ON THE C.E. DEGREES REALIZABLE IN $\Pi ^0_1$ CLASSES

Abstract

We study for each computably bounded $\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect $\Pi ^0_1$ class where all c.e. members have degree a. We also show that every $\Pi ^0_1$ set of c.e. indices is realized in some perfect $\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some $\Pi ^0_1$ class as exactly those with a computable representation.

1 Introduction

This paper is concerned with computably bounded $\Pi ^0_1$ classes. Of course we can consider these classes, up to Turing degree, as being a collection of infinite paths through a computable binary tree. They have deep connections with computability theory in general, as well as reverse mathematics, algorithmic randomness, and many other areas. For example, see [3].

The meta-question we want to address in this paper concerns realizing c.e. degrees as members of $\Pi ^0_1$ classes. One of the fundamental theorems in this area [8, Theorem 3] says that each computably bounded $\Pi ^0_1$ class has a member of computably enumerable degree. That is, if $\alpha $ is the left- or right-most path of a $\Pi ^0_1$ class P, then there is a c.e. set W such that $W\equiv _T \alpha $ .

Definition 1.1. We will say that a c.e. degree ${\mathbf w}$ is realized in a $\Pi ^0_1$ class P iff there exists some $\beta \in P$ with $deg_T(\beta )={\mathbf w}$ . A set of c.e. degrees S is realised in P if for every c.e. degree $\mathbf {w}$ , $\mathbf {w}\in S$ if and only if $\mathbf {w}$ is realised in P.

Our fundamental question is “What sets of c.e. degrees can be realized in a $\Pi ^0_1$ class?” In this paper, we will give a characterization of the sets of c.e. degrees that can be realized. In turn, this also leads to the related question of representing index sets, and it turns out that these two notions coincide. In more detail, recall that a set $I\subseteq \mathbb {N}$ is an index set if it is closed under Turing equivalence, that is, for every $W_e\equiv _T W_j$ , $e\in I \Leftrightarrow j\in I$ . Index sets are another central area of classical computability theory, and many early results calculated the complexity of such sets. For example, Cof $=\{e\mid W_e $ is cofinite $\}$ is a well known $\Sigma _3^0 m$ -complete index set. But our concern here is the corresponding set of degrees in an index set, and how we might represent them most simply. For example, $\{e\mid W_e\equiv _T \emptyset '\}$ is a well-known $\Sigma _4^0$ -complete index set, but we can represent it by a single index for $\emptyset '$ , if we allow closure under Turing degrees. With this in mind, define for $S\subseteq \mathbb {N}$ the index set that S represents, $G(S)=\{e: (\exists j\in S)~W_e\equiv _T W_j\}$ . Note that if S is $\Sigma ^0_4$ then $G(S)$ is also $\Sigma ^0_4$ , while the complexity of $G(S)$ generally cannot be reduced even if S is of lower arithmetical complexity. For instance $G(\{\emptyset '\})$ is $\Sigma ^0_4$ -complete, as we have seen, so that sometimes the complexity of S can be much simpler than that of $G(S)$ . Thus, our main task in this paper is also to investigate, for a given index set I, what the minimal complexity of S such that $G(S)=I$ can be.

In the same way that we did for degrees, we will say that a c.e. set W is realizable in a $\Pi ^0_1$ class P, if $deg_T(W)$ is realizable in P. Henceforth in this paper, all $\Pi ^0_1$ classes are effectively closed subsets of the Cantor space.

Definition 1.2. If P is a $\Pi ^0_1$ class, let $W[P]=\{e:W_e$ is realizable in $P\}$ .

For any $\Pi ^0_1$ class P, $W[P]$ is clearly an index set. We let $\boldsymbol {W}[P]=\{\boldsymbol {a}:\boldsymbol {a}$ is c.e. and realizable in $P\}$ . The obvious upper bound on the complexity of $W[P]$ can be easily calculated.

Proposition 1.3. If P is a $\Pi ^0_1$ class, then $W[P]$ is $\Sigma ^0_4$ .

Proof

$$ \begin{align*} e\in W[P]\Leftrightarrow (\exists j\exists k)(\forall n)(\exists m)(\exists s>n)~\Big[\Phi_j^{W_e}\upharpoonright{m}[s]\downarrow~\wedge~W_e[s]\upharpoonright{\varphi_j(m)}\text{ is correct}\\ ~\wedge~ \Phi_j^{W_e}\upharpoonright{m}[s]\text{ is extendible in }P[s]~\wedge~ \Phi_k^{\Phi_j^{W_e} \upharpoonright{m}[s]}\upharpoonright{n}=W_e\upharpoonright{n}\Big]. \end{align*} $$

Note that the predicate within the square brackets is $\Delta ^0_2$ .

Note that this upper bound is sharp, in the sense that $W[P]$ is $\Sigma ^0_4$ -complete for some P. For instance if we let P be a class of Martin-Löf random reals, or P be the class of all $PA$ degrees. Recall that every c.e. Martin-Löf random or PA degree must have degree ${\mathbf 0'}$ (see, for example, [5, Sections 2.21 and 8.2]).

With these broad questions in mind, we now state some specific questions that shall be addressed in this paper:

1. (1) Does Proposition 1.3 reverse, that is, is every $\Sigma ^0_4$ index set realizable in a $\Pi ^0_1$ class?

2. (2) If not, can we characterize the index sets S which can be realized?

3. (3) Is every upper and every lower cone of c.e. degrees realizable?

4. (4) What about the c.e. degrees realizable in $\Pi ^0_1$ classes of restricted rank?

We remark that our results go hand in hand with the analogous index set questions about representability. We say that a set S of indices represents an index set I if $I=G(S)$ . We shall also address the analogous questions about index sets:

1. (1) Can every $\Sigma ^0_4$ index set be represented by a $\Sigma _3^0$ set or lower?

2. (2) If not, can we characterize the index sets I which can be represented by $\Sigma _3^0$ sets, or even by computable sets?

3. (3) Is every upper and every lower cone of c.e. degrees representable by a set of lower complexity?

The only related results we are aware of are early results about representing index sets and particularly those corresponding to ideals in the Turing degrees. For example, Yates [11, 12] showed that if $D<_T C$ are c.e. and if S is $\Sigma _3^C$ , then there is a computable collection of c.e. sets $\{W_{f(k)}\mid k\in {\mathbb N}\}$ such that $D\le _T W_{f(k)}\le _T C$ for all k, and where $e\in S$ is equivalent to $W_{f(e)}\equiv _T C$ . This was used for an alternative proof of Sacks Density Theorem (see, e.g., [10, Chapter XII]). Yates also showed that a collection of c.e. sets ${\mathcal C}$ containing all finite sets is $\Sigma _3^0$ iff there is a computable collection $\{W_{f(d)}\mid d\in {\mathbb N}\}$ which equals ${\mathcal C}$ . He used this to show the classical result that the index of the lower cone $\{e\mid W_e\le _T A\}$ is $\Sigma _3^0$ iff A is Turing complete or low $_2$ . More recently, Barmpalias and Nies [1] proved that for an ideal I in the c.e. degrees:

1. (1) If I is uniformly superlow generated ¹ then it has a superlow upper bound in the c.e. degrees.

2. (2) If I is uniformly low generated then it has a low c.e. upper bound.

3. (3) If I is a $\Sigma _3^0$ generated proper ideal, then it has a low $_2$ c.e. upper bound (see also [5, Chapter 11.11]).

4. (4) If I is a $\Sigma _4^0$ generated proper ideal, then it has an incomplete upper bound.

Our paper sharpens some of these results. For example in Theorem 4.6 we show that if I is generated by a $\Sigma _4^0$ subset S of a Turing independent set of c.e. sets $\{A_i\mid i\in {\mathbb N}\}$ , (i.e., for all finite sets F if $i\not \in F$ , then $A_i\not \le _T \oplus _{j\in F} A_j$ ), then there is a c.e. set B selecting these sets, in that $A_i\le _T B$ iff $i\in S$ .

Jockusch and Soare [8] showed that for any c.e. degree $\boldsymbol {a}$ , the set $\{\boldsymbol {b}: \boldsymbol {b}$ is c.e. and $\boldsymbol {b}\geq \boldsymbol {a}\}$ can be realised in a $\Pi ^0_1$ class. Our first result, Theorem 2.1, shows that singletons can be realized in $\Pi ^0_1$ classes.

Using this we show in Theorem 3.1 that any set of indices which is representable by a $\Sigma _3^0$ set, is realized in a $\Pi ^0_1$ class. Moreover if the set contains an index for a computable set, we can realize this in a rank 2 $\Pi ^0_1$ class. This allows us to show that certain collections of high c.e. sets can be realized, under a suitable assumption about the uniformity of highness. Moreover, in Corollary 4.3 we establish the rather surprising result, at least to us, below:

Corollary 4.3. An index set I is realizable in a $\Pi ^0_1$ class iff I has a $\Sigma _3^0$ representation iff I has a computable representation.

Next we give an answer to the natural question of exactly which index sets have $\Sigma _3^0$ (or equivalently, computable) representations.

Theorem 4.1. Let $S\subseteq \omega $ . The following are equivalent.

1. (i) S has a computable representation, that is, $G(S)=G(R)$ for some computable set R.

2. (ii) There is a $\Pi ^0_1$ class P such that the set of c.e. degrees represented in P has index set $G(S)$ , that is, $W[P]=G(S)$ .

3. (iii) There is a perfect $\Pi ^0_1$ class P such that $W[P]=G(S)$ .

4. (iv) There is a computable function g such that for every n,

   $$\begin{align*}n\in G(S) \Leftrightarrow W_{g(n)}^{W_n} \text{ is cofinite}.\end{align*}$$

5. (v) There is a truth-table functional R such that for every n,

   $$\begin{align*}n\in G(S)\Leftrightarrow \exists a\forall b\exists c ~R^{W_n}(n,a,b,c).\end{align*}$$

Clearly this result allows us to show certain index sets are not realizable, and others such as, for example, the K-trivial c.e. sets are, as are all upper cones. Finally we turn to the question of exactly which lower cones can be realized. Although we don’t characterize this, as mentioned above, we show how to code $\Sigma _4^0$ sets into lower cones. In an early incarnation of the present paper, we conjectured that if A is a incomplete c.e. set and the lower cone below A can be realized, then A is low $_2$ . Inspired by our work, Downey and Melnikov have resolved this conjecture affirmatively [7].

Finally, we suggest investigating the question of which sets of c.e. degrees are realised in a restricted collection of $\Pi ^0_1$ classes, for instance, in the collection of thin $\Pi ^0_1$ classes, or in the collection of separating classes. For instance, the class of PA degrees is a separating $\Pi ^0_1$ class, and hence $\{\mathbf {0}'\}$ is realised by a separating class. On the other hand, not every singleton can be realised in a separating class. To see this, suppose that the separating class for the pair $A,B$ is realised by a single array computable c.e. degree $\mathbf {d}$ . Since A and B are both c.e. members of the separating class, we have that $A\in \mathbf {d}$ and $B\in \mathbf {d}$ . Downey, Jockusch, and Stob [6] proved that if $A\oplus B$ has array computable degree, and if $A,B$ are disjoint c.e. sets, then there is a Turing complete set separating A and B, a contradiction. In [4], Cholak, Downey, Greenberg, and Turetsky have shown that some uppercones can be realised, and some pairs, but the classification of what is realizable remains complex and mysterious.

2 Every c.e. singleton is realizable in a perfect $\Pi ^0_1$ class

The natural direction to begin our investigation is to look at singletons. Clearly $\{\boldsymbol {0'}\}$ can be realized by a perfect $\Pi ^0_1$ class, for instance, the class of Martin-Löf random reals, or the class of all PA degrees. Our first result is that $\{\boldsymbol {a}\}$ can be realized by a perfect $\Pi ^0_1$ class for any c.e. degree $\boldsymbol {a}$ .

Theorem 2.1. For any c.e. degree $\boldsymbol {a}$ , we can find (effectively in an index of a c.e. member of $\boldsymbol {a}$ ) a perfect $\Pi ^0_1$ class P such that $\boldsymbol {W}[P]=\{\boldsymbol {a}\}$ .

Proof Let $C\in \boldsymbol {a}$ be c.e., and fix a 1-1 enumeration $\{C_s\}_{s\in \mathbb {N}}$ of C. We build a perfect $\Pi ^0_1$ class P such that $\boldsymbol {W}[P]=\{\boldsymbol {a}\}$ using a stage by stage construction and satisfying requirements.

Before stating the requirements, we clarify our notational conventions. Recall that a function f is a modulus for the computable approximation $Z(x,s)$ if for every x and every $t>f(x)$ , $Z(x,t)=Z(x,f(x))$ . If f is nondecreasing then this means that $Z\upharpoonright {x+1}$ is stable at $f(x)$ . We let $Z_e(x,s)$ be the $e^{th}$ possible $\Delta ^0_2$ approximation in some effective listing. Each $Z_e$ is total computable with range in $\{0,1\}$ , where $Z_e(x)$ may not exist. Every $\Delta ^0_2$ set is approximated by some $Z_e$ . As usual $\Phi _e$ is the $e^{th}$ Turing functional.

We will construct P to meet the requirements

$$ \begin{align*} \mathcal{R}_e: \text{If }Z_e(x):=\lim_s Z_e(x,s)\text{ exists for every }x,\text{ and }Z_e\in P\text{ and }\Phi_e^{Z_e}\text{ is total and} \\\text{is a modulus for the approximation }Z_e(x,s),\text{ then }Z_e\equiv_T C. \end{align*} $$

Since we are only interested in those $\Phi _e$ which are modulus functions, we may assume that for any string $\nu $ and $y<x$ , if $\Phi _e^\nu (x)\downarrow $ then $\Phi _e^\nu (y)\downarrow $ and $\Phi _e^X$ is nondecreasing on its domain for every X. When we refer to $\mathcal {R}_e$ we actually mean $Z_i$ and $\Phi _j$ where $e=\langle i,j\rangle $ . To avoid introducing too many notations we will write $Z_e$ and $\Phi _e$ instead.

