Turing reducibility in the fine hierarchy

Journal Pre-proof Turing reducibility in the fine hierarchy

Alexander G. Melnikov, Victor L. Selivanov, Mars M. Yamaleev

PII:

S0168-0072(19)30129-0

DOI:

https://doi.org/10.1016/j.apal.2019.102766

Reference:

APAL 102766

To appear in:

Annals of Pure and Applied Logic

Received date:

21 June 2018

Revised date:

6 November 2019

Accepted date:

11 December 2019

Please cite this article as: A.G. Melnikov et al., Turing reducibility in the fine hierarchy, Ann. Pure Appl. Logic (2019), 102766, doi: https://doi.org/10.1016/j.apal.2019.102766.

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Turing reducibility in the fine hierarchy$ Alexander G. Melnikova,∗, Victor L. Selivanovb,c , Mars M. Yamaleevc a Massey University, Private Bag 102 904 SNCS, Albany 0745, Auckland Ershov Institute of Informatics Systems SB RAS, 6 Acad. Lavrentjev pr., 630090, Novosibirsk, Russia c Kazan (Volga region) Federal University, 18 Kremlyovskaya str., 420008, Kazan, Russia b A.P.

Abstract In the late 1980s, Selivanov used typed Boolean combinations of arithmetical sets to extend the Ershov hierarchy beyond Δ02 sets. Similar to the Ershov hierarchy, Selivanov’s fine hierarchy {Σγ }γ<ε0 proceeds through transfinite levels below ε0 to cover all arithmetical sets. In this paper we use a 0 construction to show that the Σ03 Turing degrees are properly contained in the Σωω +2 Turing degrees (to be defined); intuitively, the latter class consists of “non-uniformly Σ03 sets” in the sense that will be clarified in the introduction. The question whether the hierarchy was proper at this level with respect to Turing reducibility remained open for over 20 years. Keywords: Ershov hierarchy, arithmetical hierarchy, fine hierarchy, Turing degree 2010 MSC: Primary 03D28, Secondary 03D55

1. Introduction The study of various hierarchies of sets is central to mathematical logic. In computability theory, one investigates hierarchies of countable sets. This paper contributes to a general program in computability theory that seeks to understand various not necessarily algorithmic hierarchies using algorithmic tools such as Turing degrees. Recall that a Turing degree consists of oracles that are indistinguishable from the perspective of Turing computation. It is quite remarkable that many natural hierarchies admit several equivalent definitions, one usually being syntactical, and the other topological or computability-theoretic (or both). There are many examples of this sort, including the classical hierarchies [10, 23, 18] as well as some very recently introduced ones [17, 5]. Perhaps, the most famous illustration of this phenomenon is provided by the the well-known positive solution to Hilbert’s Xth Problem [15]: $ The first author was partially supported by Marsden Fund of New Zealand. The research of the second author was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project No. 1.13556.2019/13.1. The research of the third author was supported by the Russian Science Foundation, project No. 18-11-00028. ∗ Corresponding author Email addresses: [email protected] (Alexander G. Melnikov), [email protected] (Victor L. Selivanov), [email protected] (Mars M. Yamaleev)

Preprint submitted to Annals of Pure and Applied Logic

December 12, 2019

the Diophantine sets are exactly the computably enumerable sets which form the first level Σ01 of the arithmetical hierarchy. The second level Σ02 of the hierarchy consists of first-order ∃∀-definable subsets of (N, +, ×, 0) which are exactly the sets computably enumerable with the help of the Halting problem. The Π02 -sets are the complements of the Σ02 -sets, and Δ02 = Σ02 ∩ Π02 . The Shoenfield Limit Lemma [26] says that a set is in Δ02 exactly if it can be computably approximated with finitely many mind-changes; this effective approximability of Δ02 -sets makes them relatively accessible for computabilitytheoretic investigations. In fact, the study of Turing degrees of Δ02 -sets is one of the central topics of Turing computability theory [26]. The structure of Δ02 degrees turned out to be quite complex, with the theory of Σ01 -degrees being its most developed and well-understood subtheory [26]. If we cannot fully grasp the structure of Δ02 -degrees, it is natural to look at some finer intermediate complexity classes between the Σ01 and the Δ02 -degrees and try to extend the developed theory of Σ01 -degrees to these intermediate classes. However, the arithmetical hierarchy is too crude to allow for such a subtle distinction, we need a more sensitive sub-hierarchy. One such finer hierarchy is the difference hierarchy, also known as the Ershov hierarchy [7, 8, 9]. The hierarchy looks at the Boolean combinations of computably enumerable (c.e. for short) sets and measures the complexity of the Boolean forms. The hierarchy has an equivalent “dynamic” definition which measures the number of mind-changes needed to compute a Δ02 -set via the Shoenfield Limit Lemma. 0 The c.e. sets form the first level Σ−1 1 = Σ1 of the Ershov hierarchy, while the differences of c.e. sets – also known as d-c.e. sets – form its second level Σ−1 2 . } proceeds through all (notations for) computable ordinals The hierarchy {Σ−1 α to cover all Δ02 -sets. There has been many deep results on the first few nontrivial levels of the hierarchy, especially the d-c.e. degrees, see the PhD thesis of Wu [29] and the more recent surveys by Arslanov and Kalimullin [1] and Lempp [12]. Less is known about the higher transfinite levels of the difference hierarchy, but there have been some recent impressive results on the closely related difference hierarchy introduced by Downey and Greenberg, see their book [5]. A survey on this hierarchy can be also found in [6]. Since the investigation of the difference hierarchy from the perspective of Turing degrees has been rather fruitful, one naturally seeks an extension – or at least an analogy – of such results beyond the Δ02 degrees. Clearly, one naive possibility would be looking at relativised versions of the results on Σ−1 α sets. However, lots of subtle information such as a dynamic approximation will be lost after a relativisation, thus such relativised results seem to be of little interest. We need a more sensitive hierarchy than the relativised Ershov hierarchy. In [19], using some special jump operators Selivanov introduced a natural extension of the Ershov hierarchy to sets beyond Δ02 , we omit the definition. The more intuitive description of the hierarchy can be found in [21], we outline the main idea. Consider the typed Boolean combinations over variables {vi,k }i,k∈ω , where for each fixed k the variables {vi,k }i∈ω range over Σ0k -sets. Then for every typed Boolean term t, form the class Σ(t) = {t(vi,j ) : vi,j ∈ Σ0j }. Each such term has a natural notion of rank which is an ordinal below ε0 = ω sup{ω, ω ω , ω ω , . . .}. We omit further formal details since they are irrelevant to our main result; see [22]. Similarly to the Ershov hierarchy, Selivanov defined classes Σα for α < ε0 according to the rank of the typed term defining the class. He also proved that the definition is robust and the hierarchy is proper with 2

