Quantifying the Reduction of Avoidance to Complexity – General Case

I. 3.1 (Partial) Recursive Functions and Sets

IV.4 Quantifying the Reduction of Avoidance to Complexity – General Case

Remark IV.3.7. We assumed without loss of generality that mσ = 2⋅#(σ) by choosing an appropriate enumeration of the partial recursive functions. In general, ifθ∶N→Nis a total recursive, injective function with recursive coinfinite image (as in the case of n↦ 2n), then for any total recursive function g∶N→ N there is an admissible enumeration ˜ϕ₀,ϕ˜₁, . . .such that ˜ϕ_θ(e)≃ϕ_g(e) for alle∈N.

h(n) ∶=exp₂(s(n+1) ⋅ (n+1) −s(n) ⋅n).

If there ares∶N→ [1,∞)and rationalε>0 such that (i)imh⊆N, (ii)lim_n→∞s(n+1)/s(n) =1, and (iii)˜j is an order function, then COMPLEX(f) ≤_w LUA(q) for any order function q∶N → [0,∞) such that, for almost alln∈N,

q(exp₂((1−ε)⁻¹⋅ [s(n+1) ⋅ (n+1) −j(s(n+1) ⋅ (n+1))] ⋅`(n))) ≤`(n).

For the remainder of this subsection, we use the following notation:

Notation IV.4.2. kand`denote recursive functionsN→ [1,∞). h,K,L, andH are defined by, forn∈N, h(n) ∶=k(n) ⋅`(n)and

H(n) ∶=h(0) ⋅h(1) ⋯h(n−1), K(n) ∶=k(0) ⋅k(1) ⋯k(n−1), L(n) ∶=`(0) ⋅`(1) ⋯`(n−1).

It is most convenient forhto take image inNso that∣hⁿ∣ =H(n), so we make the following conventions:

Convention IV.4.3. kand`will always be such thathis an order function of the formN→N. As a result,

∣hⁿ∣ =H(n) =K(n) ⋅L(n).

Additionally, ifα∈ (1,∞), then DNR(α) = {X∈N^N∣ ∀n(X(n) <α∧X(n) ≄ϕn(n))} =DNR(⌈α⌉).

Notation IV.4.4. d^his a fixed universal left r.e. supermartingale on h^N.

Among the explicit uses ofhin the proof of Theorem IV.3.1, Uses I, II, and III only depended on writing hin the formh(n) =k(n) ⋅`(n) =2ⁿ⋅ (n+1)and observing that ifd^h(σ) ≤n! for someσ∈hⁿ, then there are at most 2ⁿ many immediate successorsτ ofσsuch that d^h(τ) ≥ (n+1)!. This last observation holds in general:

Proposition IV.4.5. For all n∈ N and all σ∈hⁿ such that d^h(σ) ≤L(n), there are at most k(n)-many immediate extensionsτ of σsuch that d^h(τ) >L(n+1).

Proof. Suppose for the sake of a contradiction that there are more thank(n)immediate extensions τ of σ such thatd^h(τ) >L(n+1). Thus,

∑

i<h(n)

d^h(σ^⌢⟨i⟩) >k(n) ⋅L(n+1) =h(n) ⋅L(n) ≥h(n) ⋅d^h(σ)

which contradicts the fact thatd^h is a supermartingale.

Now we turn our attention to Use IV.

Proposition IV.4.6. Letg∶N→ [0,∞)be an order function. ThenX ∈h^Nisg-random inh^Nif the following two conditions hold:

(i) L(n) ≤K(n)

n−g(n)

g(n) for almost alln.

(ii) d^h(X↾n) ≤L(n)for almost alln.

Proof. It suffices to show thatd^h does notg-succeed onX, i.e., that lim sup_nd^h(X↾n) ⋅ ∣hⁿ∣

g(n)

n −1< ∞. For all sufficiently largen,

L(n) ≤K(n)^{g(n)n −1} ⇐⇒ L(n) ≤K(n)

1−g(n)/n g(n)/n

⇐⇒ L(n)^g(n)/n≤K(n)^1−g(n)/n

⇐⇒

L(n)^g(n)/n K(n)^1−g(n)/n ≤1

⇐⇒

L(n)^{g(n)n −1+1} K(n)^{−(g(n)n −1)}

≤1

⇐⇒ L(n)(K(n)L(n))^{g(n)n −1}≤1

⇐⇒ L(n)∣hⁿ∣

g(n) n −1

≤1 Ô⇒ d^h(X↾n) ⋅ ∣hⁿ∣

g(n) n −1

≤1.