If Z is in P and is of c.e. degree, then the premise of $\mathcal {R}_e$ will hold for some e. We will construct P to be the limit of total function trees (see [10]). Namely, we let $T_s:2^{<\omega }\mapsto 2^{<\omega }$ be a uniform sequence of total computable functions, such that:

• • For every s, $T_s$ preserves incomparability and inclusion.

• • For every s and $\sigma $ , there is a $\tau $ such that $T_{s+1}(\sigma )=T_s(\tau )$ .

• • For every $\sigma $ , $\lim _s T_s(\sigma )$ exists.

• • For every $\sigma $ and $i=0,1$ , $T_s(\sigma * i)\supseteq T_s(\sigma )* i$ .

The fourth condition is new and says that $T_s$ has to split at the first place possible. This is not necessary but we adopt this to simplify notation. As before $P=[\lim _s T_s]$ is a perfect $\Pi ^0_1$ class.

2.1 Notations

The construction is carried out on a 2-branching construction tree. We refer to the finite strings on the construction tree as nodes, which we denote by letters early in the Greek alphabet such as $\alpha ,\beta ,\gamma ,$ etc. We refer to the finite strings in the domain and range of T as strings, and we use letters later in the Greek alphabet to denote them, such as $\sigma ,\tau ,\nu ,$ etc. Nodes of length e are assigned requirement $\mathcal {R}_e$ , and this has two outcomes, $\infty $ to the left of f. We write $\alpha <_L\beta $ if $\alpha $ is strictly left of $\beta $ . We say that $\alpha $ is of higher priority than $\beta $ if $\alpha \subset \beta $ or $\alpha <_L\beta $ holds. We say that $\alpha $ is stronger than $\beta $ if $\alpha <_L\beta $ , $\alpha \supseteq \beta *\infty $ or $\alpha * f\subseteq \beta $ holds. $\beta $ will not be allowed to injure a stronger node $\alpha $ , even though traditionally $\alpha $ may be of lower priority.

To show that $Z_\alpha \equiv _T C$ we will not build the reductions explicitly. Instead each $\alpha $ maintains two parameters $\sigma _\alpha $ and $root_\alpha $ to help guide this reduction. The intention is that $\sigma _\alpha $ is the string such that $T_s(\sigma _\alpha )$ is currently an initial segment of $Z_\alpha $ which $\alpha $ believes is a true initial segment. We always keep $root_\alpha \subseteq \sigma _\alpha $ if both are defined, and that $root_\alpha \downarrow $ iff $\sigma _\alpha \downarrow $ . For each $k\leq |\sigma _\alpha |-|root_\alpha |$ , we let $root_\alpha ^k=\sigma _\alpha \upharpoonright {(|root_\alpha |+k)}$ , the k-bit extension of $root_\alpha $ along $\sigma _\alpha $ . C will be coded into $Z_e$ above $root_\alpha $ . In particular, we code $C(k)$ in $T_s(root_\alpha ^{k+2})$ , and if k enters C we force $Z_\alpha $ to change below $T(root_\alpha ^{k+2})$ .

For $\alpha ,\sigma \in 2^{<\omega }$ we say that $\sigma $ is $\alpha $ -good if whenever $\sigma _\alpha \downarrow $ , we have $\sigma \not \subseteq \sigma _\alpha $ . The three possibilities are:

• • $root_\alpha $ and $\sigma $ are incomparable,

• • $root_\alpha \subseteq (\sigma \cap ~\sigma _\alpha )\subset \sigma _\alpha $ , or

• • $\sigma \supset \sigma _\alpha $ .

Note that any extension of a good string is good. By $\sigma ^-$ we mean the string $\sigma $ with the last bit removed, and is defined to be the empty string if $|\sigma |=0$ . We divide stage s into substages $u<s$ , in which a different node gets to act at each substage. If a node $\alpha $ is visited at (sub)stage s, we let $s^-$ denote the (sub)stage of the previous visit to $\alpha $ (it will be clear from context which $\alpha $ we are referring to). If $\alpha $ is visited at a stage s, and $\beta $ is a node such that $\beta * f\subseteq \alpha $ , we let $\ell _\alpha (\beta )=\max \{|\tau |:\tau \subseteq \sigma _\beta ~\wedge ~T(\tau )\subset Z_\alpha \}$ . This measures how much of $T(\sigma _\beta )$ is currently an initial segment of $Z_\alpha $ . If $\sigma _\beta \uparrow $ then this is taken to be $0$ . We let $\ell _\alpha =\max \{\ell _\alpha (\beta ):\beta * f\subseteq \alpha \}$ . A (sub)stage t is $\alpha $ -expansionary, if $\alpha $ is visited at t, and $\ell _\alpha [t]>\max \{\ell _\alpha [t']:t'<t $ where $\alpha *\infty $ is visited at $t'\}$ . For any $\alpha ,x$ and numbers $s<t$ , we say that $Z_\alpha [s]\upharpoonright {x}\equiv Z_\alpha [t]\upharpoonright {x}$ if $Z_\alpha [s']\upharpoonright {x}$ is constant for all $s\leq s'\leq t$ . If $\rho \supset \eta $ by moving $\eta $ to $\rho $ , we mean to obtain the modification $T'$ of T by letting $T'$ copy T at all inputs not extending $\eta $ . Then let $T'(\eta *\sigma )=T(\rho *\sigma )$ for every $\sigma \in 2^{<\omega }$ .

2.2 Proof idea

We assume that $C=\emptyset $ , and consider the general case later. We consider $\mathcal {R}_0$ with the highest priority. The basic strategy for $\mathcal {R}_0$ is simple, in fact every $\mathcal {R}_e$ follows roughly the same basic strategy with minor modifications: We begin by letting $\sigma _0=\emptyset $ , and search for some $\rho \supset {\sigma _0}* i$ such that $Z_0\supset T(\rho )$ and $\Phi _0^{T(\rho )}$ looks like a modulus function for $Z_0$ . That is, at the current stage approximation, $\Phi ^{T(\rho )}_0(|T(\sigma * i)|) \downarrow $ and there is no contradiction observed for it to be a correct modulus for $Z_0 \upharpoonright (|T(\sigma * i)|+1)$ . While searching for $\rho $ , $\mathcal {R}_0$ will prevent the rest of the construction from changing T on $\sigma _0$ . If $\rho $ is found we keep the split at $\sigma _0$ and kill off all other branches of T between $T({\sigma _0}* i)$ and $T(\rho )$ by moving ${\sigma _0}* i$ to $\rho $ . We then append i to $\sigma _0$ , define $X_0\upharpoonright {|T(\sigma _0)|}=Z_0\upharpoonright {|T(\sigma _0)|}$ . We then repeat. If ever during this process the approximation to $Z_0 \upharpoonright |T(\sigma _0)|$ changes after we have already defined $X_0 \upharpoonright |T(\sigma _0)|$ , then we shift $\emptyset $ to $\sigma _0$ and consider $\mathcal {R}_0$ satisfied, with no further action required (since we have assured that the modulus is wrong). If this never happens but still $\sigma _0$ is extended finitely often then $\mathcal {R}_0$ only restraints a finite part of T, and $\mathcal {R}_0$ is satisfied vacuously. If $Z_0$ is a path on P with modulus $\Phi _0^Z$ , then we be able to extend $\sigma _0$ infinitely often and so $X_0$ is total and equal to $Z_0$ . Thus $Z_0$ is computable since $X_0$ is.

From the above description we see that $\mathcal {R}_0$ either acts finitely often (and only restrains a finite part of T), or it acts infinitely often and it restrains other requirements from moving strings extended by $\sigma _0$ . Note that it is important for $\mathcal {R}_0$ to restrain the rest of the construction in this way because if we allow other requirements to delete $\tau =T(\sigma _0)$ after it has been recorded by $X_0$ , then $Z_0$ is free to change and become different from $X_0$ . Unlike before we cannot force $\mathcal {R}_0$ to lose by making every path in P extend $\tau $ (since $\tau $ has already been deleted).

We now consider how $\mathcal {R}_1$ can act consistently with $\mathcal {R}_0$ ’s demands. If $\mathcal {R}_0$ acts finitely often then $\mathcal {R}_1$ begins a new version after $\mathcal {R}_0$ ’s final action, which will never be interrupted. Suppose $\mathcal {R}_0$ acts infinitely often. Then we know that $X_0=Z_0=T(\lim \sigma _0)\in P$ , and $\mathcal {R}_0$ will restrain more of $Z_0$ as the construction proceeds. In this case $\mathcal {R}_1$ will only begin running its basic strategy if it is able to define $\sigma _1$ so that $Z_1\supset T(\sigma _1)$ and is incomparable with $T(\sigma _0)$ . Note that if $\mathcal {R}_1$ is unable to do this then $Z_1=Z_0$ which we know is computable through the actions of $\mathcal {R}_0$ . On the other hand suppose $\mathcal {R}_1$ starts its basic strategy, and begins building $X_1$ incomparable with $X_0$ . Suppose $Z_1$ then moves at a stage t below a length for which we have defined $X_1$ . As in the basic strategy we need to kill $\mathcal {R}_1$ . This time due to priority we kill off every part of the tree except for $T(\sigma _0)[t]$ and $T(\sigma _1)[t]$ , and all other nodes above which $\mathcal {R}_1$ has been killed. Unlike the case for $\mathcal {R}_0$ this is not an immediate win for $\mathcal {R}_1$ , even though $\mathcal {R}_1$ will no longer be active above $T(\sigma _1)[t]$ we may have $Z_1\supset T(\sigma _0)[t]$ . Hence we may need to restart $\mathcal {R}_1$ at a different part of the tree. If $\mathcal {R}_1$ is killed and restarted infinitely often then $Z_1=Z_0$ which is computable through the actions of $\mathcal {R}_0$ . Otherwise $\mathcal {R}_1$ begins building $X_1$ incomparable with $X_0$ without interruption, and hence $Z_1=X_1$ is computable.

If we now consider the third requirement $\mathcal {R}_2$ , then the strategy for $\mathcal {R}_2$ will have the obvious modifications. $\mathcal {R}_2$ will build $X_2$ incomparable with $X_0$ and $X_1$ . Each time we kill off $\mathcal {R}_2$ we have to leave $T(\sigma _0)$ and $T(\sigma _1)$ on the tree. If $\mathcal {R}_2$ is killed and restarted infinitely often then $Z_2$ is equal to either $Z_0$ or $Z_1$ and hence computable. Considering three requirements add a certain complexity to the situation: It is now possible for $\mathcal {R}_1$ to be killed and restarted infinitely often (hence $R_1=R_0$ ) and this can affect $\mathcal {R}_2$ in the following way. For instance, $X_1\cap X_0= X_2\cap X_0$ may hold, and when $\mathcal {R}_1$ is killed we need to initialize $\mathcal {R}_2$ since $X_2$ is taken off the tree and hence $Z_2$ can now change freely. If this happens infinitely often then we also have $Z_2=Z_0$ and hence $Z_2$ is computable. Otherwise $Z_2$ will extend $T(\sigma )$ for some $\sigma $ where $\mathcal {R}_1$ is killed. Then $Z_2$ will begin building $X_2$ in this part without interference.

The construction will be implemented on a priority tree. This is not necessary, but will make organization easier. Outcome f of a node $\alpha $ stands for the situation where $\alpha $ is killed finitely often, while outcome $\infty $ means that $\alpha $ is killed infinitely often. This proof has the unusual feature where a node $\alpha $ may initialize $\gamma \supseteq \alpha * f$ infinitely often, even when $\alpha $ has stopped playing outcome $\infty $ . Hence a node $\beta $ on the true path of the construction may be injured infinitely often by some $\alpha * f\subseteq \beta $ . We ensure that if this is the case, then the requirement assigned to $\beta $ will be met through some node which is extended by $\beta $ , as in the discussion above. Each node $\alpha $ will respect the nodes $\beta $ extending $\alpha *\infty $ , even though $\beta $ is traditionally of lower priority. However $\beta $ will only get to act at those stages where $\alpha $ ’s current attempt has just been destroyed, and $\beta $ will only be allowed to act infinitely often if $\alpha $ is injured infinitely often. Hence the portion of the tree restrained by $\beta $ does not grow unless $\alpha $ is injured again (hence allowing $\beta $ to be visited).

Now we consider the case where C is arbitrary. We have to code C into $Z_e$ . We modify the strategy outlined above in the following way. A node $\alpha $ will now look for an appropriate string $root_\alpha $ so that $T(root_\alpha )$ is incomparable with all strings which are restrained by stronger requirements. Once found $\alpha $ will begin a new attempt at demonstrating that $C\equiv _T Z_\alpha $ where the coding locations are above $T(root_\alpha )$ . Initially we start off with $root_\alpha =\sigma _\alpha $ . Each time $\alpha $ extends $\sigma _\alpha $ we set a new coding location for C. If n enters C for a small k, then we have to force $Z_\alpha \upharpoonright {T(root^{k+2}_\alpha )}$ to move. We do this by removing the branch $T(root^{k+2}_\alpha )$ from the tree. Hence $Z_\alpha $ has to move or else it stays out of P. To see that $Z_\alpha $ is computable in C, we note that as above, $Z_\alpha $ cannot change unless C changes and we kill off $T(root^{k}_\alpha )$ for some k. If $Z_\alpha $ changes without a corresponding C change, then we have not yet removed $T(\sigma _\alpha )$ from the tree, so we can now force a win for $\alpha $ above the string $root_\alpha $ by killing off every string extending $T(root_\alpha )$ except for $T(\sigma _\alpha )$ . If $\alpha * f\subseteq \beta $ and if $Z_\alpha \neq Z_\beta $ then $\alpha $ will only initialize $\beta $ finitely often. In fact $\beta $ will be able to carry out its basic strategy with $T(root_\beta )\supset Z_\alpha \cap Z_\beta $ . On the other hand if $\alpha $ initializes $\beta $ infinitely often then $Z_\alpha =Z_\beta $ and $\alpha $ will be responsible for building the reduction $Z_\alpha \equiv _T C$ .