respect to m-degrees. As we have mentioned above, all natural and useful hierarchies tend to admit several equivalent definitions, and the Selivanov’s fine hierarchy is not an exception. There are several equivalent ways to define the hierarchy, one of which, remarkably, uses a slight variation of the Wadge’s operator bisep [23]; this gives a technical connection between the fine hierarchy in computability theory, the Wadge hierarchy [27] in descriptive set theory, and the Wagner hierarchy [28] in automata theory. Selivanov called the new hierarchy the fine hierarchy. The hierarchy is “fine” in the sense that it is provably more sensitive than just the relativised Ershov hierarchy within Δ0n , for any n > 2. The relativised version of the Ershov hierarchy at a level Δ0n , n > 2, does not take into account the mind-changes at the lower levels, thus erasing much of the subtle complexity related to dynamic approximations. Recall that we were searching for a natural extension of the Ershov hierarchy beyond Δ02 -sets. The fine hierarchy seems to be the most natural choice here. Whenever we have a hierarchy of sets, the general expectation is that the more complex classes of the hierarchy should contain more powerful oracles, up to Turing equivalence. If this is indeed the case we say that the hierarchy is proper (i.e., does not collapse) with respect to Turing degrees. In the early 1970s, Cooper [4] constructed a d.c.e. set which is not Turing equivalent to any c.e. set. Cooper’s result can be extended to all levels of the Ershov hierarchy [11, 20], illustrating that the Ershov hierarchy is proper from the perspective of Turing reducibility. Which levels of the fine hierarchy are proper with respect to Turing degrees? One would hope for some inductive proof – e.g., by the complexity of the typed terms – that would extend the argument of Cooper [4] to all levels of the fine hierarchy, but this seems to be highly problematic. Somewhat unexpectedly, the situation with the fine hierarchy is more complicated than with the Ershov hierarchy. It seems that any construction of a set in Σα not Turing equivalent to a set in any simpler class would necessarily require a guessing at the level of the highest type of the variable involved in the defining term. The dynamic feature which is so attractive about the fine hierarchy also makes the task quite complicated. The question of which levels of the hierarchy are proper with respect to Turing degrees remained unresolved, and essentially completely open, for a few decades. As we will see, the problem of properness of the fine hierarchy with respect to Turing degrees requires an essential use of advanced degreetheoretic techniques. Only some partial progress has been done into this direction. For instance, it is known that all levels Σλ+1 (λ < ε0 is a limit ordinal) are not proper and that level Σω+2 is proper [25], also each Σω+n+2 , n < ω, is proper [24]; see [23, 25] for further results. However, there are still many levels of the fine hierarchy for which the situation with the Turing degrees remained, and still remains, unknown. We note that the results listed above used constructions of complexity at most 0 . The class Σωω +1 collapses with Σωω under Turing reducibility [25]. The first level at which a 0 guessing seems necessary is Σωω +2 vs. Σωω +1 , making the question of properness at this level of a special importance and also of some independent technical interest. We explain what these classes Σωω +2 and Σωω +1 are below. The sets in the class Σωω +1 have the same Turing degrees as the Σ03 -sets [25]. 3

Thus, we will not define the class; instead, we will be working with Σ03 sets. The next simplest class that can potentially contain more oracles is Σωω +2 . What are these sets? To enter a set from this class, each number x ∈ ω goes through the following guessing procedure, in the spirit of mind-changes in the Ershov hierarchy. Initially, we wait for the guessing procedure to claim x ∈ S in a c.e. fashion; if this never happens then it is guaranteed that x is kept out of S forever. However, if the procedure eventually claims that x ∈ S it will be not necessarily be final. At a later stage, x can be extracted from S again, in a c.e. fashion, and if this ever happens it will be given a choice of two independent guessing procedures, Σ03 and Π03 . More specifically, one of the two predicates will be tested on x, and x is truly in the set only if it passes the guessing that it chose. The choice is made by enumerating the witness, again in a c.e. way, to one of two special c.e. sets; and if x finds itself in the first c.e. set, then its final membership becomes a uniformly Σ03 -question, and in the second set the actual membership will be a Π03 -question. The set A is in charge of the Σ03 -guessing and B is in charge of the Π03 -guessing, so S = A ∪ B ∪ C; the formal definition will be given in Section 2, see also the diagram below. c.e. 2-c.e. Ay X x` “out” C “in” `  ` ` “in”  x B 9 “out” “in”  c.e.

“out” x

The set A is Σ03 and the set B is Π03 . In some sense, the guessing on the membership for such sets is still at the level of only three quantifiers, but we get to dynamically choose between a Σ03 and a Π03 -guessing procedure. These sets are almost Σ03 , there is only a subtle difference. As we noted above, all previous results on the fine hierarchy used arguments of complexity at most 0 . The reader will perhaps agree that, because of the dynamic nature of the above definition, the use of a 0 -construction of some sort is likely necessary in any proof of our main result below. Theorem. There exist a set in the class Σωω +2 which is not Turing equivalent to any set in Σωω +1 . The seemingly unavoidable complexity of guessing required to prove the theorem is very likely the main reason of why the problem has been open for so long. However, a very careful choice of the setup and a thoughtful use of selfreferential predicates on the tree of strategies allowed us to significantly reduce the combinatorial complexity of the guessing. But it is still a 0 proof, with true Π03 -outcomes and non-standard priority order on the nodes of the tree. We conjecture that our ideas can be pushed to cover all levels of the form Σωγ +2 . The hope is that we can potentially describe all proper levels of the fine hierarchy, but it will likely require a complicated iterated priority method, such as the ones introduced and developed by Lempp and Lerman [13], Ash [2], Marcone and Montalb´an [14], and Montalb´an [16]. We also hope that both the 4