Thus, lim sup_nd^h(X↾n) ⋅ ∣hⁿ∣

g(n)

n −1< ∞, as desired.

Given f(n) =n−j(n), Corollary IV.2.19 suggests we find a g of the formg(n) =n− (1−ε)^j(s_s⁽₍ⁿ_n^)⋅₎ⁿ⁾ for some ε >0, where ∣hⁿ∣ = H(n) =2^s(n)⋅n. Proposition IV.4.6 suggests that if we wish for the X ∈ h^N we construct to be (strongly)g-random, then we might as well start with Kand define Lby

L(n) ∶=K(n)^n−g(n)g(n) .

k and ` are then defined by k(n) ∶= K(n+1)/K(n) and `(n) ∶= L(n+1)/L(n) for n ∈ N. Note that H(n) =K(n)L(n) =K(n)^n/g(n) for alln∈N.

Lemma IV.4.7. If t∶N → [0,∞) is an order function, then the function ∶N→ [0,∞) defined by r(n) ∶=

(n+1) ⋅t(n+1) −n⋅t(n)forn∈Nis a recursive function which dominates an order function.

Proof. Thatris recursive is immediate. For alln∈N,

r(n) = (n+1) ⋅t(n+1) −n⋅t(n) =n⋅ (t(n+1) −t(n)) +t(n+1).

Sincetis nondecreasing,n⋅ (t(n+1) −t(n)) ≥0, and sot(n+1) ≤r(n)for alln∈N. The function ˜t∶N→ [0,∞) defined by ˜t(n) ∶=t(n+1)forn∈Nis an order function such that ˜t≤_domr.

Proposition IV.4.8. Let j,f, s,ε, and ˜j satisfy the conditions of Theorem IV.4.1. Let g(n) ∶=n−˜j(n) andK(n) =2^s(n)⋅g(n).

(a) For all n∈N,k(n), `(n), h(n) ≥1.

(b) ` andhdominate order functions.

(c) IfX∈h^N is such thatd^h(X↾n) ≤L(n)for almost alln, thenX is stronglyg-random inh^N. (d) IfX∈h^N isg-random inh^N, then π^h(X) ∈ [0,1]isf-random in[0,1].

Proof. The functionsK, L, H, k, `, hcan all be written as powers of two whose exponents involves,g, and ˜j:

K(n) =2^s(n)⋅g(n), k(n) =exp₂(s(n+1) ⋅g(n+1) −s(n) ⋅g(n)), L(n) =2^{s(n)⋅˜j(n)}, `(n) =exp₂(s(n+1) ⋅˜j(n+1) −s(n) ⋅˜j(n)), H(n) =2^s(n)⋅n, h(n) =exp₂(s(n+1) ⋅ (n+1) −s(n) ⋅n).

Condition (iii) of Theorem IV.4.1 implies ˜j is an order function. The hypothesis thatj(n) ≤nfor alln∈N implies ˜j(n) ≤n for all n∈ N as well, so g is a nondecreasing function such that g(n) ≤ n for all n∈ N. Condition (i) implies imh⊆N.

(a) Thats,g, and ˜j are all nondecreasing implies thatk,`, andhare bounded below by 1.

(b) Lemma IV.4.7 shows thathand` each dominate order functions.

(c) This is simply Proposition IV.4.6.

(d) Using Condition (ii) of Theorem IV.4.1, this is simply Corollary IV.2.19.

It remains to generalize the construction of X in Theorem IV.3.1 and establish our bounds on q as a function ofj.

Proof of Theorem IV.4.1. Fix a rational ε>0 and observe that fulfillment of the conditions ons, j, andε do not depend the value ofε.

d^h is left r.e., so uniformly in σ∈h^∗ we can simultaneously and uniformly approximate d(σ^⌢⟨i⟩) from below for all i<h(n). Uniformly inσ∈hⁿ, letmσ be such that for allx<k(n), ϕm_σ(x) ↓=iif and only if σ^⌢⟨i⟩is thex-th immediate successorτ ofσfound with respect to the aforementioned procedure such that d(τ) >L(n).