2.3 Formal construction

At stage $s=0$ we let $T_0$ be the identity function, and initialize $\alpha $ (i.e., set $\sigma _\alpha $ and $root_\alpha $ undefined) for every $\alpha $ . In the construction $\alpha $ may be initialized when another node of higher priority acts, or when $\alpha $ is visited and we see the approximation for $Z_\alpha $ move below $\sigma _\alpha $ . The former we call an external initialization.

All searches performed in stage s are limited to numbers and strings of length $<s$ . At stage $s>0$ suppose $T=T_{s-1}$ is given. We define $\delta _s$ , the approximation to the true path of length s. At substage $u<s$ we define $\delta _s(u)$ . Whenever a parameter is referenced we mean the value of the parameter at the instance (or substage) they are mentioned. Since the construction takes place on a priority tree, to make housekeeping easier we will assume that when each node $\alpha $ is visited for the kth time in the construction, it only looks at $Z_\alpha (x,k)$ . That is, during the construction and verification, when we refer to $Z_\alpha [t]$ and $\Phi ^{Z_\alpha }[t]$ , we mean the values when both are evaluated at k.

Suppose we have defined $\delta _s\upharpoonright {u}$ for some $u<s$ . Let $\alpha =\delta _s\upharpoonright {u}$ . If $\alpha $ was initialized externally since the last visit to $\alpha $ , or if $\sigma _\alpha \uparrow $ and s is $\alpha $ -expansionary, we let $\delta _s(u)=\infty $ , initialize every node to the right of $\alpha *\infty $ and end the substage. Otherwise there are two cases:

1. (1) $\sigma _\alpha \uparrow $ : Find a string $\eta $ where $|\eta |>\max \{|\alpha |,s_0\}$ such that $\eta $ is $\beta $ -good for every $\beta $ stronger than $\alpha $ . Here $s_0<s$ is the largest stage such that $\alpha $ was initialized externally. We also assume that $s_0$ is at least as large as all historical $|root_\alpha |$ . We require that there is some $\rho \supset \eta $ such that $\Phi _\alpha ^{T(\rho )}(|T(\eta )|)\leq s$ , and $Z_\alpha [s]\supset T(\rho )$ . Furthermore we need $\eta $ to be correct with respect to $\rho $ : i.e., for every $x\leq |T(\eta )|$ , we have $Z_\alpha [\Phi _\alpha ^{T(\rho )}(x)]\upharpoonright {x}\equiv Z_\alpha [s]\upharpoonright {x}$ . If $\eta $ is found (we always pick $\eta $ of minimal length), we move $\eta $ to $\rho $ . Let $\sigma _\alpha =root_\alpha =\eta $ .

2. (2) $\sigma _\alpha \downarrow $ : There are three subcases. Pick the first subcase that applies, and perform the actions described there.

   1. (a) Killing: Suppose that $Z_\alpha [s]\nsupseteq T(\sigma _\alpha )\upharpoonright {(|T(\sigma _\alpha ^-)|+1})$ . Then $Z_\alpha $ has moved below a length which $T(\sigma _\alpha )$ had promised would never change. Now note that $Z_\alpha $ can never extend $T(\sigma _\alpha )$ correctly again, and we can now thin the tree accordingly: Move $root_\alpha $ to $\sigma _\alpha $ , and initialize $\alpha $ .

   2. (b) Coding C: If there is some $k\leq |\sigma _\alpha |-|root_\alpha |-2$ such that $k\in C_{s}-C_{s^-}$ , we move $root_\alpha ^{k+1}$ to ${(root_\alpha ^{k+1})}* d$ where $d=1-\sigma _\alpha (|root_\alpha ^{k+1}|)$ . That is, we move $root_\alpha ^{k+1}$ to its immediate extension which is not along $\sigma _\alpha $ . This ensures that the old value of $T(root_\alpha ^{k+2})$ is removed from the tree. Now trim $\sigma _\alpha $ to have length $|root_\alpha ^{k}|$ .

   3. (c) Extending $\sigma _\alpha $ : There is some $i\in \{0,1\}$ and some $\rho \supset {\sigma _\alpha }* i$ such that $\Phi _\alpha ^{T(\rho )}(|T({\sigma _\alpha }* i)|)\leq s$ , $Z_\alpha [s]\supset T(\rho )$ and ${\sigma _\alpha }* i$ is correct with respect to $\rho $ . If i and $\rho $ are found, move ${\sigma _\alpha }* i$ to $\rho $ and extend $\sigma _\alpha $ by one digit (i.e., append i to $\sigma _\alpha $ ).

If none of the cases above apply let $\delta _s(u)=f$ and go to the next substage. Otherwise one of the above applies. Then we must have moved some node $\eta $ to some $\rho \supset \eta $ . For every node $\beta \supseteq \alpha * f$ such that $root_\beta $ is comparable with $\eta $ we initialize $\beta $ . We let $\delta _s(u)=f$ . Go to the next substage.

2.4 Verification

For each $\alpha $ , we write “step $\alpha .2$ (a)” to refer to step 2(a) of the construction during substage $|\alpha |$ of some stage, in which $\alpha $ is visited and given attention. Correspondingly we use $\alpha .1$ , $\alpha .2$ (b) and $\alpha .2$ (c) in the obvious way. We use s to refer to stages in the construction, and $t,u$ to refer to substages. If $t_1<t_2$ then $t_1$ is a substage before $t_2$ , though they are not necessarily both substages of the same stage. Unless otherwise specified, if P is a parameter then $P[t]$ refers to the value of P at the end of substage t. We begin by verifying several facts about the construction.

Lemma 2.2.

1. (a) At the end of every substage, for every $\alpha $ stronger than $\beta $ , if $\sigma _\beta \downarrow $ then $root_\beta $ is $\alpha $ -good.

2. (b) At the end of a substage t, suppose $\sigma _\alpha \downarrow $ and $\tau \subseteq \sigma _\alpha $ . Let $t'$ be the largest such that $t'\leq t$ where $|\sigma _\alpha [t'-1]|<|\tau |$ (hence at $t'$ we most recently assigned the value of $\sigma _\alpha $ to extend $\tau $ ). Then no prefix of $\tau $ is moved strictly between $t'$ and t.

3. (c) For any $\alpha $ once $T(root_\alpha )=\tau $ is killed under step $\alpha .2$ (a), it is impossible for $T(root_\alpha )\supseteq \tau $ again.

4. (d) $T_s$ satisfies all the conditions we required of a sequence of total function trees. In particular, each $\sigma $ is moved finitely often.

Proof (a): Note that if $\beta $ plays outcome $\infty $ at t or if we move left of $\beta $ , then $\sigma _\beta [t]\uparrow $ . Let $t'<t$ be the substage where $root_\beta $ is defined. At $t' root_\beta $ is $\alpha $ -good. Between $t'$ and t if $\alpha $ tries to make $\sigma _\alpha \supseteq root_\beta $ then that same action must initialize $\beta $ .

(b): Suppose the node $\beta $ moved some prefix of $\tau $ . Then $\alpha $ cannot be stronger than $\beta $ because of (a). Since $root_\alpha $ is comparable with the node being moved, hence if $\beta $ is stronger than $\alpha $ it is easy to see that this would cause $\alpha $ to be initialized. Finally suppose that $\alpha =\beta $ . Steps $\alpha .1$ , $\alpha .2$ (b), and $\alpha .2$ (c) are not possible since $|\sigma _\alpha |\geq |\tau |$ holds between $t'$ and t. Step $\alpha .2$ (a) is not possible because this will cause $\alpha $ to be initialized.

(c): Suppose for a contradiction that $\alpha .2$ (a) is taken at $t_1$ , and $T(root_\alpha )\supseteq \tau $ at some $t_2>t_1$ . Hence at $t_1$ we have $Z_\alpha [t_1]\nsupseteq T(\sigma _\alpha )\upharpoonright {(|T(\sigma _\alpha ^-)|+1})$ . Let $t_3<t_1$ be the largest substage where $\alpha .1$ or $\alpha .2$ (c) was taken. This means that at the end of $t_3$ , we have $\Phi _\alpha ^{T(\sigma _\alpha )}(|T(\sigma _\alpha ^-)|+1)\leq t_3$ , and $Z_\alpha [t_3]\supset T(\sigma _\alpha )=\tau $ . By part (b) no prefix of $\sigma _\alpha [t_3]=\sigma _\alpha [t_1-1]$ is moved between $t_3$ and $t_1$ . This means that $Z_\alpha $ has changed below the length $|T(\sigma ^-)[t_3]|+1$ between substages $t_3$ and $t_1$ . There must be some substage between $t_1$ and $t_2$ , call it $\hat {t}$ , for which $root_\alpha $ is defined such that (by applying part (b) again) $T(root_\alpha )[\hat {t}]\supseteq \tau $ under step $\alpha .1$ . This is however impossible given that $root_\alpha [\hat {t}]$ cannot be correctly defined at $\hat {t}>t_1$ .

(d): At every substage we modify the tree by moving an $\eta $ to an extension $\rho \supset \eta $ , so it is clear that $T_{s+1}(\sigma )$ is on $T_s$ . Suppose for a contradiction that $\sigma $ is a minimal node moved infinitely often. Since $|root_\alpha |>|\alpha |$ for every $\alpha $ , it follows that if $\sigma $ is moved infinitely often, there is a highest priority $\alpha $ which moves $\sigma $ infinitely often. Suppose $\alpha $ moves $\sigma $ under $\alpha .2$ (a) at t. Then $T(\sigma )[t]=\tau $ , and we may assume that no proper prefix of $\sigma $ is moved after t. Hence $T(\sigma )\supseteq \tau $ holds, and by part (c) $root_\alpha \not \supseteq \sigma $ after t.

If $\alpha $ moves $\sigma $ under $\alpha .2$ (b) then it must be because k has entered C for some $k<|\sigma |$ , so $\alpha .2$ (b) acts finitely often to move $\sigma $ . Suppose $\sigma $ is moved under $\alpha .2$ (c) at $t_1$ . Then the only way to move $\sigma $ under $\alpha .2$ (c) again at some $t_2>t_1$ is for $|\sigma _\alpha [t_2-1]|<|\sigma |=|\sigma _\alpha [t_1-1]|$ to hold. If $\alpha $ gets initialized externally between $t_1$ and $t_2$ then $root_\alpha $ will be later picked to have length larger than $t_1>|\sigma |$ and so $\alpha .2$ (c) cannot move $\sigma $ at $t_2$ . Therefore $\sigma _\alpha $ must get shorter due to some action of $\alpha $ . If $\alpha .2$ (a) applies between $t_1$ and $t_2$ then a prefix of $\sigma $ has to be moved, contradicting the minimality of $\sigma $ . If $\alpha .2$ (b) applies then it must be due to k entering $\emptyset '$ for some $k\leq |\sigma |$ which can only happen finitely often. Hence $|\sigma _\alpha [t_2]|<|\sigma |=|\sigma _\alpha [t_1]|$ cannot hold and so case $\alpha .2$ (c) can only move $\sigma $ finitely often.

Finally assume that $\sigma $ is moved under $\alpha .1$ infinitely often. This means that $\alpha $ has to be initialized between each such movement of $\sigma $ . Again this initialization cannot be external, otherwise $root_\alpha $ will be picked to have long length. It is possible for $\alpha .2$ (a) to initialize $\alpha $ , but by above this only happens finitely often.

Hence $P=[\lim _s T_s]$ is a perfect $\Pi ^0_1$ class. We let $TP$ denote the true path of the construction, defined in the usual way to be the leftmost path visited infinitely often during the construction.

Lemma 2.3. Suppose that $\alpha * f\subset TP$ . The following hold.

1. (a) $|root_\alpha |$ is bounded.

2. (b) Steps $\alpha .1$ and $\alpha .2$ (a) are taken finitely often.

3. (c) $\max \{\ell _\alpha [t]\mid \alpha $ is visited at $t\}<\infty $ .

4. (d) If the $\mathcal {R}_\alpha $ hypotheses fail then $\alpha $ acts finitely often.

Proof (a): Assume the contrary. Since $\alpha $ is only initialized externally finitely often, it is easy to see that at infinitely many substages t when we pick $root_\alpha $ , we have ${root_\alpha }^-$ is not $\beta $ -good for some stronger $\beta $ . This must be some $\beta * f\subseteq \alpha $ , since nodes to the left of $\alpha $ and extending $\alpha *\infty $ are never visited again. Hence we have $\ell _\alpha \geq |{root_\alpha }^-|$ at these t, which means that some large t has to be $\alpha $ -expansionary, and $\sigma _\alpha \uparrow $ , which means that $\alpha *\infty $ has to be played at t, a contradiction.

(b): By (a) and Lemma 2.2(d) it follows that $\alpha .1$ is taken finitely often. Consequently $\alpha .2$ (a) is taken finitely often.

(c): Suppose the contrary that $\lim \sup \ell _\alpha =\infty $ and $\alpha $ plays outcome $\infty $ finitely often. Let $s_0$ be large. Note that after $s_0$ , $\alpha $ cannot be initialized externally. By (b) we kill $\alpha $ under $\alpha .2$ (a) finitely often. Hence either $root_\alpha $ remains undefined forever, or it receives a final definition. In the former case we will eventually play outcome $\infty $ . In the latter case we have $\ell _\alpha>|root_\alpha |$ holds eventually at some visit to $\alpha $ . Hence $\sigma _\beta \supseteq root_\alpha $ for some $\beta * f\subseteq \alpha $ , otherwise we would kill $\alpha $ at that visit. Hence we have a contradiction to Lemma 2.2(a).