hierarchy and our methods might find interesting applications in degree theory and computable structure theory. For example, is there a natural syntactical description of the subsets of a computable structure that are relatively intrinsically Σα , in the sense of [3]? 2. The main strategy As we noted above the Turing degrees of Σωω and Σωω +1 -sets coincide with the degrees of Σ03 -sets. Before we proceed to the strategy we give the formal definition of Σωω +2 -sets. Definition. A set S is Σωω +2 if S = A ∪ B ∪ C, where A is a Σ03 -set, B is a Π03 -set, and there exist c.e. sets E0 , E1 , E such that E0 ∩ E1 = ∅, E0 ∪ E1 ⊆ E, C = E\(E0 ∪ E1 ), A ⊆ E0 , and B ⊆ E1 . The naive basic idea of the strategy is obvious, but its implementation will not be straightforward. We shall exploit the extra freedom that a Σωω +2 -set has in the choice of the Σ03 vs. Π03 -guessing and try to “do the opposite” to what a given Σ03 -set D suggests on a given witness. The main difficulty will be related to the Σ03 -guessing on what D says. It has to be dynamic because we also need to build a c.e. partition of the universe that incorporate the Σ03 - vs. Π03 guessing pieces. In fact, one of the main roles of the tree of strategies will be merely in showing that the diagonalization was successful along the true path. Proving that the set that we constructed is indeed in Σωω +2 will require a global argument which takes into account the actions of all strategies, including those off the true path. It is not that easy to explain informally what exactly will have to be done. However, the diagonalization strategy is (relatively) straightforward. We therefore begin with a detailed description of the basic diagonalization strategy in isolation. We then discuss the situation of only two strategies, and then the case of several strategies. Only then we give the formal construction. An experienced reader will hopefully have no problem in seeing the general technical idea quite early in the proof. 2.1. The requirements We have to construct disjoint c.e. sets E0 , E1 , a Σ03 -set A ⊆ E0 , a Π03 -set B ⊆ E1 , and a c.e. set E such that (E0 ∪E1 ) ⊂ E, and ensure that S = A∪B∪C is not Turing equivalent to any Σ03 -set D, where C = E\(E0 ∪ E1 ). To prove the theorem, we need to satisfy the requirements Rk = RΦ,Ψ,D : A ∪ B ∪ C = ΦD ∨ D = ΨA∪B∪C , where Φ, Ψ, D = Φk , Ψk , Dk is the kth element in a fixed computable enumeration of all such triples; here Φk , Ψk are computable functionals and Dk are Σ03 -sets (represented by their Σ03 -indices) in some fixed uniform computable enumeration. 2.2. One strategy in isolation For simplicity, assume that we enumerate elements into C instead of E, having in mind that an element may later be extracted from C. Also, under this convention this means that whenever we extract an element from C, we 5

immediately put it either into E0 or into E1 . All these sets are under our control. A strategy for RΦ,Ψ,D . (1) Choose a fresh witness x and keep it out of S. As usual, “fresh” means that the number that we pick has never been seen in the construction so far. Keeping out of S means that we currently do not enumerate it into the c.e. set E.

(2) Wait for a stage s0 and a string τ of length < s0 such that Φτ (x)[s0 ] = (A ∪ B ∪ C)(x)[s0 ] = 0 ∧ ΨA∪B∪C  |τ |[s0 ] = τ. If ΦD (x) = (A ∪ B ∪ C)(x) ∧ ΨA∪B∪C  ϕD (x) = D  ϕD (x) is indeed the case then such a string must exist. Otherwise, if D does not satisfy the condition then the requirement will be met. Note that, in general, τ does not need to be an initial segment of D.

(3) Put x into C at stage s0 + 1. (Enumerate x into E.) In particular, we will have C(x)[s0 + 1] = 1 and C(x)[s0 ] = 0. Thus, Φτ (x)[s0 ] = 0 = (A ∪ B ∪ C)(x)[s0 ] and ΨA∪B∪C  |τ |[s0 ] = τ , but Φτ (x)[s0 + 1] = 0 = 1 = (A ∪ B ∪ C)(x)[s0 + 1]. Perhaps, D  |τ |[s0 ] = D  |τ | = τ but not necessarily so. Nonetheless, we make an attempt of diagonalization by changing (A ∪ B ∪ C)(x)[s]. We will see that this attempt will either succeed, or we will find a new string σ with a similar property but incomparable with τ . In the former case the requirement will be met.

(4) Wait for a stage s1 > s0 and a string σ of length < s1 such that Φσ (x)[s1 ] = (A ∪ B ∪ C)(x)[s1 ] = 1 ∧ ΨA∪B∪C  |σ|[s1 ] = σ. As above, either σ exists or the requirement is met. Let zx be least such that τ (zx ) = σ(zx ). Note that τ and σ must be incomparable, and ΨA∪B∪C (zx )[s0 ] ↓= τ (zx ) = ΨA∪B∪C (zx )[s1 ] ↓= σ(zx ). Thus, the value ΨA∪B∪C (zx ) is now under our control. We will use these two strings to attack against D at zx . Note that we have not yet used the Σ03 -guessing anywhere in the strategy.

(5) Fix zx least such that τ (zx ) = σ(zx ). Consider the two cases: (C1) τ (zx ) = 1, (C2) τ (zx ) = 0. – If (C1) holds, then enumerate x into E0 . (In particular, extract from C.) At the following stages s > s1 define A(x)[s] = D(zx )[s]. – If (C2) holds, then enumerate x into E1 . At the following stages s > s1 define B(x)[s] = 1 − D(zx )[s]. In (C1), the computation ΨA∪B∪C (zx )[s0 ] ↓= τ (zx ) = 1 halts under the assumption that x is not in our set A ∪ B ∪ C, while τ (zx ) = ΨA∪B∪C (zx )[s1 ] ↓= σ(zx ) = 0 halts under the assumption that x is in the set (recall we put x into C at stage s0 + 1). In particular, if we can ensure that zx ∈ D iff x ∈ A then the requirement will be met.

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Formally, we let A decide on the membership of x by enumerating x ∈ E0 . We will dynamically correct the Σ03 -definition of A(x) (which initially was set equal to 0) to mimic the Σ03 -procedure for D(zx ). This can be done with all possible uniformity. The

case (C2) is dual to (C1). In particular, we have ΨA∪B∪C (zx ) = ΨA∪B∪C (zx )[s0 ] ↓= τ (zx ) = 0 under the assumption that x is not in our set A ∪ B ∪ C. So we will make sure that zx ∈ D iff x ∈ / B. This is done by allowing B(x) to dynamically copy the Π03 -definition of D(zx ) from this stage on.

