Bushy Trees

I. 3.1 (Partial) Recursive Functions and Sets

VI.1 Bushy Trees

order functionq∶N→ (1,∞)such that q○uis slow-growing and for which

Avoid^C1(p₁) ≰_wAvoid^ψ1(q) and Avoid^C2(q○u) ≰_wAvoid^ψ2(p₂).

Theorem VI.2.1 is then an easy consequence of Theorem VI.3.7.

In Section VI.4, we use Theorem VI.2.1 to deduce the following implications concerning LUAslow. Theorem VI.4.1. LUA_slow is not of deep degree.

Theorem VI.4.2. There is no order function q∶N→ (1,∞)such thatLUAslow≡wLUA(q).

Theorem VI.4.3. SC≰wLUAslow.

In Section VI.5, we prove the following variant of Theorem VI.2.2 whereqbeing slow-growing is strength- ened to Avoid^ψ2(q○u)being of deep degree.

Theorem VI.5.1. Suppose p1∶N→ (1,∞) and p2∶N→ (1,∞) are order functions, u∶N→ N is a strictly increasing order function, ψ₁ and ψ₂ are universal partial recursive functions, and ψ₃ and ψ₄ are partial recursive functions. Then there exists a order function q∶N→ (1,∞) such that Avoid^ψ2(q○u) is of deep degree and

Avoid^ψ1(p1) ≰wAvoid^ψ3(q) and Avoid^ψ2(q○u) ≰wAvoid^ψ4(p2).

IfB is notn-big aboveσ, thenB is said to ben-small aboveσ.

Our arguments are based on the idea that there are ‘bad’ sets of strings which we wish to avoid. If we can ensure that those ‘bad’ sets of strings are sufficiently small, then we can construct a realX ∈N^Nnone of whose initial segments lie in those ‘bad’ sets.

In [16] several fundamental combinatorial lemmas are identified which are reproduced below, sometimes with minor modifications suited to our needs.

Lemma VI.1.3 (Concatenation Property). [16, Lemma 2.6]SupposeA⊆N^∗ isn-big above σand⟨Aτ⟩τ∈A

is a family of subsets ofN^∗ indexed byA. IfAτ isn-big aboveτ for everyτ∈A, then⋃τ∈TAτ isn-big above σ.

Proof. For eachτ∈A, letT_τ be a finiten-bushy tree aboveτ all of whose leaves lie inA_τ. LetT be a finite n-bushy tree above σall of whose leaves lie in A. Define ˆT to be the tree obtained by taking the union of T with the treesT_τ whereτ is a leaf ofT in A. We claim that ˆT isn-bushy aboveσ. Because every string in ˆT extends an element of T, it follows that every string in ˆT extends σ. Now suppose ρ is a string in ˆT extendingσ and which is not a leaf. We consider three cases:

Case 1: Ifρis a member ofT and not a leaf of T, then the fact thatT isn-bushy aboveσimplies it has at leastnimmediate extensions in T⊆Tˆ.

Case 2: If ρ is a leaf of T, then ρ=τ for some τ ∈A; τ extends itself (improperly), so Tτ being n-bushy above τ implies it is either a leaf ofTτ (and hence of ˆT) or it has at leastn immediate extensions in T_τ⊆Tˆ.

Case 3: Ifρis not a member ofT, thenρ∈T_τ for someτ∈A. The argument then follows exactly as in Case 2.

It only remains to show that the leaves of ˆT lie in⋃τ∈AAτ. Indeed, the leaves of ˆT are exactly the leaves of Aτ for allτ∈A, which each lie inAτ, respectively, and hence in⋃τ∈AAτ.

Lemma VI.1.4(Smallness Preservation Property). [16, Lemma 2.7]Suppose thatB, C⊆N^∗,m, n∈N, and σ∈N^∗. IfB ism-small aboveσ andC isn-small aboveσ, thenB∪C is(n+m−1)-small above σ.

Proof. Suppose for the sake of a contradiction thatB∪Cis not(n+m−1)-small aboveσ, i.e.,(n+m−1)-big aboveσ. Then there exists a finite treeT which is(n+m−1)-bushy aboveσall of whose leaves lie inB∪C.

We will label each element of T by either a ‘B’ or a ‘C’, starting with leaves and working our way towards the stemσ. Label a leaf ofT ‘B’ if it lies in B and ‘C’ otherwise. If an extension τ ofσ inT has not been

labeled but all of its proper extensions have been, then label τ ‘B’ if at leastm of its immediate successors have ‘B’ label, and ‘C’ otherwise (by the pigeonhole principle, there must be at leastnimmediate successors labeled ‘C’). Continue in this way untilσitself has been labeled.