(d): Suppose the $\mathcal {R}_\alpha $ hypotheses fail. By (b) either $root_\alpha \uparrow $ at almost every stage or it receives a final value. If the former holds then we are done, so suppose $root_\alpha $ is stable with value $\sigma $ , and let $T(\sigma )$ be the final position of $T_s(\sigma )$ . Suppose for a contradiction that $\alpha $ acts infinitely often. Hence $\alpha .2$ (c) has to be taken infinitely often, otherwise $|\sigma _\alpha |$ is bounded and so $\alpha .2$ (b) is taken finitely often. Hence for each k, $root_\alpha ^k$ must receive a final value, call it $\sigma _k$ .

Suppose first of all that $Z_\alpha $ does not exist. Then there is some least i such that for each n, $Z_\alpha (i)[t]=n$ for infinitely many $\alpha $ -stages t. Since $|T(\sigma _i)|>i$ , then after $T(\sigma _{i+1})$ is stable $Z_\alpha $ has to change at position i, and we will kill $\alpha $ under $\alpha .2$ (a), contradicting the fact that $root_\alpha $ is stable. Next suppose that $Z_\alpha $ exists and is not in P. Then at almost every visit to $\alpha $ , $Z_\alpha [t]$ is not on $[T_t]$ . Consequently $\alpha .2$ (c) is taken finitely often. Now suppose that $Z_\alpha $ exists and is in P, but $\Phi _\alpha ^{Z_\alpha }(i)\uparrow $ . At substage t we will set $root_\alpha ^i$ to have its final value. At the end of t we must have $\Phi _\alpha ^{T(root_\alpha ^i)}(i)[t]\downarrow $ and $Z_\alpha [t]\supset T(root_\alpha ^i)$ . By assumption Z has to change below $T(root_\alpha ^i)$ , and $T(root_\alpha ^i)$ will retain its value until then. This means that when $\alpha $ is next visited it will be killed giving a contradiction. Finally suppose $\Phi _\alpha ^{Z_\alpha }$ is total but not a modulus of convergence. Hence for some i, $Z_\alpha \upharpoonright {i}$ has to change after $\Phi _\alpha ^{Z_\alpha }(i)$ . As above at the end of substage t we note that $root_\alpha ^i$ is correct. Hence Z has to change below $T(root_\alpha ^i)$ after t, because otherwise $\Phi _\alpha ^{Z_\alpha }(i)=\Phi _\alpha ^{T(root_\alpha ^i)[t]}(i)$ , contradicting the fact that $Z_\alpha \upharpoonright {i}$ has to change after $\Phi _\alpha ^{Z_\alpha }(i)$ . Hence at the next visit to $\alpha $ we will kill $\alpha $ , another contradiction.

Lemma 2.4. Let $\alpha $ be on $TP$ .

1. (a) If the $\mathcal {R}_\alpha $ hypotheses hold and $\alpha * f\subset TP$ then for each k, $root_\alpha ^k$ will be defined permanently and $T(root_\alpha ^k)\subset Z_\alpha $ .

2. (b) $\mathcal {R}_\alpha $ is satisfied.

Proof Fix $\alpha $ on $TP$ , and assume that (a) and (b) holds for every $\beta \subset \alpha $ . Suppose the $\mathcal {R}_\alpha $ hypotheses hold. Consider the case when $\alpha *\infty \subset TP$ . Since (a) holds trivially we show (b) holds for $\alpha $ . We have that $\alpha $ is either initialized externally infinitely often, or there are infinitely many $\alpha $ -expansionary stages. Suppose the former holds. Let $\beta $ be a node such that $\beta * f\subseteq \alpha $ and $\beta $ initializes $\alpha $ infinitely often. By Lemma 2.3(d) the $\mathcal {R}_\beta $ hypotheses must hold. If $Z_\alpha =Z_\beta $ then $Z_\alpha =Z_\beta \equiv _T C$ , and we are done. Otherwise suppose $Z_\alpha (i)\neq Z_\beta (i)$ for some i. Without loss of generality let $Z_\alpha \supset \rho * 0$ and $Z_\beta \supset \rho * 1$ . Hence at almost every visit to $\alpha $ we have $Z_\alpha [t]\supset \rho * 0$ and $Z_\beta [t]\supset \rho * 1$ . Take a sufficiently large t where $\beta $ initializes $\alpha $ . The actions of $\beta $ at t must move some string $\eta $ which is comparable with $root_\alpha $ . We may assume that $|\eta |>i+1$ and $|root_\alpha |>i+1$ . By Lemma 2.3(b) we may assume that this action is either $\beta .2$ (b) or $\beta .2$ (c). Since $|root_\alpha |>i+1$ we may assume that $T(root_\alpha )[t-1]\supset \rho * 0$ . Since $\eta $ is comparable with $root_\alpha $ we also have $T((\eta ^-)^-)[t-1]\supset \rho * 0$ . Since $Z_\beta [t]\supset \rho * 1$ this means that at t, $\beta $ must have chosen to kill instead, a contradiction.

Suppose instead there are infinitely many $\alpha $ -expansionary stages. Let $\beta * f\subseteq \alpha $ be such that $\ell _\alpha (\beta )[t]>\max \{\ell _\alpha (\beta )[t']:\alpha *\infty $ is visited at $t'<t\}$ for infinitely many t in which $\alpha *\infty $ is visited. Hence $\lim \sup \ell _\alpha (\beta )=\infty $ , and by Lemma 2.3(d) the $\mathcal {R}_\beta $ hypotheses must hold and $Z_\alpha =Z_\beta $ . Since $\mathcal {R}_\beta $ is satisfied we have $Z_\alpha =Z_\beta \equiv _T C$ .

Now we consider the case when $\alpha * f\subset TP$ . We first show (a) for $\alpha $ . Clearly $\alpha $ is only initialized finitely often. We first claim that $root_\alpha $ is defined permanently at some stage. Pick a sufficiently large substage t, and we claim that $root_\alpha $ will be defined at t (and hence retain this value permanently). The only thing preventing us from finding a suitable $\eta $ under $\alpha .1$ is the requirement of being $\beta $ -good. If $\beta $ is to the left of $\alpha $ or extends $\alpha *\infty $ , then $\beta $ acts finitely often and so $\sigma _\beta $ varies finitely and will not be a problem. If $\beta * f\subseteq \alpha $ and the $\mathcal {R}_\beta $ hypotheses fail then $\beta $ also acts finitely often. If the $\mathcal {R}_\beta $ hypotheses hold and $Z_\beta =Z_\alpha $ then by induction hypothesis (a) on $\beta $ , we have $\ell _\alpha \rightarrow \infty $ contradicting Lemma 2.3(c). On the other hand if $Z_\beta \neq Z_\alpha $ then $\sigma _\beta \not \subset Z_\alpha [t']$ holds at almost every $t'$ , and will not be a problem to defining $\eta $ . Hence we will be able to define $root_\alpha $ permanently. It is clear that $Z_\alpha \supset T(root_\alpha )$ , because otherwise $Z_\alpha \cap T(root_\alpha )\subseteq T({root_\alpha }^-)$ since $Z_\alpha \in P$ . Consequently we will kill $\alpha $ contradicting the fact that $root_\alpha $ is stable. It is then easy to verify (inductively on k) that for each $k>0$ , $root_\alpha ^k$ is defined permanently and $T(root_\alpha ^k)\subset Z_\alpha $ .

We now verify (b) for $\alpha $ , i.e., $C\equiv _T Z_\alpha $ . To compute $C(k)$ , find t large enough such that $T(root_\alpha ^{k+2})[t]\subset Z_\alpha [t]$ , and $Z_\alpha $ is correct up to $|T(root_\alpha ^{k+2})[t]|$ . If k enters C after t, then at the next visit to $\alpha $ we will remove the old value $T(root_\alpha ^{k+2})[t]$ from P, which contradicts $Z_\alpha \in P$ . Hence $k\in C$ iff $k\in C[t]$ . Now to compute $Z_\alpha \upharpoonright {i}$ from C, we look for t large enough such that $\alpha $ is visited, $T(root_\alpha ^{i+1})[t]\subset Z_\alpha [t]$ and $C[t]\upharpoonright {i+1}$ is stable. Since no coding of $C\upharpoonright {i+1}$ takes place after t, it follows that $root_\alpha ^{i+1}[t]$ is stable. By Lemma 2.2(b) $root_\alpha ^{i+1}[t]$ is never moved after t. Consequently $T(root_\alpha ^{i+1})[t]$ is stable, and since $|T(root_\alpha ^{i})|>i$ we conclude that $Z_\alpha [t]\upharpoonright {i}=Z_\alpha \upharpoonright {i}$ , because otherwise we would kill $\alpha $ after t.

This ends the proof of the theorem. We note that the construction of P from $\mathbf {a}$ is effective.

3 Realizable index sets

In this section we investigate which index sets can be realized. The proof of Theorem 2.1 shows that any computable sequence of c.e. degrees containing $\boldsymbol {0}$ can be realized in a perfect $\Pi ^0_1$ class; the uniformity of the proof of Theorem 2.1 allows us to produce a class $P_i$ realising each degree $\mathbf {a_i}$ in a computable set S, and we can realise $S\cup \{\mathbf {0}\}$ by taking the class $\{ 0^i*P_i\}_{i\in \omega }$ . We wish to improve on this. In Theorem 3.1, we show that generally any index set generated by a $\Sigma ^0_3$ set S can be realized, even if S does not contain an index for $\emptyset $ . If S contains an index for $\emptyset $ then (Corollary 3.4) we can also realize $G(S)$ in a rank $2$ class. Note that this is the best possible rank, since a rank $1 \Pi ^0_1$ class only realizes finitely many degrees. If S is not required to contain $\emptyset $ and is non-empty, then we can always realize $G(S)$ by a perfect $\Pi ^0_1$ class. We also give an example of a $\Pi ^0_3$ set S where $G(S)$ cannot be realized in any $\Pi ^0_1$ class (Theorem 3.5).

Theorem 3.1. For any non-empty $\Sigma ^0_3$ set S, there is a perfect $\Pi ^0_1$ class P such that $W[P]=G(S)$ .

Proof Let $V_{e,i}$ be a uniform sequence of c.e. sets such that $S=\{e:\exists i|V_{e,i}|=\infty \}$ . Fix an $e_0\in S$ , and let $T_{j_0}$ be the computable tree that gives $W[T_{j_0}]=G(\{e_0\})$ in Theorem 2.1. If $\sigma _0\subset \sigma _1\subset \cdots \subset \sigma _n$ is a sequence of finite strings, and f is a total computable function, we define the tree $Tree(f,\sigma _0,\ldots ,\sigma _n)$ to be the tree which is the result of copying $T_{f(n)}$ above ${\sigma _n}* 0,{\sigma _n}* 1$ and copying $T_{f(m)}$ above ${\sigma _m}* 1$ for each ${\sigma _m}* 0\subseteq \sigma _n$ . Formally take the downwards closure of the set of strings $\{{\sigma _n}* 0* T_{f(n)}\}\cup \{{\sigma _n}* 1* T_{f(n)}\}\cup \{{\sigma _m}* 1* T_{f(m)}:m<n$ and ${\sigma _m}* 0\subseteq \sigma _n\}$ . We can extend this to an infinite $\sigma $ -sequence $Tree(f,\sigma _0,\sigma _1,\ldots )$ in a natural way, by copying $T_{f(m)}$ above ${\sigma _m}* 1$ for each m such that ${\sigma _m}* 0\subseteq \sigma _{m+1}$ . The tree $Tree(f,\sigma _1,\ldots )$ might not be computable unless the sequence of $\sigma _m$ is computable.

In the following we fix $\langle f, e,i,\rangle $ , where $e,i\in \mathbb {N}$ and f is a total computable function. We describe the construction of a $\Pi ^0_1$ class $P_{f,e,i}$ (the relationship $\langle e,i,f\rangle \mapsto P_{f,e,i}$ is an effective one). For ease of notation we temporarily drop $f,e,i$ . We build P by defining a computable tree U, in stages. At stage s we define U up till level s. To make U computable we only enumerate strings of length s at stage s. We also define a sequence of finite strings $\sigma _{0,s}\subset \sigma _{1,s}\subset \cdots $ . The intention is that if $|V|=\infty $ then $\cup _n\lim _s\sigma _{n,s}$ will be a path in $P=[U]$ which is of the same Turing degree as $W_e$ , with all other paths in P having degrees of paths occurring in the $T_{f(n)}.$

If $|V|< \infty $ , we still make sure that P is perfect, now containing only paths whose degrees are realized in the $T_{f(n)}$ . We do this by aiming to make $P=[Tree(f,\lim \sigma _0,\lim \sigma _1,\ldots )]$ , by making $U=Tree(f,\lim \sigma _0,\lim \sigma _1,\ldots )$ except on the dead ends introduced by the coding of $W_e$ .