Finally, whenever the strategy finds a suitable Ψ-computation, it attempts to preserve the initial segment which was used in the computation, with the exception of x whose membership will be decided according to the instructions at (5) above. This ends the description of one strategy in isolation. 2.2.1. The outcomes The outcomes of one strategy in isolation are as follows. Σ: The outcome is played if τ and σ have been defined and τ (zx ) = 1. In this case we will produce a Σ03 -definition for the witness membership. Π : The outcome is played if τ and σ have been defined and τ (zx ) = 0. In this case we will produce a Π03 -definition for the witness membership. f in1 : The outcome is played if τ has not yet been defined. In this case the witness is currently kept outside of the set C. f in2 : The outcome is played if σ has not yet been defined. In this case the witness will be kept in C until a suitable σ is found. It is clear that one strategy in isolation ensures that the requirement is met in each of the four cases. Note that the outcomes above are finitary in nature, although the complexity of recognising whether the witness is in the set will be at the level of Σ03 or Π03 at worst. The outcomes and the strategy itself will have to be further modified to incorporate a Π03 -guessing. 3. An informal description of the construction 3.1. Combining two strategies Clearly, the main problem with two strategies, one working below the other, is that the weaker priority strategy will have to guess on the membership of the witness of the higher priority strategy. Remark 3.1. Before we start, we note that the recursion theorem will be used throughout (and typically without explicit reference) to justify some of the uniform changes in the tree and in the strategies. However, the recursion theorem it is not really absolutely necessary for this proof. Although various predicates will be frequently modified, it will be quite clear what these uniform modifications will be, so the s-m-n theorem will actually suffice, but that would require some thought. The recursion theorem makes this combinatorics implicit.

Suppose we have only two strategies, ρ and ρ , one working below the other, with ρ having a higher priority. According to its instructions, ρ does not really have to know anything about its Σ03 -set Dρ to search for strings τ and σ (see 7

the previous section). Recall that the Σ03 -definition of the set will be used only after some suitable σ and τ strings are found (if ever). Then, if successful in its search, the strategy will attempt to preserve an initial segment of the set. Nonetheless, S = A ∪ B ∪ C may be effected by the actions of the higher priority ρ which may have its witness xρ below the restraint of ρ . The definition of the membership relation for the witness xρ depends on the outcome of ρ and could be Π03 at worst. For instance, consider the case when ρ is below Σ of ρ. This means that ρ has found its strings and has adopted a Σ03 -definition for the relation xρ ∈ S; the index of the Σ03 -predicate is given to us dynamically in the construction. For ρ , seeing whether xρ ∈ S or not is a Σ03 vs. Π03 piece of information. To make the guessing comprehensible, we split the outcome Σ further into the suboutcomes ΣΠ2 ,0 < ΣΠ2 ,1 < . . . < ΣΠ2 ,n < . . . . . . < ΣΠ3 , / S, and ΣΠ2 ,n measuring the number of witnesses t with ΣΠ3 saying that x ∈ corresponding to n in x ∈ S ⇐⇒ ∃n ∃∞ t P (x, t, n), where P is the recursive kernel of the Σ03 membership predicate for x. As usual, in ∃n ∃∞ t P (x, t, n) we can assume that such an n, if it exists, is unique. We can uniformly transform the computable kernel of the Σ03 -predicate into one for which the uniqueness of ∃-witnesses holds. By uniformity, we can assume that the index for P is known by the recursion theorem. Similarly, we further split the outcome Π. The combined ordering will be: the Σ suboutcomes < the Π suboutcomes < f in2 < f in1 , where the ordering on the Σ and the Π suboutcomes has already been discussed. The tree of strategies will be defined in accordance with this ordering. Its formal definition will be given later, but it is rather standard. 3.1.1. The first naive attempt to coordinate nodes Naively, it may seem that all we have to do is to split the second strategy into different and independent versions ρ , ρ , . . . and associate these versions with different nodes below ρ. Of course, each version of the second strategy will have its own witness. For instance: • ρ below ΣΠ2 ,n believes that xρ ∈ S and will act accordingly when active, • ρ below ΣΠ3 believes that xρ ∈ / S and will act accordingly when active, and similarly for the other outcomes. As it is typical in such constructions, we intend to visit the Π02 -outcome ΣΠ2 ,n whenever the predicate “fires” again on n. The ΣΠ3 -outcome will be visited in-between the stages at which the Π02 outcomes are played. Without loss of generality, we will assume that for each n the predicate fires at least once. This in particular means that, once the strategy chooses to put its witness into E0 and unless it will be initialised, the ΣΠ3 -outcome will be played by the strategy infinitely often even if it is not the true outcome.

8

This naive set-up brings us to the following potential conflict. It could be the case that ρ below a Π03 -outcome of ρ, say ΣΠ3 , has already chosen its witness. The other strategy ρ under a Π02 outcome of ρ, say ΣΠ2 ,n , must be able to live with this because the Π03 -outcome will be visited infinitely often. Note that ρ may have a witness much smaller than the witness of ρ , thus potentially upsetting the restraint or the computation of ρ . Even worse, it seems that guessing on “xρ ∈ S” could be Π03 at worst. 3.1.2. Coordinating nodes in presence of only two requirements To circumvent the potential problem discussed above, the node ρ below ΣΠ2 ,n of the highest priority ρ will be associated with a state (related to ρ ) which is a symbol from the set {Σ, Π, f in2 , f in1 }. The state provides ρ with the information on the current outcome of ρ which is below the Π03 -outcome of ρ. We shall also adopt the following two uniform modifications which are dual to each other: Modification 3.2. The basic strategy associated with a node γ will be modified as follows. If the strategy chooses the Σ03 -attack and thus is playing its Σ/S outcome, then we uniformly replace the Π03 non-membership definition of xγ ∈ from the basic strategy for γ by the conjunction of this definition with the statement saying that γ is on the true path, and that the strategy will never be initialised again. The index of the new predicate is uniform in the index of the Σ03 -set measured by the strategy and the index of the construction (which is known by the Recursion Theorem). The Π03 -suboutcome ΣΠ3 of Σ will be measuring whether the new predicate holds, and the Π02 -suboutcomes ΣΠ2 ,n will therefore be measuring the Σ03 -negation of this new predicate. In particular, if γ eventually settles on Σ and is off the true path then xγ ∈ S. Modification 3.3. The basic strategy associated with a node γ will be modified as follows. If the strategy chooses the Π03 -attack and thus is playing its Πoutcome, then we uniformly replace the Π03 membership definition of xγ ∈ S from the basic strategy for γ by the conjunction of this definition with the statement saying that γ is on the true path, and that the strategy will never be initialised again. The Π03 -suboutcome ΠΠ3 of Π will be measuring whether the new predicate holds, and the Π02 -suboutcomes ΠΠ2 ,n will therefore be measuring the Σ03 -negation of this new predicate. In particular, if γ eventually settles on / S. Π and is off the true path then xγ ∈ Assuming the modifications, consider the following cases. In these cases ρ is below one of the Π02 -suboutcomes of the Σ-outcome of ρ, and ρ is below ΣΠ3 of ρ. In each case the state of ρ is associated with ρ . The idea is that ρ will believe that it is on the true path, and therefore it will assume that ρ is off the true path; the modifications above will allow ρ to use its state associated with ρ to guess about the status of xρ in a strictly finitary fashion, as follows. Case 1: ρ has state f in1 and thus believes that ρ will never find a suitable string τ (see the instructions of the basic strategy in isolation) and therefore / S. xρ ∈ Case 2: ρ has state f in2 and believes that ρ has found its τ , but will never find a suitable σ and therefore its witness will be permanently kept in S. 9