Let #∶h^∗→Nbe the inverse of the enumeration ofh^∗ according to the shortlex ordering. In particular, for almost alln∈Nand allσ∈hⁿ,

#(σ) ≤ ∣h⁰∣ + ∣h¹∣ + ⋯ + ∣hⁿ∣ =

n

∑

i=0

2^s(i)⋅i< (n+1) ⋅2^s(n)⋅n.

By potentially modifying our enumerationϕ0, ϕ1, ϕ2, . . .of partial recursive functions, we can assume without loss of generality thatm_σ=2#(σ). Letm^∗_n= (n+1) ⋅2^s(n)⋅n+1, so that 1+sup{m_σ∣σ∈hⁿ} ≤m^∗_n for almost alln.

Let Ψ_n∶DNR(`(n)) → P<sub>h</sub><sup>1,k</sup><sub>(n</sub><sup>(</sup><sub>)</sub><sup>n)</sup> be uniformly recursive functionals realizing the reductions P<sub>h</sub><sup>1,k</sup><sub>(n</sub><sup>(</sup><sub>)</sub><sup>n)</sup> ≤<sub>s</sub> DNR(`(n)) from Proposition IV.3.4, and letU_n∶N→ P_fin(N)be the associated recursive functions (so that

∣Un(i)∣ ≤k(n)(^h_`(⁽_nⁿ₎⁾)). We are principally interested in initial segments ρof elements of lengthm^∗_n (in fact, we are only concerned with the values at the inputsmσ forσ∈hⁿ), so that:

(1) ρ(mσ) <h(n) =h(∣σ∣).

(2) For allx<k(n), ifϕm_σ(x)↓, thenρ(mσ) ≠ϕm_σ(x).

DefineU∶N→ P_fin(N)by settingU(n) ∶= ⋃_i<m^∗

nU_n(i)forn∈Nand subsequently defineu∶N→Nby u(0) ∶=0,

u(n+1) ∶=u(n) + ∣U(n)∣.

Finally, defineψ∶ ⊆N→Nby letting, for n∈Nandj< ∣U(n)∣,

ψ(u(n) +j) ≃ϕj-th element ofU(n)(0).

By construction, for any Z ∈Avoid^ψ(`(n)), Z↾u(n+1) can be used to compute an initial segment of an element ofP_h^1,k_(n⁽₎ⁿ⁾of lengthm^∗_n, and this is uniform inn.

Ifp∶N→Nis a recursive order function satisfying

p(u(n+1)) ≤`(n)

for almost alln, then uniformly innandZ∈Avoid^ψ(p),Z↾u(n+1)can be used to compute an initial segment of P_h^1,k_(n⁽₎ⁿ⁾ of length m^∗_n. Given Z ∈Avoid^ψ(p), define G∶N→ Nby setting the value of G(mσ) according to this uniform process for each σ∈h^∗; for n not of the form m_σ (which can be recursively checked), set G(n) ∶=0. Then defineX ∈h^N recursively by

X(0) ∶=G(m_⟨⟩),

X(n+1) ∶=G(m_⟨X(0),X(1),...X(n)⟩).

We claim that d^h does not g-succeed onX. We start by showing that d^h(X↾n) ≤L(n) by induction.

Forn=0, this follows sinced^h(X↾n) =d^h(⟨⟩) =1. Now suppose for our induction hypothesis thatd(X↾n) ≤ L(n). By construction, for x<k(n), if ϕ_mX↾n(x)↓ =i thenX(n+1) =G(m_X↾n) ≠i; in combination with

the induction hypothesis and Proposition IV.4.5, it follows thatd(X↾ (n+1)) ≤L(n+1). By our definition ofLand Proposition IV.4.6,d^h does notg-succeed onX. Equivalently, X is (strongly) g-random in h^N.

Proposition IV.4.8 shows that π^h(X) = Y ∈ [0,1] is f-random. In other words, COMPLEX(f) ≤_w Avoid^ψ(p). By Proposition IV.4.8,` dominates an order function, and hence there exists a recursive order functionpsatisfyingp(u(n+1)) ≤`(n). Ifq∶N→Nis a recursive order function such that for alla, b∈Nwe haveq(an+b) ≤p(n)for almost alln, then

COMPLEX(f) ≤wAvoid^ψ(p) ≤wLUA(q).