The plan: The idea is to break down the membership question $e\in S$ into infinitely many $\Pi ^0_2$ questions (about the cardinality of $V_{e,i}$ ). We approximate the truth of $e\in S$ locally by examining if $|V_{e,i}|=\infty $ , and building a class $P_{e,i}$ . We use $P_{e,i}$ to denote $P_{f,e,i}$ when f is simply the function $f(x)=j_0$ . At the end we combine the different $\Pi ^0_1$ classes $P_{e,i}$ together. Locally if $|V_{e,i}|=\infty $ , then we must make some $\alpha \in P_{e,i}$ such that $\alpha \equiv _T W_e$ . The standard way of doing this is to make $\alpha $ the complement of the retraceable $\Pi ^0_1$ set associated with $W_e$ ; we refer the reader to Cenzer, Downey, Jockusch, and Shore [2], where they showed that $\{\boldsymbol {w},\boldsymbol {0}\}$ is realizable for any $\boldsymbol {w}$ . The problem is that this gives a rank $1$ class in which some $\alpha \equiv _T W_e$ sits, and contains computable (in fact, isolated) paths. Inside $P_{e,i}$ we have to be careful not to introduce c.e. degrees which are not in $G(S)$ . The only reason why the isolated paths show up is because we have to allow a different path to be extendible in the tree; this is to allow for the later coding of some $n\in W_e$ . Observe that for the purpose of making $\alpha \equiv _T W_e$ , it would not matter if we keep many paths (associated with $n\in W_e$ ) extendible on the tree instead of only two. Hence if $\alpha _s$ is the current approximation of an initial segment of $\alpha $ , we would copy the tree $T_{j_0}$ above every node distinct from $\alpha _s$ which we currently want to keep extendible. If later on some m enters $W_e$ and we need to switch $\alpha _{s+1}$ to a different string $\sigma $ , then we simply take the leftmost extension of $\sigma $ of length s. On the other hand if $\sigma $ is such a string which is never used, then we will copy the entire tree $T_{j_0}$ above $\sigma $ . Doing so puts $deg(W_{e_0})$ into $W[P_{e,i}]$ —no other c.e. degree is introduced. Of course we only extend $\alpha _s$ if new numbers enter $V_{e,i}$ . If $|V_{e,i}|<\infty $ then we will copy $T_{j_0}$ above every terminal node.

Following the plan above gives effectively the class $P_{e,i}$ for each pair $e,i$ , which introduces no new c.e. degree and contains $deg(W_e)$ iff $|V_{e,i}|=\infty $ . Finally we want to glue the separate $P_{e,i}$ ’s into a single class P. A slight technical issue arises if we take the simple amalgamation $P=\{0^n 1* P_{n}:n\in \mathbb {N}\}$ ; even though none of the $P_n$ introduces new c.e. degrees, however, the support $X=\emptyset $ will be in P. To get around this problem we nest the construction of each $P_x$ within the outer construction of $P_{e_0}$ . This will create X in P with Turing degree $X\equiv _T W_{e_0}$ , which supports the trees associated with $P_0,P_1,\ldots $ .

Construction of U: At stage $1$ declare $\sigma _0=\langle \rangle $ and $\sigma _1,\sigma _2,\ldots $ undefined, and enumerate $\langle \rangle ,0,1$ into U. For a $\sigma \in U$ , when we say we copy the next level of $T_{j}$ above $\sigma $ at stage s we mean the following. For each $\sigma *\tau \in U$ such that $|\sigma *\tau |=s-1$ , we enumerate $\sigma *\tau * 0$ into U if $\tau * 0$ is in $T_{j}$ ; similarly for $\sigma *\tau * 1$ . Clearly if $\sigma \in U_{|\sigma |}$ and we issue this command for $\sigma $ at every stage after $|\sigma |$ , then $T_{j}$ will be copied successfully above $\sigma $ ; i.e., $U\cap \{\tau :\tau \supseteq \sigma \}=\sigma * T_{j}$ . A string is said to require copying at stage s, if it is either of the form ${\sigma _m}* 1$ for some convergent ${\sigma _m}* 0\subseteq \sigma _{m+1}$ , or it is ${\sigma _m}* 0$ or ${\sigma _m}* 1$ for the maximal convergent $\sigma _m$ . When we say we copy $T_f$ above such a $\sigma $ we mean that we copy the next level of $T_{f(m)}$ for the corresponding m. The actions at stage $s+1$ involve the following. Look for the largest n such that $\sigma _n\downarrow $ . See if there is some least $2m<n$ such that $m\in W_{e,s+1}-W_{e,s}$ :

1. (i) m exists. Let $\tau $ be the leftmost string of length s extending ${\sigma _{2m}}* 1$ , such that $\tau \in U$ . Move $\sigma _{2m+1}=\tau $ and put both $\tau * 0,\tau * 1$ in U. Declare $\sigma _{2m+2},\sigma _{m+3},\ldots $ undefined.

2. (ii) m does not exists. If $s+1\in V$ we let $\tau $ be the leftmost string of length s extending ${\sigma _n}* r$ which is in U. Here $r=0$ if n is odd, and $r=W_{e,s+1}(\frac {1}{2}n)$ if n is even. Define $\sigma _{n+1}=\tau $ and enumerate both $\tau * 0,\tau * 1$ into U, otherwise if $s+1\not \in V$ do nothing.

Copy $T_{f}$ above every string which needs copying, and go to the next stage. This ends the construction. It is easy to see that U is a computable tree.

Lemma 3.2. If $|V|<\infty $ then $[U]=[Tree(f,\tilde {\sigma }_0,\ldots ,\tilde {\sigma }_n)]$ for a finite collection of strings.

Proof Let n be the largest such that $\tilde {\sigma }_n=\lim _s \sigma _{n,s}$ exists. It is easy to check that $[U]\subseteq [Tree(f,\tilde {\sigma }_0,\ldots ,\tilde {\sigma }_n)]$ , since after a large enough stage the only thing we do in the construction is to copy $T_f$ above a string which requires copying. To see the reverse inclusion, fix an $m<n$ and consider the stage s where $\tilde {\sigma }_m$ is picked as $\sigma _{m,s}$ . Then $|\tilde {\sigma }_m|=s-1$ and we may assume that $(\tilde {\sigma }_m)* 0\subseteq \tilde {\sigma }_n$ . Since $(\tilde {\sigma }_m)* 1\in U_{s}$ it follows that we will successfully copy $T_{f(m)}$ above $(\tilde {\sigma }_m)* 1$ . A similar argument follows for $m=n$ .

Lemma 3.3. If $|V|=\infty $ and $[T_{f(m)}]\neq \emptyset $ for every m, then $\tilde {\sigma }_m=\lim _s\sigma _{m,s}$ exists for every m, and $\cup _m \tilde {\sigma }_m\equiv _T W_e$ . Furthermore $[U]=[Tree(f,\tilde {\sigma }_0,\tilde {\sigma }_1,\ldots )]$ , and $(\tilde {\sigma }_{2m+1})* 0\subseteq \tilde {\sigma }_{2m+2}$ for all m.

Proof It is straightforward to verify that the search for $\tau $ in steps (i) and (ii) of the construction is always successful, and consequently $\lim _s\sigma _{m,s}$ exists for every m. Observe that $X=\cup _m \tilde {\sigma }_m$ can compute the sequence $\tilde {\sigma }_0,\tilde {\sigma }_1,\ldots $ , and we have $m\in W_e\Leftrightarrow X(|\tilde {\sigma }_{2m}|)=1$ . To see that $X\leq _T W_e$ , fix n and compute a stage s large enough so that $W_{e,s}\upharpoonright {n+1}=W_e\upharpoonright {n+1}$ and all of $\sigma _{0,s},\ldots ,\sigma _{n,s}$ are defined. These must already be at their final values, and since $n<\sigma _{n,s}$ it follows that $n\in X$ iff $\sigma _{n,s}(n)=1$ . To prove the last statement, observe that $[U]\supseteq [Tree(f,\tilde {\sigma }_0,\tilde {\sigma }_1,\ldots )]$ follows from a similar argument as in Lemma 3.2. To see that $[U]\subseteq [Tree(f,\tilde {\sigma }_0,\tilde {\sigma }_1,\ldots )]$ , observe that any $Z\in [U]$ satisfies either $Z=X$ or else there is some m such that $Z\supset (\tilde {\sigma }_m)* 1$ and $X\supset (\tilde {\sigma }_m)* 0$ . Finally note that $(\tilde {\sigma }_{2m+1})* 0\subseteq \tilde {\sigma }_{2m+2}$ clearly holds for all m.

We are ready to show that there is a perfect $\Pi ^0_1$ class P such that $W[P]=G(S)$ . Let $f(x)=j_0$ for every x; we want to combine all the classes $P_{f,e,i}$ . Let g be a computable function defined by the following: $g(2x)=j_0$ and for the odd inputs we select an index such that $[T_{g(2\langle e,i\rangle +1)}]=P_{f,e,i}$ . From Lemmas 3.2 and 3.3, and the fact that $T_{j_0}$ is perfect, we may conclude that each $T_{g(x)}$ is perfect. Since $e_0\in S$ pick $i_0$ such that $|V_{e_0,i_0}|=\infty $ , and let $P=P_{g,e_0,i_0}$ . It is not hard to verify that P itself is also perfect, and $W[P]=G(S)$ .

Corollary 3.4. Given a $\Sigma ^0_3$ set S that contains an index for $\emptyset $ , there is a $\Pi ^0_1$ class P of rank $2$ such that $W[P]=G(S)$ .

Proof Suppose S contains an index for $\emptyset $ . We want to produce P of rank $2$ . We run the argument in the proof of Theorem 3.1, except now we let $T_{j_0}=\{0^m:m\in \mathbb {N}\}$ , and set $f(x)=j_0$ for all x. Each of the $P_{f,e,i}$ has rank at most $1$ —if $V_{e,i}$ is finite then so is $P_{f,e,i}$ and if $V_{e,i}$ is infinite then $P_{f,e,i}$ has rank $1$ . We let $P=\{0^{\langle e, i \rangle } 1* P_{f,e,i}:\langle e, i \rangle \in \mathbb {N}\}$ , which is of rank at most $2$ , and $W[P]=G(S)$ .

We now turn to the question of whether every $\Sigma ^0_4$ set can be realized. An obvious counter-example is the $\Pi ^0_3$ index set $\{e:W_e$ is not computable $\}$ . In fact no $\Pi ^0_3$ set which is downwards dense in the c.e. degrees (in the sense that for any c.e. $W>_T\emptyset $ there is $e\in S$ such that $W_e\leq _T W$ ), and containing no computable set can be realized, due to an old result of Jockusch and Soare [8]. In the following theorem we give another example of a $\Pi ^0_3$ set which cannot be realized, but which contains an index for $\emptyset $ .

Theorem 3.5. There is a $\Pi ^0_3$ set S containing an index for $\emptyset $ , such that there is no $\Pi ^0_1$ class P with $W[P]=G(S)$ .

Proof For any c.e. set $L>_T\emptyset $ , one can effectively obtain a c.e. set D and reduction $\Psi $ such that $D=\Psi ^L$ and $D>\emptyset $ and $D\not \geq _T L$ . If we iterate this process starting with some noncomputable low $_2$ c.e. set L, we get computable increasing functions g and $\ell $ such that $L=W_{g(0)}>_T W_{g(1)}>_T W_{g(2)}>_T\cdots $ , and $W_{g(n)}=\Phi _{\ell (n)}^L$ for all n. Let $T_e$ be the eth primitive recursive tree. Since the $W_{g(e)}$ are uniformly computable from L via $\ell $ , it follows that there are L-computable relations $R_1$ and $R_2$ such that

$$\begin{align*}R_1(e,i,x,s)&\Leftrightarrow \Phi_i^{W_{g(e)}}\upharpoonright{x}[s]\downarrow \text{and is on } T_e,\\R_2(e,i,j,x,m,s,\sigma)&\Leftrightarrow \Phi_i^{W_{g(e)}}\upharpoonright{m}[s]\downarrow=\sigma\textrm{ and }\Phi_j^\sigma\upharpoonright{x}[s]\downarrow=W_{g(e)}\upharpoonright{x}.\end{align*}$$

We define $n\not \in S$ iff whenever $n=g(e)$ , then there are some $i,j$ for which:

1. (i) $\Phi _i^{W_{g(e)}}$ is total and is in $[T_e]$ , and

2. (ii) $\Phi _j(\Phi _i^{W_{g(e)}})$ is total and equals $W_{g(e)}$ .

The quantifier for e is bounded since g is increasing. Clause (i) can be expressed as $\forall x\exists sR_1(e,i,x,s)$ . Since this is $\Pi ^0_2(L)$ , and L is low $_2$ it is also $\Sigma ^0_3$ . Similarly clause (ii) can be expressed as $\forall x\exists m\exists s\exists \sigma R_2(e,i,j,x,m,s,\sigma )$ which is also $\Sigma ^0_3$ . Hence S is $\Pi ^0_3$ , and it is easy to verify that there can be no $\Pi ^0_1$ class P such that $W[P]=G(S)$ . To see that S can be made to contain $\emptyset $ , note that $S\cup \tilde {S}$ is also good for any $\Pi ^0_3$ set $\tilde {S}$ where $G(\tilde {S})\cap G(W_{g(n)})=\emptyset $ for every n.

Note that S is clearly non-empty, since empty index sets are realized by the empty $\Pi ^0_1$ class. The above counterexample relies on the fact that for a low $_2$ set L, $\Pi ^0_2(L)=\Sigma ^0_3$ .

So far all the examples we have of a set S that can be realised are those with a $\Sigma ^0_3$ definition (such as singletons). We wish to show that some S can be realized which has no $\Sigma _3^0$ definition. The next result does this.

If A is a high set then by Martin [9], there is an A-computable function $\Gamma ^A$ which dominates every computable function, meaning that if f is computable, then for almost all x, $\Gamma ^A(x)>f(x).$ Thus, if A is high then some $\Gamma =\Phi _j$ satisfying this will exist. We will call such a j a domination index for A. As usual, the use of $\Gamma ^A(x)[s]$ is denoted by $\gamma (x,s)$ .

Theorem 3.6. Consider a computable collection $\mathcal {C}$ of pairs of the form $\langle e,j(e)\rangle ,$ where j is a computable function such that each $W_e$ has high degree, and $j(e)$ is a domination index for $W_e$ . Let S be any $\Sigma _4^0$ subset of $\mathcal {C}$ , and fix an index $e_0$ for $\emptyset $ . Then there is a rank 2 $\Pi _1^0$ class P with such that $W[P]=G\left (S\cup \{e_0\}\right )$ .