Case 3: ρ has state Σ and believes that ρ has found both σ and τ , and the membership of its witness xρ has become a Σ03 -fact. In this case ρ below ΣΠ2 ,n assumes that the witness of ρ is in the set S. Case 4: ρ has state Π and believes that the version of ρ below the Π03 -outcome has chosen a Π03 -definition for the membership of its witness. Then ρ below ΠΠ2 ,n assumes that the witness of ρ is not in the set. Note that the guess of ρ Cases 3 and 4 rely on Modifications 3.2 and 3.3 and on its belief that ρ is off the true path. The state of a strategy ρ is changed as soon as the outcome of ρ changes, and even if ρ is currently not being played at the stage. Also, the strategy ρ will be initialised as soon as it changes its state. It abandons its current witness and when it becomes active again (if ever) it restarts its basic module by picking a new, fresh and large witness. We will discuss the initialisation in detail later. We have explained how ρ below ΣΠ2 ,n of ρ can handle the witness of ρ below ΣΠ3 of the same ρ; the case of ρ below ΠΠ2 ,n and ρ below ΣΠ3 is identical. More generally, if ρ to the left of ρ but cannot initialise ρ then it will use its state to guess about xρ ∈ S. It is crucial that this guess is strictly finitary; that is, it does not require anything apart from the current state and the position of the nodes in the tree. The state can change only at most twice. Nonetheless, even in presence of only two requirements and only three nodes ρ, ρ , ρ there could be further tensions. For example, when ρ is to the left of ρ then ρ will likely have to guess whether xρ ∈ S. We could allow ρ initialise ρ if the witness of ρ interferes with the strategy of ρ ; the membership relation on the abandoned witness x of ρ will be decided by ρ and will be chosen in a way that makes the computations of ρ halt; i.e., if ρ needs x ∈ S to define τρ then it declares x ∈ S. In other words, ρ takes the control over x while ρ will later choose its new witness xρ fresh and large if necessary. One crucial observation here is that it is not really important that ρ is to the left of ρ or vice versa. As long as we decide which of the two has a stronger priority, either way it will work. Indeed, initialisation is not related to any complex effects coming from the tree or from the Π03 -guessing. Our next task is to generalise these ideas to the case of several requirements. 3.2. Several requirements In presence of many requirements we need to cope with further tensions that were not visible in the case of only two requirements. For example, a strategy will have to monitor several witnesses of other strategies, but it cannot possibly handle infinitely many such witnesses. 3.2.1. Priority To resolve the above mentioned difficulty and some other problems, we introduce a priority order on the nodes in the tree, as follows. Definition 3.4. Introduce an effective priority order, of order-type ω, on all the nodes in the tree. The order must satisfy the property: if ρ is a prefix of ρ then the priority of ρ is stronger than that of ρ.

10

It will not really matter how exactly we define the priority order, as long as the prefix condition is satisfied. For example, it could be consistent with the natural enumeration of all strings in the tree; a formal definition will be given later. 3.2.2. States Every node ρ on the tree will have at most finitely many nodes of higher priority. Suppose there are exactly m such nodes. Each node ρ will be associated with a state which is an m-tuple of symbols Σ, Π, f in2 , f in1 . The ith symbol in the tuple reflects the most recent outcome of the ith node of higher priority. Note that the Σ and Π outcomes of these higher priority nodes will not be subdivided into sub-outcomes when listed in a state. A current state of ρ allows it to have the best available guess about the witnesses of at most m higher priority nodes in the spirit of 3.1.2 which are not prefixes of ρ. More specifically, those higher priority nodes that are playing Π and f in1 will be assumed to have their witnesses out, and for those which have recently been playing Σ and f in2 their witnesses will assumed to be in. The current witnesses of weaker priority strategies will be assumed to be out or in the set, depending what makes the computation of the higher priority strategy halt. The stronger priority strategy will be searching for a suitable initial segment of the set S that makes its computation halt. This initial segment may contain witnesses xρ for lots of weaker ρ . It is too hard to guess on the membership status of xρ , there are too many of such weaker strategies. Therefore, the stronger priority strategy ρ will be searching for some computation assuming that the membership status of such xρ is up to the stronger strategy ρ to decide, if necessary. As soon as a new suitable computation is found by ρ (if ever) the weaker priority nodes will be instantly initialised and their former witnesses will be handled by ρ. If later ρ is itself initialised in a similar fashion by a higher priority ρ0 then these abandoned witnesses will be handled by ρ0 , etc. We give a bit more detail. Write α ≺ β if the priority of α is weaker than the priority of β, and we write rα to denote the restraint of α meaning that it currently attempts to preserve the interval [0, rα ]. Two nodes, ρ and ρ , are said to have a conflict at a stage if one of the two has a witness below the restraint of the other. Note it does not have to be known whether the witness is in or out of S. Consider the cases: (C1): α ≺ β and xβ ≤ rα but β is not an initial segment of α. Then α will use its state to have a guess about xβ ∈ S. (C2): α ≺ β , xβ ≤ rα , and β is an initial segment of α. Then the guess of α will be consistent with the outcome of β above α; more specifically: – If βf in1 ⊆ α then α will assume xβ ∈ / S. – If βf in2 ⊆ α then α will assume xβ ∈ S. – If βΣΠ2 ,n ⊆ α then α will assume xβ ∈ S. / S. – If βΣΠ3 ⊆ α then α will assume xβ ∈ / S. – If βΠΠ2 ,n ⊆ α then α will assume xβ ∈ 11

– If βΠΠ3 ⊆ α then α will assume xβ ∈ S. (C3): α ≺ β and xα ≤ rβ . Initialise the weaker priority α in this case. We will initialise the node whenever it changes its state; this will also cover case (C3). As before, the state can be changed even if the node is not currently being played. Remark 3.5. Note that the notion of initialisation above is somewhat suspicious, because we did not initialise anything to the right of the current true path in the spirit of the standard infinite injury proof. We used only the states and conflicts, all being finitary in their nature. Quite interestingly, we will show that this degenerate initialisation will be sufficient.