To get a more explicit condition onq, we find an upper bound forau(n+1) +b. For all nandi,

∣Un(i)∣ ≤k(n)(h(n)

`(n)) ≤k(n) (h(n)e

`(n) )

`(n)

=e^{`(n)</sup>k(n)<sup>`(n)+1}=exp₂((log₂k(n) +log₂e)`(n) +log₂k(n)).

Thus, for alln,

∣U(n)∣ ≤ ∑

i<m^∗_n

∣Un(i)∣

≤m^∗_n⋅max

i Un(i)

≤ (n+1) ⋅2^s(n)⋅n+1⋅exp₂((log₂k(n) +log₂e)`(n) +log₂k(n))

≤exp₂(log₂H(n) + (log₂k(n) +log₂e)`(n) +log₂k(n) +log₂(n+1) +1). For anyaandband almost alln,

log₂(au(n+1) +b) =log₂(a ∑

m≤n

∣U(m)∣ +b)

≤log₂(a(n+1) ⋅ ∣U(n)∣ +b)

≤log₂(3an⋅ ∣U(n)∣)

≤log₂H(n) + (log₂k(n) +log₂e)`(n) +log₂k(n) +2 log₂(n+1) +log₂(3a) +1.

Substituting log₂H(n)and log₂k(n)with expressions in terms ofs, g, and ˜j gives

log₂(au(n+1) +b) =s(n) ⋅n+ (s(n+1) ⋅g(n+1) −s(n) ⋅g(n) +log₂e) ⋅`(n) +s(n+1) ⋅g(n+1)

−s(n) ⋅g(n) +2 log₂(n+1) +log₂(3a) +1

≤2s(n) ⋅g(n+1) + (s(n+1) ⋅g(n+1) −s(n) ⋅g(n) +log₂e) ⋅`(n) +s(n+1) ⋅g(n+1)

−s(n) ⋅g(n) +s(n) ⋅g(n+1) +log₂(3a) +1

≤s(n+1) ⋅g(n+1) ⋅ (`(n) +4)

=s(n+1) ⋅ (n+1− (1−ε)j(s(n+1) ⋅ (n+1))

s(n+1) ) ⋅ (`(n) +4)

≤ (s(n+1) ⋅ (n+1) − (1−ε)j(s(n+1) ⋅ (n+1))) ⋅ (`(n) +4)

≤ 1

1−ε(s(n+1) ⋅ (n+1) −j(s(n+1) ⋅ (n+1))) ⋅`(n)

= 1

1−εf(s(n+1) ⋅ (n+1)) ⋅`(n)

for almost alln, where the final line follows from the fact that lim_n→∞j(n)/n=0. Thus, ifqsatisfies q(exp₂((1−ε)⁻¹⋅f(s(n+1) ⋅ (n+1)) ⋅`(n))) ≤`(n)

then COMPLEX(f) ≤wLUA(q).

Example IV.4.9. Supposej(n) =√

nlog₂nand lets(n) =n. Simplifying`gives

`(n) =exp₂(2(1−ε) (

√

(n+1)²log₂(n+1) −

√

n²log₂n))

=exp₂(2(1−ε)log₂((n+1) ⋅ (1+ 1 n)

n

))

= ((n+1) ⋅ (1+ 1 n)

n

)

2(1−ε)

. This provides the bounds

(n+1)^2(1−ε)≤`(n) ≤ (n+1)^2(1−ε)⋅e^2(1−ε) and consequently

(1−ε)⁻¹⋅f((n+1)²) ⋅`(n) ≤ (1−ε)⁻¹⋅ [(n+1)²−2(n+1)log₂(n+1)] ⋅ (n+1)^2(1−ε)⋅e^2(1−ε)

≤ e^2(1−ε)

1−ε ⋅ (n+1)^2(2−ε)

≤ (n+1)^4−2ε. Thus, Theorem IV.4.1 implies COMPLEX(λn.n−

√nlog₂n) ≤_wLUA(q)whenever

q(exp₂((n+1)^2(2−ε))) ≤ (n+1)^2(1−ε).

