map $x\mapsto E^{\mathit{ce},x}$ a reduction from $T(E)$ to $\sim_{c}$.

In the theory of countable Borel equivalence relations, a fundamental result by Adams and Kechris revealed the intricacy of the structure of $\leq_{B}$ on countable Borel equivalence relations.

Theorem 4.8** (Adams-Kechris, [AS00]).**

The partial order of Borel sets under inclusion can be embedded in the quasi-order of Borel reducibility of countable Borel equivalence relations, i.e., there is a map $A\mapsto E_{A}$ from the Borel subsets of $\mathbb{R}$ to countable Borel equivalence relations such that $A\subseteq B\iff E_{A}\leq_{B}E_{B}$. In particular it follows that any Borel partial order can be embedded in the quasi-order of Borel reducibility of countable Borel equivalence relations.

This theorem disclosed at once many features of $\leq_{B}$ on countable Borel equivalence relations, like — for instance — that it features antichains of size $2^{\aleph_{0}}$ and chains of size $\aleph_{1}$.

\thref

redu can be viewed as something similar for the theory of equivalence relations on $\mathbb{N}$. Indeed, we know from computability theory that there are many orders that we can embed into the Turing degrees.

Theorem 4.9** (Sacks, [Sac61]).**

Every partial order of cardinality $\leq\aleph_{1}$ in which every downward cone is countable can be embedded into the Turing degrees.

Corollary 4.10**.**

Let $\mathrm{ER}$ be the set of equivalence relations on $\mathbb{N}$. Every partial order of cardinality $\leq\aleph_{1}$ in which every downward cone is countable can be embedded into $(\mathrm{ER}/{\sim_{c}},{\leq_{c}})$.

We also know that there are antichains of Turing degrees of size $2^{\aleph_{0}}$ (for example, that given by minimal Turing degrees).

Corollary 4.11**.**

There are $2^{\aleph_{0}}$ equivalence relations on $\mathbb{N}$ that are mutually $\leq_{c}$-incomparable.

5 Part II, locally?

After showing that part I of uniform Martin’s conjecture is the consequence of a local phenomenon, it comes natural to ask whether this is also the case for part II.

In [Bec88], Becker reproved part II of uniform Martin’s conjecture in a particularly perspicuous way: he used the descriptive set-theoretic notion of “reasonable pointclass” and proved that, under $\mathsf{AD}$, every UTI $f>_{M}\operatorname{id}_{{2^{\mathbb{N}}}}$ is Turing equivalent on a cone to a $\Gamma$-jump operator

[TABLE]

for some reasonable pointclass $\Gamma$. Reasonable pointclasses are indeed lightface pointclasses that can be relativized to arbitrary $x\in{2^{\mathbb{N}}}$ and admit universal sets. For example, the Turing jump $x\mapsto x^{\prime}$ is a $\Sigma^{0}_{1}$-jump operator, the relativization of Kleene’s $\mathcal{O}$, $x\mapsto\mathcal{O}^{x}$, is a $\Pi^{1}_{1}$-jump operator, and so on.

Part II of uniform Martin’s conjecture then follows from the link between the ordering $\leq_{M}$ on pointclass jump operators and Wadge reducibility $\leq_{W}$ on ${2^{\mathbb{N}}}$. Recent work by Kihara and Montalbán [KM18] improved Becker’s result, pushing even further this connection.

Thus, we might ask whether these results arise locally. In fact, Becker’s theorem, and a fortiori Kihara and Montalbán’s, tell us that, up to Turing equivalence on a cone, there exist no other UTI functions besides constant functions, identity function and pointclass jump operators (under $\mathsf{AD}$), so it is natural to ask whether any UTI functions that have nothing to do with constant functions, identity function and pointclass jump operators can exist locally.

Question 5.1**.**

Fix a Turing degree $\bm{x}$, and consider the smallest family $\mathcal{J}_{\bm{x}}$ of functions $f:\bm{x}\to{2^{\mathbb{N}}}$ that contains

• •

all constant functions from $\bm{x}$ to ${2^{\mathbb{N}}}$

• •

$\operatorname{id}_{\bm{x}}$* and $\bm{x}\ni x\mapsto\bar{x}$, where \bar{x}:i\mapsto\bigl{(}1-x(i)\bigr{)}*

• •

all pointclass jump operators defined on $\bm{x}$

and such that, if $f_{0},f_{1},\dots$ are in $\mathcal{J}_{\bm{x}}$, then $f_{0}\oplus f_{1}$, $\bigoplus_{n}f_{n}$ and $(i,j)\mathbin{{}^{s}\odot_{T}}f_{0}$ are in $\mathcal{J}_{\bm{x}}$, too, for all $(i,j)$ such that $(i,j)\mathbin{{}^{s}\odot_{T}}f_{0}(x)$ is defined for each $x\in\bm{x}$.555Of course, these operations are meant to be pointwise, e.g. $(f_{0}\oplus f_{1})(x)=f_{0}(x)\oplus f_{1}(x)$. Every function in $\mathcal{J}_{\bm{x}}$ is UTI. Is $\mathcal{J}_{\bm{x}}$ the set of all UTI functions from $\bm{x}$ to ${2^{\mathbb{N}}}$?