3.2.3. Abandoned witnesses We must clarify what happens with a witnesses after it is abandoned by its strategy. Let x = xρ be such a witness. After it is abandoned by ρ, the old witness may be put back to the set and then extracted, and then put back again, etc., in the Δ02 -manner. This process will be regulated by strategies having higher priority than ρ, as follows. When a higher priority strategy ρ is searching for a halting computation it decides on the membership status of x and assumes it is either in or out of the set, according to what makes the computation halt. The membership status of x is then preserved with the priority of ρ . However, ρ may itself be later initialised by a higher priority node ρ , and in this case the membership status of x may be changed and preserved with priority ρ , etc. We will see that every strategy on the tree will be initialised at most finitely many times, and therefore this process will eventually terminate, and the membership of x will be finally stably defined. As we will explain later in the construction, this behavior of the abandoned witnesses can be formalised in terms of sets A and B; in other words, we will show that the set that we construct is in the right complexity class. This finishes the intuitive description of some elements of the construction. We are ready to give the formal construction and its verification. 4. Construction 4.1. The tree of strategies The tree of strategies T will be the infinitely branching tree {ΣΠ2 ,n , ΣΠ3 , ΠΠ2 ,n , ΠΠ3 , f in1 , f in2 : n ∈ ω}<ω , in which the strings are partially ordered lexicographically based on the order ΣΠ2 ,0 < . . . < ΣΠ2 ,n < . . . < ΣΠ3 < ΠΠ2 ,0 < . . . < ΠΠ2 ,n < . . . < ΠΠ3 < f in2 < f in1 . The set of strings (nodes) of the tree T clearly has a computable enumeration in which α  β appears earlier than β. We fix any such enumeration of nodes. Let the index of a string α be the stage at which α appears in this enumeration. We say that α has a weaker priority than β, written α ≺ β, if the index of β in the enumeration is smaller than the index of α (cf. 3.2.1). Also, suppose ρ has exactly mρ different β such that ρ ≺ β. We shall associate ρ with its state which, in general, will be an mρ -tuple of symbols Σ, Π, f in2 , and f in1 . Initially, when T is defined and before the construction begins, we let the state of each ρ be equal to the mρ -tuple consisting entirely of f in1 . 12

4.2. The strategy associated with ρ Suppose ρ is a node (string) of T . Then ρ will be associated with a modified version of the basic strategy for the |ρ|th requirement; see Section 2. The basic strategy relies on the following assumptions in all its computations: (1) If ρ  ρ but ρ is not an initial segment of ρ, then the relation xρ ∈ S will be guessed by ρ based on the respective parameter in the state of ρ. More specifically, if the respective parameter is Σ or f in2 then ρ will assume / S. xρ ∈ S, and otherwise (if it is Π or f in1 ) ρ will assume xρ ∈ (2) If ρ is an initial segment of ρ, then xρ ∈ S will be guessed naturally based on the outcome of ρ above ρ: • If ρf in1 ⊆ ρ then ρ will assume xρ ∈ / S. • If ρf in2 ⊆ ρ then ρ will assume xρ ∈ S. • If ρΣΠ2 ,n ⊆ ρ then ρ will assume xρ ∈ S. • If ρΣΠ3 ⊆ ρ then ρ will assume xρ ∈ / S. • If ρΠΠ2 ,n ⊆ ρ then ρ will assume xρ ∈ / S. • If ρΠΠ3 ⊆ ρ then ρ will assume xρ ∈ S. (3) If ρ ≺ ρ and ρ has already picked its witness xρ or is controlling several witnesses of some even weaker priority strategies, then ρ will declare that these witness(es) in or out of S, depending on what makes the desired computation of the stronger ρ halt; of course, only if such a potential computation is discovered. (If ρ has no specific preference on the membership relation for such a witness, then it will declare such a witness to be not in S.) From this point on, the old witness of ρ and all other abandoned witnesses previously preserved with priority ρ will now be preserved with priority of ρ; only some strategy having its priority even higher than ρ will be allowed to change their membership status. The actions of (the strategy for) ρ will be similar to those discussed in Section 2, but based on the guesses above and with the following modifications below (cf. 3.2 and 3.3). (a) When the strategy ρ searches for its strings σρ and later τρ , it can redefine the membership status of any witness whose membership status is controlled by a weaker priority strategy; this also applies to the abandoned witnesses which are preserved with priority weaker than the priority of ρ (see also (3) above). This new membership definition will then be preserved with priority of ρ from further change. (b) Suppose the strategy chooses to play Π. Then replace the original Π03 definition for xρ ∈ S from Section 2 with the conjunction of the original definition with the predicate saying that ρ is on the true path. (c) Suppose the strategy chooses to play Σ. Then replace the naive original / S with the conjunction of the original definition Π03 -definition for xρ ∈ with the predicate saying that ρ is on the true path. (d) In both Σ and Π cases we further uniformly adjust the recursive kernel of the new Π03 -predicate so that the resulting predicate has the unique existential witness property. (In particular, there is exactly one Π02 -witness in the Σ03 -case.) Also, each approximated Π02 predicate will be viewed as a predicate of the form ∃∞ nR(x, n), and we further dynamically and uniformly adjust its kernel so that for each x there exists at least one n such that R(x, n) holds. 13

4.3. The current true path In the previous two sections we have already explained how each of the outcomes of a strategy ρ is played. We give a brief summary: (1) The outcomes f in1 and f in2 are finitary and are played when ρ waits for τρ and for σρ , respectively (see 2.2.1). (2) The Σ and Π outcomes are played only if both σρ and τρ have been found and ρ has chosen to adopt either a Σ03 - or a Π03 -definition for its witness xρ (respectively). In the former case only the Σ-suboutcomes of ρ will ever be played, and in the latter case only its Π-suboutcomes will be played, unless ρ is initalised. Suppose ρ plays its Σ outcomes. This is done as follows: • ΣΠ2 ,n is played if the nth column of the measured predicate fires again. • The ΣΠ3 is played in-between the stages at which the respective ΣΠ2 ,n -outcomes are played. We also assume that for every n the Π02 -predicate fires at least once. The case of Π is the same, up to a change of notation. The definition of the current true path is standard. Simultaneously with the current true path and for every node in the tree ρ not necessarily along the current true path, define the state of ρ to be the tuple of the most recently seen outcomes of the higher priority strategies; if a strategy has not yet been played or has recently been initialised then put f in1 at the respective position of the tuple. 4.4. Initialisation When a node is initialised it abandons its current witness. At the next stage at which it becomes active again (if ever) it will pick a new fresh and very large witness and restarts its basic strategy with this new witness. Initialise a node ρ whenever it changes its state. If α is the highest priority strategy because of which the state of ρ had to be changed, then the membership of the previous witness of ρ and of all other abandoned witnesses currently controlled by ρ will now be decided in the construction by α, and will be restrained with the priority of α. As noted before, the state of ρ can be changed even if ρ is currently not being played. 4.5. Construction At stage 0 initialise all strategies. At stage s, let the strategies along the current true path act according to their (modified) instructions. We also keep updating the membership predicates for all witnesses, including those which were abandoned or which belong to nodes that are not currently active, according to the instructions of the strategies which currently control these witnesses. If an abandoned witness z of some ρ is currently not below the restraint of any strategy ρ having its priority higher than ρ, then z will be declared out of the set S.