In particular, if S is $\Sigma ^0_4$ -complete, then $S\cup \{e_0\}$ is an example of a realizable set which is not $\Sigma ^0_3$ .

Proof For simplicity of notation, we will pretend that $\mathcal {C}=\omega $ and also write $\Phi _{j(e)}=\Gamma _e$ for the domination function for e.

In Theorem 3.1, we realized each $\Sigma _3^0$ set of e’s. Here we will be dealing with a $\Sigma _4^0$ set S. Thus, $e\in S$ iff $\exists p \forall m \exists s \forall t R(e,p,m,s,t)$ . For a fixed p, should the $\Pi _3^0$ condition $\forall m \exists s \forall t R(e,p,m,s,t)$ hold, we need to code $W_e$ .

Again, the idea will be to devote a unique part of the $\Pi _1^0$ class, to the pair $\langle e,p\rangle .$ These indices indicate that we are concerned with $W_e$ (with domination function $g_e^{W_e}),$ and p is the $\Sigma _4^0$ witness.

We will be making the class to be of rank 2. Concentrating on this fixed pair, $\langle e,p\rangle $ we are thinking of the part of the class being a rank 1 subclass extending some fixed $\sigma *1=\nu $ , say. The idea is again to more-or-less follow the coding of Cenzer, Downey, Jockusch, and Shore [2], where they showed that $\{\mathbf {w},\mathbf {0}\}$ is realizable for any $\mathbf {w}$ . In this construction, it is convenient to allow $\langle m,0\rangle $ to correspond to the (initial) coding location for “ $m\in W_{e,s}$ ”. That is, initially we will think of $\nu *0^{\langle m, j\rangle }$ as the coding location $c(\langle m,j\rangle ,s)$ ; and in the construction, $c(\langle m,0\rangle ,s)$ for “ $m\in W_e$ ”. Should m enter $W_{e,s}$ , and conditions are opportune, we would code by changing this to $\nu *0^{\langle m, 0\rangle -1}*1^{s-\langle m,0\rangle }*0^i$ and re-assign the coding location for $\langle m',j'\rangle $ to be $c(\langle m', j'\rangle , s)$ accordingly for $\langle m,0\rangle <\langle m',j'\rangle $ . (Of course all other paths in this subclass are isolated ones of the form $\rho *1^\omega $ , as in Theorem 3.1.) Again this makes a rank 1 subclass extending $\nu $ .

As we see below, there are other reasons we might move a coding location. In the construction, we would only allow $c(n,s)$ to move if $W_{e,s}\upharpoonright \gamma (n,s)$ changes. This is to keep the degree below that of $W_e$ . Without loss of generality we will assume that for $i\le s$ , $\gamma (i,s), \Gamma ^{W_e,s}(i)\downarrow [s]<s.$

We only want to code $W_e$ if $\Sigma _4^0$ is correct. We do not want to code $W-e$ should the $\Pi _4^0$ outcome be correct. Thus, each time $\Sigma _3^0$ of the triple corresponding to $e,p,m$ in $\neg R(e,p,m,s,t)$ looks correct, what we would like to do is to move the coding location for $c(\langle m+i,0\rangle s)$ for $i\ge 0$ spontaneously to $\nu *0^{\langle m,0\rangle -1}*1^{s-\langle m,0\rangle }*0^i$ . We call this action kicking. Then, if $e\not \in S$ , so that the $\Pi _4^0$ outcome is correct and has witness m for this p, eventually all the paths in in the class extending $\nu $ will be computable, as they will be driven to infinity. If $e\in S$ , and p is the correct $\Sigma _4^0$ witness, then since each triple $e,p,m$ has a $\Sigma _2^0$ witness, for each m the above process will fire only finitely often, and hence only move the coding location for $m\in W_e$ a finite number of times.

Now, of course there is a small problem with all of this, as it upsets the computation of the coding locations from $W_e$ . In the $\Sigma _3^0$ case in Theorem 3.1, locations $c(m,s)$ only moved when $m'\le m$ entered $W_{e,s'} $ for some $s'\le s$ .

The solution is to use the highness of $W_e$ . At the same time as the above, we will be monitoring $\Gamma _e^{W_{e,s}}(k)$ for $k\le s$ , and building a sequence of potential computable functions $f_k$ . The reader should note that if the $\Pi _4^0$ outcome is true, it is enough the force almost all coding markers to infinity. That is, even if $\langle e,p,m\rangle $ is the witness of the failure of the $\Sigma _4^0$ condition, it is enough that some $c(n',s)\to \infty $ for some $n'\ge \langle m,0\rangle $ .

So suppose that the we have a fixed $e,p,m$ , and m seems to be a witness $\forall s\exists t\neg R(e,p,m,s,t)$ because this fires. At this stage we would define $f_{\langle e,p,m\rangle }(\langle m,0\rangle ) $ ² to be large and fresh, and continue to do this for $f_{\langle e,p,m\rangle }(\langle m,k\rangle )$ (i.e., for more and more k each time it continues to fire) until we see some $W_e$ change on $\gamma (\langle m,k\rangle )$ to make $g_e^{W_e}(\langle m,k\rangle )>f_{\langle e,p,m\rangle }(\langle m,k\rangle )$ , which it must as $\Gamma ^{W_e}$ is dominant, and $f_{\langle e,p,m\rangle }$ will be total if it fires infinitely often. Note that in the construction, for each k where we get a $W_e$ change on $\gamma (\langle m,k\rangle )$ at some $t\ge s$ , we would kick the coding markers $c(z,t)$ for $z\ge \langle m,k\rangle $ . Additionally, we would kick all coding markers $c(\langle m', q\rangle ,s)$ for $m'>m,k $ to also extend $c(\langle m, s\rangle t+1)$ . Here s is the stage where we first saw a $\Gamma ^{W_e}$ change for some $\langle m,k\rangle $ , and hence this is last defined value of $f_{\langle e,p,m\rangle }$ . Notice that now the original ordering of the coding markers is no longer valid. If we need to kick for m because the $\Pi _4^0$ outcome seems correct, then we will move all coding markers for $m'>m,k$ , and they will now be above $c(\langle m,s+1\rangle , t+1)$ . If the process begins anew, and k was the place where $\Gamma ^{W_e}$ is dominant above $\langle m, k\rangle $ , then if it fires again, all coding markers corresponding to $m'>m,k$ will again be moved as we’d get a change of $W_e$ on $\gamma (\langle m,s+1\rangle ,t+1).$

The key point is that if $\langle e,p,m\rangle $ fires infinitely often, than for almost all $m'$ coding markers corresponding to $m'$ will be driven to infinity, as required. The remaining details work as in the $\Sigma _3^0$ case.

4 Characterizing the sets which can be realized in a $\Pi ^0_1$ class and computable representations of index sets

Recall that for $S\subseteq \omega $ , we defined $G(S)=\{e\mid W_e\equiv _T W_i$ for some $i\in S\}$ . We say that a set $S\subseteq \omega $ represents a set of c.e. degrees D if $G(S)=\{e\mid deg(W_e)\in D\}$ . Theorem 3.1 says that every $\Sigma ^0_3$ representable set of c.e. degrees is represented in some $\Pi ^0_1$ class. We show that in fact the converse holds; that every set of c.e. degrees realized by a $\Pi ^0_1$ class has a computable representation. Hence the index sets which are realized by some $\Pi ^0_1$ class are exactly those with a computable representation. In fact, we shall be able to classify the sets S with a computable representation. Notice that all $G(S)$ which are represented in some $\Pi ^0_1$ class have complexity between $\Sigma ^0_3$ and $\Sigma ^0_4$ . Condition (v) below describes the exact syntactic criterion for an index set to have a computable representation; the condition “ $ \exists a\forall b\exists c ~R^{W_n}(n,a,b,c)$ ” is clearly between $\Sigma ^0_3$ and $\Sigma ^0_4$ (in the sense that every $\Sigma ^0_3$ predicate can be written in this form, and its natural complexity upper bound is $\Sigma ^0_4$ ). Thus, we characterise the index sets with a computable representation as exactly those which are “ $\Sigma ^0_3$ relative to the index being tested”.

Theorem 4.1. Let $S\subseteq \omega $ . The following are equivalent.

1. (i) S has a computable representation, that is, $G(S)=G(R)$ for some computable set R.

2. (ii) There is a $\Pi ^0_1$ class P such that the set of c.e. degrees represented in P has index set $G(S)$ , that is, $W[P]=G(S)$ .

3. (iii) There is a perfect $\Pi ^0_1$ class P such that $W[P]=G(S)$ .

4. (iv) There is a computable function g such that for every n,

   $$\begin{align*}n\in G(S) \Leftrightarrow W_{g(n)}^{W_n} \text{ is cofinite}.\end{align*}$$

5. (v) There is a truth-table functional R such that for every n,

   $$\begin{align*}n\in G(S)\Leftrightarrow \exists a\forall b\exists c ~R^{W_n}(n,a,b,c).\end{align*}$$

Proof (iii) $\Rightarrow $ (ii): Immediate.

(iv) $\Rightarrow $ (v): It is easy to see that (iv) is equivalent to the condition “there is a computable function $\hat {g}$ such that for every n, $n\in G(S)\Leftrightarrow \Phi ^{W_n}_{\hat {g}(n,m)}$ is total for some m”. For any inputs $X,n,a,b$ and c, let $R^X(n,a,b,c)$ hold iff $\Phi ^{W_n}_{\hat {g}(n,a)}(b)[c]\downarrow $ with use u and $W_n[s]\upharpoonright u\subset X$ . Then $R^X$ is clearly total and X-computable for all X, and has the required property.

(v) $\Rightarrow $ (iv): Let $\Phi ^X_{\hat {g}(n,a)}(b)\downarrow $ iff $R^X(n,a,b,c)$ holds for some c.

(ii) $\Rightarrow $ (v): The predicate within square brackets in the proof of Proposition 1.3 is computable in $W_e$ . As above, this is obviously witnessed by a truth-table functional by comparing $W_e[s]$ with the oracle.

(i) $\Rightarrow $ (iii): This is Theorem 3.1.

(iv) $\Rightarrow $ (i): Fix $\hat {g}$ such that for every e, $e\in G(S)\Leftrightarrow \Phi ^{W_e}_{\hat {g}(e,q)}$ is total for some q. If $S=\emptyset $ then there is nothing to do, so fix a c.e. set D with an index in S.

For each pair $e,q$ such that $e,q\in \omega $ , we build (effectively) a c.e. set $V_{e,q}$ . At the end we will take R to be the set of all indices for the sets $V_{e,q}$ (ranging over all pairs $e,q$ ) and show that R represents $G(S)$ . R is of course computable.

Fix a pair $e,q$ . We describe how to build $V=V_{e,q}$ . We also define markers $m_0[s]<m_1[s]<\cdots $ for V. At each stage s define $n_i[s]=\varphi ^{W_e}_{\hat {g}(e,q)}(i)[s]$ , if this converges. We assume that $n_{i+1}[s]\downarrow \Rightarrow n_i[s]\downarrow $ and is less than $n_{i+1}[s]$ . We shall ensure that $m_k[s]\downarrow $ if and only if $n_k[s]\downarrow $ for all $k<s$ . At stage $0$ set all our parameters undefined and set $V=\emptyset $ . Now assume we are at stage $s+1$ . Let $z\in W_e[s+1]-W_e[s]$ be the least such, if it exists. If $z<n_k[s]$ , pick the least such k with $n_k[s]\downarrow $ and enumerate $m_k[s]$ into V. Note that $n_0[s+1],\ldots ,n_{k-1}[s+1]$ are all still defined since there has been no $W_e$ change below them, and therefore $m_0,\ldots , m_{k-1}$ are all still defined. For every number x larger than $m_{k-1}$ that has been assigned to code $D(i)$ for some i, we set $D(i)=V(x)$ . See if $n_k[s+1]\downarrow $ , and if so set $m_k$ to be a large fresh number, and assign a new number less than $m_k$ to code the next $D(i)$ . Otherwise set $m_k\uparrow $ .

If z does not exist, or if z is larger than any $n_k[s]$ seen at the previous stage s, we do the following. For the least k such that $n_k[s]\uparrow $ , and for every number x larger than $m_{k-1}$ that has been assigned to code $D(i)$ for some i, we set $D(i)=V(x)$ . Assign a new number to code the next $D(i)$ . Check and see if $n_{k}[s+1]\downarrow $ . If so, set $m_k$ to be a large fresh number, otherwise do nothing else. This ends the definition of V.

Now we state some properties about V. There are two possibilities:

1. (I) $\Phi _{\hat {g}(e,q)}^{W_e}$ is total. Then $\lim _s n_k[s]$ and $\lim _s m_k[s]$ exist for every k. In this case it is easy to check that $W_e\equiv _T V$ . To compute $W_e\leq _T V$ , fix x and wait for a stage large enough such that $m_x[s]\downarrow $ and V is correct up to $m_x[s]$ . Then $x\in W_e$ iff $x\in W_e[s]$ . To compute $V\leq _T W_e$ we wait for a stage large enough such that $n_k[s]\downarrow $ and $W_e$ is correct up to $n_k[s]$ . This means that $m_k[s]\downarrow $ and the construction will never again change V below $m_k[s]$ .

2. (II) $\Phi _{\hat {g}(e,q)}^{W_e}$ is not total. Then there must be some least k such that either $\lim _s n_k[s]=\lim _s m_k[s]=\infty $ , or $n_k[s], m_k[s]\uparrow $ for almost every s. Let r be the final value of $m_{k-1}$ . (If $k=0$ set $r=0$ ). In the first case there are infinitely many stages where $m_k$ is enumerated into V, and at every such stage we will assign for every i, some number larger than r to code $D(i)$ . We will also be able to ensure that $V(x)=D(i)$ for every $x>r$ assigned to code some $D(i)$ . In the second case $m_{k}[s]\uparrow $ for almost all s, and the construction will attend to the coding of D above r at almost every stage. In either case, we see that every bit of V larger than r is either assigned to code some $D(i)$ , or is never again modified after $m_k$ is lifted past it. Thus, $V\equiv _T D$ .