If σ has been labeled ‘B’, then the set T_B of all extensions of σ(along with the initial segments of σ) labeled ‘B’ is a finite tree which ism-bushy aboveσ. Otherwise, the setT_Cof all extensions ofσ(along with the initial segments ofσ) labeled ‘C’ isn-bushy aboveσ. In either case, we reach a contradiction.

Lemma VI.1.5(Small Set Closure Property). [16, Lemma 2.8, essentially]SupposeB⊆N^∗isk-small above σ. LetC= {τ∈N^∗∣B isk-big aboveτ}. ThenCisk-small aboveσand is k-closed, i.e., ifC isk-big above a string ρ, thenρ∈C.

Moreover, the upward closure of C isk-small above σandk-closed.

Proof. Suppose for the sake of a contradiction thatC isk-big aboveσ. Then, by Lemma VI.1.3,B isk-big aboveσ, yielding a contradiction.

The same reasoning can be applied to show that if C isk-big above a stringρ, then B isk-big aboveρ and henceρ∈C.

Now consider the upward closure C^↑∶= {ρ∈ {0,1}^∗ ∣ ∃τ ∈C(τ ⊆ρ)} of C. The following lemma shows thatC^↑ is similarlyk-small above σandk-closed.

Lemma VI.1.6. SupposeB⊆N^∗ andσ∈N^∗. Then B isk-big aboveσ if and only if its upward closureB^↑ isk-big above σ.

Proof. IfBisk-big aboveσ, then any finitek-bushy treeT aboveσrealizing this also shows thatB^↑isk-big aboveσ.

Conversely, suppose B^↑ is k-big above σ, and let T be a finite k-bushy tree above σ whose leaves are withinB^↑. Let

T˜= {τ∈T ∣τ∈B^↑∧τ↾ (∣τ∣ −1) ∉B^↑}.

T˜ is a tree, as if τ ∈T and σ⊂−−τ, then σis an element of T∖B^↑, andB^↑ being upward closed implies no initial segment ofσis inB^↑, so σ∈T.˜

Definition VI.1.7 (k-Closure). IfB ⊆N^∗ is k-small above σ, then its k-closure is the upward closure of the set{τ∈N^∗∣B isk-big aboveτ}.

In addition to the above lemmas, we also collect a series of facts which either follow quickly from those above lemmas or else follow immediately from the definitions.

Lemma VI.1.8. Supposeσ∈N^∗ andB, C⊆N^∗ are given.

(a) IfB is `-big aboveσ andk<`, thenB isk-small aboveσ.

(b) IfB is k-small aboveσ andk<`, thenB is`-small aboveσ.

(c) IfB=B1∪B2∪ ⋯ ∪Bn isn⋅k-big aboveσandn, k>0, then there existsi∈ {1,2, . . . , n} such that Bi

isk-big aboveσ.

(d) IfB⊆C andB isk-big above σ, thenC isk-big aboveσ.

(e) IfB⊆C andC isk-small above σ, thenB isk-small aboveσ.

(f ) IfB isk-small aboveσandk-closed andCisk-big aboveσ, then there exists aτ∈C∖B which extends σ.

Proof.

(a) IfT is a finite tree which is`-bushy aboveσand all of whose leaves lie inB, then T isk-bushy above σ. Thus,B isk-big aboveσ.

(b) IfB is`-big aboveσ, then (a) above shows thatB isk-big aboveσ, a contradiction.

(c) Suppose for the sake of a contradiction that Bi is k-small above σ for every i ∈ {1,2, . . . , n}. By repeated applications of Lemma VI.1.4 we find that B isn⋅k− (n−1) = (n⋅ (k−1) +1)-small above σ. n⋅ (k−1) +1<n⋅k, so (a) above gives a contradiction.

(d) A finitek-bushy tree aboveσwhose leaves are in B is a finite k-bushy tree aboveσ whose leaves are in C⊇B.

(e) IfB isk-big aboveσ, then (d) above impliesC isk-big aboveσ, a contradiction.

(f) Suppose for the sake of a contradiction that there is noτ ∈C∖B extendingσ. Because C is k-big above σ, there exists a k-bushy treeT aboveσ all of whose leaves lie inC. But every leaf ofT is an extension of σ in C, which by hypothesis implies it lies in B, so T is a k-bushy tree above σ all of whose leaves lie in B, contradicting the hypothesis thatB isk-small aboveσ.