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5. Verification First, we will show that all the diagonalization requirements are met, and then we argue that the set S that we have constructed belongs to the desired class of the fine hierarchy. The set S will consist of all witnesses of strategies, including abandoned witnesses, which will be eventually declared to be in the set. The true path is the left-most path visited infinitely often. Clearly, the top node is on the true path. We need to check that the true path is infinite. The below lemma is central to the verification. Lemma 5.1. 1. Every node in the tree is initialised at most finitely many times in the construction. 2. Every node in the tree has its state eventually stable. Proof. We prove the lemma by induction on priority. The root of the tree satisfies the lemma. It has the highest priority, and its state is the empty string. Thus, the strategy will never be initialised after stage 0. It will follow its instructions and will eventually permanently settle on either Σ or Π or f in2 or f in1 ; in this lemma we do not need to consider the sub-outcomes of the former two outcomes. Now suppose we have proven the lemma for the first n nodes having the highest priority. Let ρ be the next highest priority node. There is a stage after which the first n nodes forever settle on either Σ or Π or f ini , i = 1, 2. This means that they have computed their strings (if they could) and have their final witnesses in place, and their restraints will never be changed again. It follows that after this stage ρ cannot be initialised. After ρ becomes active again (if ever), it will choose its witness larger than the restraints of all strategies having higher priority. Note that ρ will be able to control the membership relation for all other witnesses in the construction, including abandoned ones, which are not restrained by the finitely many higher priority strategies. Thus, it will follow its instructions without any further interruption, and it will eventually settle on one of its outcomes Σ or Π or f ini , i = 1, 2. Lemma 5.2. The true path is infinite. Proof. We prove the lemma by induction on the length of the true path δ. The root of the tree is clearly on the true path. We show that the true path has length at least n + 1, where n ≥ 1. By the previous lemma, we can fix a stage after which the first n nodes on the true path will eventually be never initialised after the stage. Let ρ be the n-th node. The true outcome of ρ is the outcome that correctly reflects the outcome of the guessing performed by the strategy ρ. Since the strategy will never be initialised, no witness of any other strategy (including abandoned witnesses) can further interfere with the instructions of ρ. The strategy will respect the restrains of the finitely many higher priority strategies, which we also assume to have settled. If ρ never finds a string τ then it will permanently settle at / S. If it will find τρ but will never define f in1 and will permanently declare xρ ∈ σρ , then ρ will be permanently playing f in2 whenever visited. In each of these cases ρf ini , for the respective i, will be on the true path.

15

Otherwise, assume that both σρ and τρ are eventually found by ρ. Then the definitions of σρ and τρ are permanent, and ρ will settle on either Π or Σ; however, recall that these outcomes were subdivided into suboutcomes. Regardless of what exactly the suboutcomes measure, they split the respective predicate into logically exclusive cases which cover all possibilities. Therefore, either one of the Π02 -suboutcomes will be visited infinitely often, and in this case there will be exactly one such, or the Π03 -suboutcome will be the left-most outcome of ρ visited infinitely often. If o is this left-most (sub)outcome of ρ visited infinitely often, then ρo is also on the true path. Lemma 5.3. Every diagonalisation requirement is met along the true path. Proof. Recall that the original predicate for the membership of the witness xρ had to be modified. Nonetheless, provided that ρ is on the true path, the modifications described in 4.2 lead to logically equivalent predicate. All we have to check is that the computation of ρ is correct, in the sense that its guess about the initial segment of the set is actually correct. We use by Lemma 5.1 and Lemma 5.2 throughout. Assume s is large enough, so that ρ is never initialised after stage s. The strategy of ρ relies on knowledge about various witnesses, including abandoned witnesses, which may or may not be in S. Suppose ρ is some other strategy. We must argue that ρ will have a correct guess about xρ ∈ S after stage s. It must also have a correct guess about the membership of all abandoned witnesses which could be controlled by ρ . The proof proceeds by cases on the position of ρ with respect to ρ: (1) ρ is a prefix of ρ. Then the guess about xρ is provided by the true path. If ρ additionally controls several abandoned witnesses, then their current membership status will never be changed again since ρ will never act again and will never be initialised again. By induction, all these guesses provided to ρ by ρ will be correct. (2) ρ has a weaker priority than ρ. Then ρ ignores the witness of ρ and assumes it is in or out, depending on what suits ρ better. The same applies to the abandoned witnesses currently controlled by ρ , if there are any. But ρ will be initialised if a suitable computation is found by ρ, and all its witnesses will be handled by ρ after the stage. It will be up to ρ to decide what happens with the witnesses previously controlled by ρ ; the decision will then be preserved with priority ρ. By the choice of stage s, this preservation will be permanent, and therefore ρ will have the correct guess about the membership status of all these witnesses. (3) ρ has a stronger priority than ρ but is off the true path. Then after stage s the strategy ρ will never be initialised. It will either never be played ever again, or it will eventually keep playing one of its outcomes Σ, Π, f in1 or f in2 . In either case the state of ρ which records the last played outcome of ρ will never change again. Suppose ρ is off the true path and is eventually never played again in the construction. If the last outcome played by ρ was either f in1 or f in2 then the state of ρ will also have the respective f ini in the position corresponding to ρ . In either case the state will provide ρ with the correct guess about xρ ∈ S. If either Σ or Π was the outcome which was last played by ρ , then the definition of xρ has been declared according 16