Now we verify that $G(R)=G(S)$ . Suppose that $V_{e,q}$ is included in R. Then either (I) or (II) holds for the construction of $V_{e,q}$ . In case (II) $V_{e,q}\equiv _T D$ and so $V_{e,q}$ is represented in $G(S)$ . In case (I) $\Phi _{\hat {g}(e,q)}^{W_e}$ is total and $V_{e,q}\equiv _T W_e$ . This also means that $V_{e,q}$ is represented in $G(S)$ . Hence $R\subseteq G(S)$ .

Now let $e\in G(S)$ . Suppose q is such that $\Phi _{\hat {g}(e,q)}^{W_e}$ is total. But this means that case (I) holds for $V_{e,q}$ and the index of $V_{e,q}$ is in R. Hence $G(S)\subseteq G(R)$ .

Corollary 4.2. If S is $\Sigma ^0_3$ then there is some computable R such that $G(S)=G(R)$ . However not every $\Pi ^0_3$ set has a computable representation.

Corollary 4.3. An index set I is realisable in a $\Pi ^0_1$ class iff I has a $\Sigma ^0_3$ representation iff I has a computable representation.

Corollary 4.4. If S and T have computable representations then so do $G(S)\cap G(T)$ and $G(S)\cup G(T)$ .

Corollary 4.5. The following classes of c.e. degrees have a computable representation and can be realized in some ( $perfect$ ) $\Pi ^0_1$ class:

• • The set of all superlow c.e. degrees.

• • The set of all K-trivial c.e. degrees.

• • Any $\Sigma ^0_4$ subset of a computable collection of high indices with an index for $\boldsymbol {0}$ .

• • Any uppercone of c.e. degrees, that is, the set $\{\boldsymbol {b}\mid \boldsymbol {b}$ is c.e. and $\boldsymbol {b}\geq \boldsymbol {a}\}$ for any c.e. degree $\boldsymbol {a}$ .

• • Any lowercone of c.e. degrees below a low $_2$ c.e. degree, that is, the set $\{\boldsymbol {b}\mid \boldsymbol {b}$ is c.e. and $\boldsymbol {b}\leq \boldsymbol {a}\}$ for any low $_2$ c.e. degree $\boldsymbol {a}$ .

Proof The first two have obvious $\Sigma ^0_3$ definitions. The third is just Theorem 3.6. For the fourth, let $A\in \boldsymbol {a}$ be a c.e. set. Then $W_n\geq _T A$ if and only if $\exists e\forall x\forall s\exists t>s ~\Phi _e^{W_n}(x)[t]\downarrow =A(x)[t]$ and $W_n[t]\upharpoonright {\varphi _e(x)[t]}$ is correct. For the fifth, note that $W_n\leq _T A$ if and only if $\exists e~\Big (\Phi _e^A$ is total and $\forall x\forall s\exists t>s ~\Phi _e^A(x)[t]\downarrow =W_n(x)\Big )$ . Since A is low $_2$ , the statement “ $\Phi _e^A$ is total” is $\Sigma ^0_3$ .

Finally, we wish to consider which lowercone of c.e. degrees has a computable representation. From Corollary 4.5 we know that every low $_2$ lowercone has a computable representation. We will next show that not every lowercone of c.e. degrees has a computable representation, and will follow from the next theorem. The next result says that given any effective sequence of independent c.e. sets $A_0,A_1,\ldots $ , and any $\Sigma ^0_4$ set S, there is a c.e. set B that bounds $A_i$ for all $i\in S$ and does not bound any $A_i$ for $i\not \in S$ ; roughly speaking, it says that the S-infinite join of any independent sequence exists, and we believe that this result is of independent interest.

Theorem 4.6. Let C be a uniformly computable collection of c.e. sets $\{A_i\mid i\in \omega \}$ which are independent in that for any finite set F, if $i\not \in F$ , then $A_i\not \le _T \oplus _{j\in F}A_j$ . Let S be a $\Sigma _4^0$ set. Then there is a c.e. B such that for all $i\in \omega $ , $i\in S$ iff $A_i\le _T B$ .

Proof The proof uses the ${\mathbf 0"}$ methodology, even though each requirement shall be divided into infinitely many subrequirements, but there are no injuries along the true path. We need to meet for all e the following:

$$ \begin{align*}M_e: e&\in S \to A_e\le_T B,\\N_e: e&\not\in S \to A_e\not \le_T B.\end{align*} $$

To meet $M_e$ , let R be a computable relation representing S in that

$$ \begin{align*}e\in S\mbox{ iff }\exists x \forall m\exists s\forall t R(e,x,m,s,t).\end{align*} $$

$M_{e,x}$ denotes the attempt to meet $M_e$ at witness x. We set aside infinitely many potential columns $\{B^{\langle e,x,n\rangle }\mid n\in \omega \}, $ for the sake of $M_{e,x}$ . At each stage s, $n(e,x,s)$ denotes the current choice. As we will later see, we will in fact be using multiple positions to try to meet the same $\langle e,x\rangle $ . We will ensure that $n^*=\liminf _s n(e,x,s)$ exists for at least one choice and this column $B^{\langle e,x,n^*\rangle }$ correctly codes $A_e$ iff $e\in S$ .

4.1 Description of the strategy

For a fixed choice $n(e,x,s)$ , we do the following. For each number a, we will have a coding location $c(2a,s)$ which is coding whether $a\in A_i$ . (The odd ones will be used for the $\Sigma _4^0$ -testing.) If a enters $A_{i,s}$ put $\langle e,x,n(e,x,s),c(2a,s)\rangle $ into $B_{s+1}-B_s.$ Unless restrained, each time $R(e,x,m,-,-)$ fires (for each m), we will kick all coding locations $c(a,e,x,s)=c(a,s)$ for $a\ge 2m+1$ to fresh places, dumping $\langle e,x,n(e,x,s),c(2m+1,s)\rangle \dots \langle e,x,n(e,x,s),c(s,s)\rangle $ into $B_{s+1}-B_s$ , and reassigning new coding locations for $a\ge 2m+1$ to be $c(2m+i,s+1)=c(s,s)+i,$ for $i\ge 1.$ This will obviously be modified below, to take into account of other requirements, but the actual action is more or less similar. Note that if x is not the $\Pi _3^0$ witness that $e\in S$ , then for some m, the $\Pi _2^0$ fact $R(e,x,m,-,-)$ fires infinitely often, and hence, if this is a stable $n(e,x,s)$ , almost all of the column $\omega ^{\langle e,x,n(e,x,s)\rangle }$ will enter B.

To implement this strategy on a tree of strategies we will introduce $M_{e,x}$ at some mother node $\tau (e,x)=\tau $ , and at various places in the cone below $\tau (e,x)$ , we will have sub-requirements $\tau _m(e,x)$ testing whether $R(e,x,m,-,-)$ fires infinitely often, i.e., whether the predicate $\forall s\exists t R(e,x,m,s,t)$ holds. This node $\tau _m(e,x)$ will have outcomes $\infty <_L f$ , corresponding to the truth value of the predicate $\forall s\exists t R(e,x,m,s,t)$ . The way the mother node $\tau (e,x)$ and its children nodes $\tau _m(e,x)$ work is the following. The mother node $\tau (e,x)$ is assigned a column, say p of B. It wants to code $A_e$ into column p of B if and only if the $\Pi ^0_3$ predicate $\forall m \exists s\forall t \neg R(e,x,m,s,t)$ . It measures this $\Pi ^0_3$ predicate by distributing the task amongst its children nodes; as usual, each child $\tau _m(e,x)$ measures the $\Sigma ^0_2$ fact $\exists s \forall t\neg R(e,x,m,s,t)$ . The mother node $\tau (e,x)$ will assign coding location for each z, coding $A_e(z)$ into some position of $B^{\langle p\rangle }$ . Its task whenever visited is to monitor if some $A_e(z)$ has changed since it last checked, and if so, put the appropriate coding marker into $B^{\langle p\rangle }$ . It also assigns a new coding location for z if the previous one has been put into B. The task of each child node $\tau _m(e,x)$ is to monitor $\exists s \forall t\neg R(e,x,m,s,t)$ . Every time the $\Pi ^0_2$ instance holds it will work towards making $B^{\langle p\rangle }$ cofinite by picking some least coding location z (above some restraint) and enumerating everything in the column $B^{\langle p\rangle }$ between $c(2z+1)$ and s. Thus in the $\Pi ^0_3$ case, each coding location is lifted finitely often by the children nodes and the mother node will successfully ensure $A_e\leq _T B^{\langle p\rangle }$ . In the $\Sigma ^0_3$ case, some child node will force $B^{\langle p\rangle }$ to be cofinite, and obviously in this case $A_e\not \leq _T B^{\langle p\rangle }$ . Note that in the $\Sigma ^0_3$ case, making $A_e\not \leq _T B^{\langle p\rangle }$ is not enough, as we in fact need $A_e\not \leq _T B$ , but it is necessary for the N-requirements to work correctly.

Below outcome $\tau _m(e,x)*\infty $ we have confirmation that x is not the witness for $e\in S$ . So we must make progress towards meeting the requirement $N_e$ (i.e., making $A_e\not \le _T B$ ) as described below. Note that we may not actually meet $N_e$ at the end, since perhaps $e\in S$ , but witnessed by some other $x'>x$ . We will describe how this works in due course, but for the time being, we shall suppose that no such $x'$ exist and that e is in fact not in S. Now, $N_e$ is divided into infinitely many subrequirements $N_{e,k}: e\not \in S\to \Phi _k^{B}\ne A_e.$

We first discuss this for $e=0$ and $k=0$ . We will be assuming that $n(0,0,s)=0$ for all s, so that $A_0$ is being coded into column $B^{\langle 0,0,0\rangle }$ for all stages. Now, below some $\tau _m(0,0)*\infty $ (where recall that $\tau (0,0)$ is the mother node devoted to $M_{0,0}$ ) we will work as follows. We wish to make $\Phi _k^B\ne A_0$ for $k=0$ . The first case would be that $\tau _m(0,0)$ is immediately below the position where we introduced the mother node $\tau (0,0)$ and therefore $N_{0,0}=\tau _m(0,0)*\infty $ are the first two requirements below $\tau (0,0)$ . The strategy for $N_{0,0}$ is relatively simple. We would allow $N_{0,0}$ to assert control of B by freezing $B\upharpoonright \varphi _k(z)$ at the first stage that we see a $B^{\langle 0,0,0\rangle }$ -correct computation such that $\Phi _k^B[s]\upharpoonright z =A_0[s]\upharpoonright z$ . Then, since $A_0$ is noncomputable, this would only have finite restraint on the overall construction, since at some stage some small a will enter $A_0$ and create a disagreement, which will be preserved forever.

More generally, let us suppose that we are dealing with $\Phi _k^B\ne A_0$ and that the requirement in charge of this, $N_{0,k}$ , is placed below, say, a coding column which is coding $A_1$ . More specifically, we are now considering the situation where $N_{0,k}$ is below the node $\tau (1,0)$ (assigned to requirement $M_{1,0}$ ), which is in turn below $\tau _m(0,0)*\infty $ .

So in this simplified set-up, $\Phi _k^B\ne A_0$ being met at $\sigma $ will need to cope with, say, the coding of $A_1$ into some column of B (say column p), and it is not reasonable that this should be restrained by $\sigma $ as the mother node $\tau (1,0)$ in charge of the coding of $A_1$ is at a higher priority place on the tree.

Now there are two possibilities. Either column p actually codes $A_1$ at the end ( $\Pi ^0_3$ outcome), or there is some $m_1$ which fires infinitely often for column p and $A_1$ ( $\Sigma ^0_3$ -outcome). In the $\Pi ^0_3$ outcome there are no difficulties with arranging for the coding of $A_1$ and $\tau (1,0)$ to be of higher priority than $\sigma $ . Following the above strategy, $\sigma $ will impose restraint on all requirements of lower priority as indicated, and the only injury would to $\sigma $ will be caused by the $A_1$ -coding done by $\tau (1,0)$ . Since $A_0\not \le _T A_1$ , the $\liminf $ of the restraint would exist. Thus, on the tree of strategies, we will represent the possible outcomes as the sequence $w<_L 0<_L 1<_L 2<_L 3<_L \dots $ with w representing the waiting outcome to see a valid length of agreement above $0$ , and the rest the $\liminf $ of the length of agreement. Note that we are using the hat convention so that if a computation with length above n occurs and the use is injured, then the length of agreement must drop below n, for at least the next $\sigma $ -stage. Naturally, we will need to prove one of these outcomes is on the true path assuming that $\sigma $ is.

Our biggest problem come from the fact that the numbers entering B can either be caused by coding of some $A_i$ set, or some $m_1$ trying to prove that some $\tau $ is incorrect. If this node $\sigma $ trying to make $\Phi _k^B\ne A_0$ is below some $\tau _{m_1}(1,0)*\infty $ , then $\sigma $ knows that $\tau _{m_1}(1,0)$ is going to make column p of B cofinite. As usual, $\sigma $ would wait for computations which are both $\tau _{m_1}(1,0)*\infty $ and $\tau _m(0,0)*\infty $ correct, and there is no real problem. However it could be that $m_1$ fires infinitely often, and $\sigma $ is above all such $\tau _{m_1}(1,0)*\infty $ -nodes. The decision of coding locations of $A_1$ inside column p of B is made at the mother node $\tau (1,0)$ . However, as $\tau (1,0)$ is committed to coding $A_1$ into its assigned column of B if and only if some $\Pi ^0_3$ predicate holds, the mother $\tau (1,0)$ will distribute this guess amongst its children nodes $\tau _k(1,0)$ for all k. Hence, in the $\Sigma ^0_3$ case we will have to let some child node $\tau _{m_1}(1,0)$ make column p of B cofinite and move its corresponding coding location to $\infty $ . If this child node $\tau _{m_1}(1,0)$ is below $\sigma $ , then there is no way for $\sigma $ to wait for a $\tau _{m_1}(1,0)*\infty $ -correct computation.