In particular, we may takeq(n) ∶= (log₂n)^β for anyβ<1/2. In other words, whereas Theorem IV.3.1 shows that COMPLEX(λn.n−α√

nlog₂n) ≤wLUA(λn.(log₂n)^β)for anyα>1 andβ<1/2, Theorem IV.4.1 shows that for the same order functionsq, we in fact are able to compute (λn.n−

√nlog₂n)-complex sequences.

Example IV.4.9 can be generalized further to address functions of the form f(n) =n−

√n⋅∆(n):

Theorem IV.4.10. Given an order function∆∶N→ [0,∞)such thatlimn→∞∆(n)/√

n=0and any rational

ε∈ (0,1),

COMPLEX(λn.n−

√n⋅∆(n)) ≤_wLUA(λn.exp₂((1−ε)∆(log₂log₂n))).

More generally, COMPLEX(λn.n−

√n⋅∆(n)) ≤_wLUA(q)for any order function qsatisfying

q(exp₂((1−ε)⁻¹⋅ [(n+1)²− (n+1) ⋅∆((n+1)²)] ⋅`(n))) ≤`(n) for almost alln∈N, where`(n) =exp₂((1−ε)[(n+1) ⋅∆((n+1)²) −n⋅∆(n²)]).

Proof. Lets(n) ∶=nandj(n) ∶=√

n⋅∆(n)for alln∈N. We show that the conditions of Theorem IV.4.1 are fulfilled with these choices ofsandj:

(i) (n+1)²−n²=2n+1 and 2²ⁿ⁺¹∈Nfor alln∈N. (ii) Immediate.

(iii) ^j(n_n²⁾ =

√n²⋅∆(n²)

n =∆(n²)shows that the functionn↦j(s(n) ⋅n)/s(n)is an order function.

The condition given by Theorem IV.4.1 requires that for someε>0 and almost alln q(exp₂((1−ε)⁻¹⋅ [(n+1)²− (n+1) ⋅∆((n+1)²)] ⋅`(n))) ≤`(n),

where`(n) =exp<sub>2</sub>((1−ε)((n+1) ⋅∆((n+1)<sup>2</sup>) −n⋅∆(n<sup>2</sup>))). Rearranging the exponent of`(n)gives a simple lower bound:

log₂`(n) = (1−ε) ⋅ (∆((n+1)²) +n⋅ (∆((n+1)²) −∆(n²))) ≥ (1−ε) ⋅∆((n+1)²) ≥ (1−ε)∆(n²).

Concerning the exponent ofq’s argument, we have the following: for almost alln,

(1−ε) ⋅ [(n+1)²− (n+1) ⋅∆((n+1)²)] ⋅`(n) ≤ (1−ε) ⋅ (n+1)²⋅exp₂((1−ε) ⋅ (n+1) ⋅∆((n+1)²)) ≤2ⁿ². Thus, COMPLEX(λn.n−

√n⋅∆(n)) ≤wLUA(q)if q(exp₂exp₂n²) ≤exp₂((1−ε)∆(n²)) and hence if the following stronger condition holds:

q(n) ≤exp((1−ε)∆(log₂log₂n)).

Remark IV.4.11. Settingq(n) ∶=exp₂((1−ε)∆(log₂log₂n))can be very inefficient; when ∆(n) =log₂n, this givesq(n) = (log₂log₂n)^1−ε, significantly slower than the lower bound onqestablished in Example IV.4.9.

A better bound can be given for well-behaved ∆ which are dominated by log₂. Suppose ∆≤domlog₂and that

c∶= lim

n→∞n[∆((n+1)²) −∆(n²)] < ∞.

Then we may give the following lower and upper bounds for`: for almost alln,

exp₂((1−ε)∆((n+1)²)) ≤`(n) ≤exp₂((1−ε)∆((n+1)²) +c) ≤2^c⋅ (n+1)^2(1−ε). Consequently, COMPLEX(λn.n−

√n⋅∆(n)) ≤_wLUA(q)wheneverq is such that, for almost alln,

q(n) ≤exp₂((1−ε)∆([(1−ε) ⋅2^−c⋅log₂n]^1/(2−ε))).