to the Σ03 - or the Π03 -attack of ρ . Although ρ will never be visited again, the index describing the Σ03 or the Π03 -predicate xρ ∈ S has been already fixed in the construction. It is crucial that ρ is off the true path and this predicate reflects it. Because of the modifications to the predicate, the state provides ρ with the correct guess about xρ ∈ S. Recall that ρ could also control several witnesses which had been abandoned by weaker priority strategies. Then ρ will decide their membership in a strictly finitary fashion at the stage, i.e. without any reference to some complex Π03 or Σ03 -predicate. This membership status will be permanently preserved with priority ρ after stage s, and thus it will be correctly guessed by ρ. Similarly, if ρ is off the true path but is played infinitely often, then after stage s the state of ρ will provide it with the correct guess about the true outcome of ρ . As before, the case of f ini (i = 1, 2) is elementary, and when the true outcome of ρ is either Π or Σ then the definition of xρ ∈ S will code the fact that ρ is off the true path. The abandoned witnesses permanently controlled by ρ will have their status decided in a finitary fashion and will never change it again after stage s. It follows that ρ can safely act according to its instructions, and its computations will eventually be correct and final. We now argue that the basic module of ρ ensures that the requirement R|ρ| is met after stage s. We have just checked that its guess about various witnesses controlled by higher priority strategies will be correct; there will be only finitely many such witnesses. As for the other witnesses, when ρ is searching for τρ it does not make any assumptions about their membership, and it searches for any halting computation on a segment extending the segment preserved by the higher priority strategies. Note that ρ will have a correct guess about this segment after stage s. Once such a computation is found, the membership status of the weaker priority witnesses will be decided so that this computation halts, and the use of this computation will be preserved by ρ. When ρ searches for its σρ , it will again be allowed to decide the membership of any weaker priority witness in the construction, and it will search for any computation whose use extends the currently preserved part of S. Again, when such a computation is found (if ever), it will perhaps mention at most finitely many weaker priority witnesses which are not already used in the computation of τρ . The strategy will then decide whether it wants these finitely many witnesses having priority  ρ to be in or out of the set, depending on what makes the computation of σρ halt. It will then preserve the use of this computation too (with the exception of xρ ); this preservation will be permanent. In particular, if no potential computation of σρ or τρ is found then, regardless on what will happen in the construction, the requirement will be met. Indeed, the membership status of the higher priority witnesses will never change again, and the computation does not assume anything about the weaker priority witnesses. Otherwise, if these two computations are eventually found by ρ, the requirement will be met because the initial segment of S necessary for diagonalization will be permanently preserved, and the strategy will achieve its goal by controlling xρ ∈ S.

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It remains to check that the set S belongs to Σωω +2 . This means that we must define: • disjoint c.e. sets E0 , E1 , • a Σ03 -set A ⊆ E0 , • a Π03 -set B ⊆ E1 , and • a c.e. set E such that (E0 ∪ E1 ) ⊂ E. We will then define S to be A ∪ B ∪ [E \ (E0 ∪ E1 )]. The interpretations of these sets are informally described in the introduction; the diagram in the introduction may help the reader to understand the rest of the verification. In particular, recall that: • if x ∈ E \ (E0 ∪ E1 ) then we have put x into the set; • if x is put into E0 then we have chosen a Σ03 -definition for “x ∈ S”; • if x is put into E1 then we have chosen a Π03 -definition for “x ∈ S”. Only numbers which were once picked as a witness by some strategy can possibly enter S. We must analyse the life cycle of each such witness beginning with the stage at which it was first introduced in the construction. Suppose x was first picked by ρ and initially was kept out of E. Then ρ may later enumerate x into E according to its instructions at (3) of its basic module. Otherwise, ρ could be initialised even before it puts x into E. In this case we immediately put x into both E and E0 , thus we promise to provide a uniform Σ03 -definition of x ∈ S by deciding if x ∈ A ⊆ E0 . This definition will be in fact merely uniformly Δ02 and will describe what happens with x if it is abandoned by ρ; see the description of I below. We shall return to this case later. For now, suppose x enumerated x into E. If ρ gets initialised before it puts x into either E0 or E1 according to its instructions, then we again immediately put x into E0 and will use the uniform Δ02 -description of what will happen to x to decide whether x ∈ A or not. Finally, suppose x is enumerated to E0 or E1 according to the instructions of ρ, and therefore the membership of x will have to be decided using a uniform Σ03 - or a uniform Π03 -procedure, respectively. The definition of the procedure is dynamic. Initially, it starts with imitating the procedure whose index is suggested by the instructions of the basic module of ρ. However, if at a later stage ρ is initialised in the construction the procedure will be adjusted to describe what happens with x in case it is abandoned by ρ. Equivalently, the Σ03 - or Π03 predicate in the instructions of ρ will have to be replaced with another predicate whose index can be assumed known ahead of time by the recursion theorem; we sketch the definition of this predicate below. Without loss of generality, suppose ρ declared x ∈ E0 , and the intended Σ03 -definition for x ∈ S initially suggested in the description of the module for ρ has the form x ∈ A ⇐⇒ ∃z∃∞ yL(z, y, x), where the index of the recursive L is known. To obtain the actual definition of x ∈ A, replace this naive definition of x ∈ A by the predicate which says: 18

• “there is no stage s at which ρ abandons x” and ∃z∃∞ yL(z, y, x); or • “there is a stage s at which ρ abandons x” and I(x), where I(x) is a uniformly Δ02 predicate which will be described below; it reflects what happens with x if it gets abandoned by ρ in the construction. Note that, assuming I has the desired properties, the overall complexity is Σ03 , with all possible uniformity allowing the use of the recursion theorem. Using this straightforward technique we can also clarify the case in which x was declared an element of E1 , and also the above-mentioned cases when ρ was initialised too early, i.e., before it had a chance to put x to either E0 or E1 . It remains to give a uniform description of the life cycle of x in the case if it is abandoned by ρ in the construction. Once x is abandoned its membership status becomes explicit at every stage of the construction. Some strategy of priority higher than the priority of ρ will explicitly declare either x ∈ S or x∈ / S, depending on what makes its computation halt. It may later be changed by an even higher priority strategy, etc. This process is fully uniform and can be described using the Limit Lemma by a (uniformly) Σ02 -predicate. Since the are only finitely many strategies having their priorities higher than the priority of ρ, there will be a stage after which none of these strategies will be ever initialised in the construction. After this stage no other strategy will have its priority high enough to change the membership status of x, and thus x will remain in or out of S, depending on what its status will be at that stage. We conclude that S belongs to the right difference class, and therefore the theorem is proved. References [1] Marat M. Arslanov and Iskander Sh. Kalimullin. A survey of results on the d-c.e. and n-c.e. degrees. In Computability and complexity, volume 10010 of Lecture Notes in Comput. Sci., pages 469–478. Springer, Cham, 2017. [2] C. Ash. Recursive labeling systems and stability of recursive structures in hyperarithmetical degrees. Trans. Amer. Math. Soc., 298:497–514, 1986. [3] C. Ash and J. Knight. Computable structures and the hyperarithmetical hierarchy, volume 144 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 2000. [4] Barry Cooper. Degrees of unsolvability. PhD thesis, Leicester University, Leicester, England, 1971. [5] Rod Downey and Noam Greenberg. A transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability. preprint, 2018. [6] Richard L. Epstein, Richard Haas, and Richard L. Kramer. Hierarchies of sets and degrees below 0 . In Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80), volume 859 of Lecture Notes in Math., pages 32–48. Springer, Berlin, 1981. [7] Yu. L. Ershov. A certain hierarchy of sets. I. Algebra i Logika, 7(1):47–74, 1968. 19

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