In this case, the child node $\tau _{m_1}(1,0)$ has lower local priority than $\sigma $ , and thus should defer to the wishes of $\sigma $ . We will arrange for controlled injury of the column p in this case. When any k fires (i.e., $\tau _k(1,0)$ sees the next instance of $\Pi ^0_2$ hold), if it is associated with some $\tau _k(1,0)*\infty \preceq \sigma $ , then we let it act as usual, since $\sigma $ must be looking for $\tau _k(1,0)$ -correct computations anyway. However, if k is sufficiently large that it is handled below $\sigma $ , the tree machinery will handle the injury. More specifically, $\tau _k(1,0)$ will be making column p of B coinfinite above the restraint r such that $\tau _k(1,0)\succeq \sigma *r$ . If $\tau _k(1,0)$ is along the true path, then every time where it is visited, the restraint currently imposed by $\sigma $ is less than r. Of course, if we move to some outcome $n<r$ of $\sigma $ later on, then the restraint imposed by $\sigma $ will drop to $n<r$ , and the corresponding sibling node $\tau _k(1,0)\succeq \sigma *n$ will make enumerate all numbers in column p of B between n and r and lift all corresponding coding markers. Notice that the $\sigma $ -restraint will only drop because something enters $A_1$ forcing a coding into column p. The upshot is that in both cases it cannot be that $\Phi _k^B=A_0$ , as either $A_0\le _T R$ , a computable set, or $A_0\leq _T A_1$ , in the latter case.

The analysis above shows how we can code a $\Pi _3^0$ set S into B, but since S is $\Sigma _4^0$ , we have multiple attempts to code, e.g., $A_0$ into B, and the above dealt with attempt $(0,0)$ . To wit, we will have infinitely many x such that $(0,x)$ potentially witnesses $0\in S$ , provided that it has the $\Pi _3^0$ outcome.

The remaining details are to put this together and argue that the combinatorics works. We turn to these details.

4.2 The priority tree

We will define $PT$ by induction on length of $\nu $ on $PT$ according to the following rules.

The tree will consist of $\tau $ -nodes, $\tau _m$ -nodes, and $\sigma $ -nodes. $\tau $ and $\tau _m$ -nodes have associated with them a pair $(e,x)$ , namely $e(\tau ),x(\tau ))$ and in the case of $\tau _m$ , this refers to its mother, which will be a $\tau $ node introduced in the tree and above it. A $\tau $ -node has a single outcome o which simply indicates a place that we are introducing an attack on $(e(\tau ),x(\tau )) $ and hence trying to code $A_{e(\tau )}$ into B. This is done via some column $n(\tau )$ chosen from ones set aside for e as indicated at the beginning of the proof. We will not explicitly mention the relevant $\langle e,x,p,q\rangle $ as this seems unnecessary, but will associated coding markers $c(\tau ,m,s)$ and m-markers $c(\tau ,2m+1,s)$ the latter concerned with whether $e\in S$ is not witnessed by $(e,x)$ . $\tau _m$ has outcomes $\infty <_L f$ . $\sigma $ -nodes will have outcomes $w<_L 0<_L 1<_L 2<_L 3<_L \dots $ . Associated with $\sigma $ is a triple $(e(\sigma ),x(\sigma ),k(\sigma )) $ also indicating which $A_e$ it is attempting to diagonalize and for which x. This information would be summarized by some pair $(\tau _m,k)$ , with $e(\tau _m)=e(\sigma )$ , and $x(\tau _m)=x(\sigma )$ .

We then work by length and generate $PT$ . Along some path we will gradually introduce the the nodes above. We will let $\lambda $ , the empty string, be devoted to $(0,0)$ and thus $\tau (0,0)=\lambda $ . This has a single outcome o. We would let this be $\tau _0(0,0)$ and have mother $\lambda $ . This has two outcomes $\infty ,f$ . Below the $\infty $ outcome we would have a $\sigma (\tau _0,0)$ -node and below the f-outcome, we would introduce a new $\tau (1,0)$ -node so that it equals $o f$ . The $\sigma $ -type node below the $\infty $ -outcome will have outcomes $w<_L 0<_L 1<_L 2<_L 3<_L \dots $ . Below each of these we would introduce a new $\tau (1,0)$ -node, and promise according to some priority ordering to introduce a new $\tau (0,1)$ -node. (This latter case differs from the outcome f of o since we are recognising that we think we have killed $(0,0)$ and need to restart $A_0$ with a new $x=1$ ; whereas below f we are thinking that $x=0$ is good.)

In general as we work down the tree, gradually introducing nodes according to some priority ordering, but remaining consistent with the information. If we hit a node $\nu $ , we will see three cases.

Case 1. $\nu $ should be assigned to a new $\tau (e,x)$ . Outcome o.

Case 2. $\nu $ should be assigned to some $\tau _m(e,x)$ . This will have outcomes $\infty ,f$ . Below the $\infty $ outcome we will need to:

• • Start introducing $\sigma (e,x',k)$ nodes for $x'\le x.$

• • Introduce some other $\tau (i,j)$ -nodes and $\tau _m(i,j)$ -nodes (but not for $i=e $ and $j\le x$ as these have been dealt with) equitably according to the list.

• • And then introduce a $\tau (e,x+1)$ -node.

Below the f-outcome, we would assign the next type of node not yet deal with at $\nu $ according to some priority list, and consistent with the information on the path. (So, for example, not a $\tau (e,y)$ -node for $y>x$ , for instance.)

Case 3. $\nu $ should be assigned to some $\sigma (\tau ,k)$ -node. It will have outcomes $w<_L 0<_L 1<_L 2<_L 3<_L \dots $ . Below each, we would assign the next type of node not yet deal with at $\nu $ according to some priority list, and consistent with the information on the path. Note that nothing will be re-started.

4.3 The construction

We begin at $\lambda $ . For any n which has entered $A_0$ at stage $s+1$ , put the coding marker for n from column $n(\lambda )$ (Column $n(\lambda )$ equals $\langle 0,0,0\rangle $ ), into B.

Play outcome o. See if $(0,0)$ has fired. If so play outcome $\infty $ , else play outcome f.

More generally, we have hit $\nu $ .

If $\nu $ is a $\tau _m(e,x)$ -node, see if $(e,x)$ has fired since the last $\nu $ -stage. Look at those $\sigma $ -nodes $\rho \prec \nu $ . Each such $\rho $ will have an outcome $\rho *n\preceq \nu $ , and an associated use $u(\rho ,s)$ of the computation $\Phi _{e(\rho )}^B\upharpoonright n-1=A_e\upharpoonright n-1$ , and possibly $\Phi _{e(\rho )}^B\upharpoonright n\ne A_e\upharpoonright n$ , since we preserve to the first disagreement using standard Sacks’ strategy. Let u be the maximum of such uses. Accordingly, there will be some $m'\ge \max \{u,m\}$ which we will use to move all coding markers $c(z,s)$ for $z\ge 2m'+1$ which have not moved since the last $\nu $ -stage.

If $\nu $ is a $\tau $ -node, then if this is the first stage we have visited $\tau $ since initialization, assign a fresh acceptable column $n(\tau )$ to $\tau $ to code $(e(\tau ),x(\tau )).$ If $\nu $ already has a column, then see if any numbers have entered $A_e$ since the last $\nu $ -stage and then enumerate the relevant coding markers into column $n(\tau )$ .

If $\nu $ is a $\sigma $ -node, then play outcome n for the current $\sigma $ -correct length of agreement (i.e., consistent with all $\tau _m*\infty \preceq \nu $ ) up to the first disagreement, with w being played if there are no computations at all.

Generate the $TP_s$ of length s in this way and initialize all nodes right of $TP_s$ at the end of stage s.

This ends the description of the construction.

4.4 Verification

We verify the construction. We argue by induction on the length of $\nu $ that $TP$ exists and the outcomes are true.

We begin with $\nu $ a $\sigma $ -node. We need to prove that the overall restraint caused by $\nu $ is finite along the true path, and that $\nu * n\prec TP$ for some n. Now go to a stage $s_0$ where we are never left of $\sigma $ . Let $e,x,k$ be the relevant $\sigma $ -parameters. Each time after stage s we see a $\nu $ -correct computation $\Phi _e^B\upharpoonright n=A_e\upharpoonright n$ (up to the first disagreement) we will play outcome n. The first time we do this, mothers of lower priority than $\nu $ below this outcome will have their coding locations set to be above the use of the computation. Furthermore, any $\tau _{m_1}$ -node below outcome n has to use a proxy $m_1'$ above the $\nu ,n$ -use for its firings. Therefore this computation will be preserved unless some coding action i entering $A_{e'}$ for some mother $\tau '\preceq \nu $ occurs after stage s. This will cause the length of agreement to drop below n at the next $\nu $ -stage, and hence if we ever play n again, mothers below this outcome will again have their coding locations reset. Since no $\tau _m*\infty $ -injury can occur for $\tau _m\preceq \nu $ as this is incorporated into $\nu $ -correctness, and $\tau _{m_1}$ extending any $\tau _{m_1}$ -node below outcome n has to use a proxy above the n-use for its firings. We can conclude that injuries only occur because of i entering $A_{e'}$ for some mother $\tau '\preceq \nu $ , after $s_0$ . So suppose that $\Phi _k^B=A_e$ at $\nu $ . For each n there will be a computation $\Phi _k^B\upharpoonright n=A_e\upharpoonright n$ at a $\nu $ -stage s which is $A_i$ -correct for all $A_i$ with mothers above $\nu $ . This computation therefore must actually be correct. But then we can conclude $A_e\le _T \oplus _{i\in F}A_i$ with F the collection of with mothers $\tau $ above $\nu $ , and $\tau _m*\infty \prec \nu $ . But this means that $A_e$ cannot be in F by the positioning of the $\sigma $ -type nodes such as $\nu $ . This contradicts the independence of $\{A_j\mid j\in {\mathbb N}\}$ . Therefore $\Phi _k^B=A_e$ at $\nu $ . Hence the $\liminf n$ must exist.

If $\nu $ is a $\tau $ node, then at some stage $\nu $ is never re-initialized, and its coding locations are set. Each time we visit $\nu $ we code as required.

If $\nu $ is a $\tau _m$ node, if it stops firing at some stage after $s_0$ , then $\nu *f$ will be on $TP$ . If m fires infinitely often, we know by induction that the uses associated with $\sigma $ -type nodes $\sigma *n\preceq \tau _m$ will have finite use, and hence from some point onwards, the proxy $m'$ for m will be fixed. Each time m fires it will move $c(2m'+1,s)$ in column $n(\tau )$ from the mother $\tau \prec \tau _m$ , and if $\tau $ is below the outcome n for such $\sigma $ -nodes, then $\tau $ will have chosen so that $m=m'$ .

Finally, if $e\in S$ , then the way we construct the priority tree means that we will put some $\tau \prec TP$ with $(e(\tau ),x(\tau ))$ being correct in that x is a witness for $e\in S$ . Below $\tau $ , every $\tau _m$ will have outcome f. Thus we will correctly code $A_e$ into B since each coding marker for $n\in A_e$ can be moved at most a finite number of times by m below n.

If $e\not \in S$ , then for each pair $(e,x)$ there will be some $\tau _m*\infty \prec TP$ with $(e(\tau ),x(\tau ))=(e,x)$ . Below each such node we will have $\sigma (e,x,k)$ -nodes $\sigma $ and these will be met as above. This means that $A_e\not \le _T B$ .

This concludes the proof of Theorem 4.6.

Corollary 4.7. Not every lowercone of c.e. degrees has a computable representation.

Proof Fix any effective independent sequence of low c.e. degrees $A_0,A_1,\ldots $ with uniform lowness indices, that is, $(A_k)'=\Phi _{F(k)}^{\emptyset '}$ for all k and some computable function F. This can be easily done by constructing the sequence directly, satisfying both lowness and diagonalization requirements. Both kinds of requirements are finitary, and the construction produces a uniform sequence of lowness indices F. Now let S be any $\Sigma ^0_4$ -complete set S. By Theorem 4.6 let B be a c.e. set such that $i\in S$ iff $A_i\leq _T B$ . If the lowercone below B has a computable representation, say witnessed by the effective sequence $V_0,V_1,\ldots $ of c.e. sets, then we see that $i\in S$ iff $A_i\leq _T B $ iff $A_i\equiv V_k$ for some k. Since $A_i$ is low, the predicate “ $A_i\leq _T V_k$ ” is $\Sigma ^0_3$ (in the variable k). Since the sequence $\{A_i\}$ is uniformly low, the statement “ $A_i\equiv V_k$ for some k” is $\Sigma ^0_3$ . This is a contradiction.

In an early draft of this paper, from several years ago, we concluded with the following question.

Question 4.8. Is there some Turing incomplete non-low $_2$ c.e. set A such that the c.e. lowercone below A has a computable representation?

However, in [7], Downey and Melnikov showed that the answer is no.

Theorem 4.9 (Downey and Melnikov [7]).

Suppose that A is c.e. and Turing incomplete, then the lowercone below A has a computable representation iff A is low $_2$ .

Funding

Csima is partially supported by an NSERC Discovery Grant. Downey is partially supported by Marsden Fund of New Zealand. Ng is partially supported by the grants MOE2015-T2-2-055 and RG131/17.