Strong f -Randomness

I. 3.1 (Partial) Recursive Functions and Sets

IV.1 Partial Randomness in h N

IV.1.2 Strong f -Randomness

A related variant of partial randomness can be defined similarly. First, from the measure-theoretic paradigm:

Definition IV.1.16(prefix-freef-weight inh^∗). Theprefix-freef-weight of a set of stringsS⊆h^∗is defined by

pwt_f(S) ∶=sup{dwt_f(A) ∣prefix-freeA⊆S}.

Definition IV.1.17 (strongf-randomness inh^N). A weakf-ML-test (in h^N) is a uniformly r.e. sequence

⟨A_k⟩_k∈Nof subsetesA_k⊆h^∗ such that pwt_f(A_k) ≤2^−k for allk∈N.

A weakf-ML-test⟨Ak⟩k∈N covers X∈h^N ifX∈ ⋂k∈NJAkK. If X is covered by no weak f-ML-test, then X isstronglyf-random (in h^N).

Like in{0,1}^N, strongf-randomness inh^N is associated with an analog of a priori complexity inh^∗.

Definition IV.1.18(continuous semimeasure onh^∗). Acontinuous semimeasure onh^∗is a functionν∶h^∗→ [0,1]such thatν(⟨⟩) =1 and for everyσ∈h^∗,

ν(σ) ≥ ∑

i∈<h(∣σ∣)

ν(σ^⌢⟨i⟩).

A continuous semimeasureνisleft recursively enumerable, orleft r.e., if it is left r.e. as a functionh^∗→R. A left r.e. continuous semimeasureν isuniversal if for every left r.e. continuous semimeasureξonh^∗ there existsc∈Nsuch thatξ(σ) ≤c⋅ν(σ)for allσ∈h^∗.

Proposition IV.1.19. There exists a universal left r.e. semimeasure Mon h^∗. Proof. The proof given in [6, Theorem 3.16.2] easily generalizes toh^∗.

Definition IV.1.20 (a priori complexity in h^∗). Fix a universal left r.e. semimeasure M. The a priori complexity of a stringσ∈h^∗ is defined by

KA(σ) =KAM(σ) ∶= −log₂M(σ).

Akin to the well-definedness of prefix-free complexity, ifN were another universal left r.e. semimeasure, then KAMand KAN differ by at most a constant.

Definition IV.1.21(strongf-complexity inh^N). X ∈h^Nisstronglyf-complex (inh^N)if there exists ac∈N such that, for alln∈N,

KA(X↾n) ≥ (log_1/2γ(X↾n)) ⋅f(X↾n) −c.

We will show that strong f-complexity is equivalent to strongf-randomness. Before doing so, we intro- duce a third approach to defining strongf-randomness/complexity, this time in terms of supermartingales as in the unpredictability paradigm.

Definition IV.1.22 (supermartingale). Asupermartingale (overh^∗) is a functiond∶h^∗→ [0,∞)such that

∑

i<h(∣σ∣)

d(σ^⌢⟨i⟩) ≤h(∣σ∣)d(σ) for allσ∈h^∗.

A supermartinagle disleft recursively enumerable, orleft r.e., if it is left r.e. as a functionh^∗→ [0,∞).

Definition IV.1.23(f-success). Supposedis a left r.e. supermartingale andX∈h^N. dis said to f-succeed onX if

lim sup

n

(d(X↾n) ⋅γ(X↾n)^n−f(X↾n)) = ∞.

The following lemma reveals the close connection between continuous semimeasuresν and supermartin- galesdsuch thatd(⟨⟩) =1.

Lemma IV.1.24. Given ν∶h^∗→ [0,1], letd_ν∶h^∗→ [0,∞)be defined byd_ν(σ) ∶= ∣h^∣σ∣∣ ⋅ν(σ)forσ∈h^∗. (a) ν is left r.e. if and only if d_ν is left r.e.

(b) ν is a continuous semimeasure if and only ifdν is a supermartingale.

(c) ν is a universal left r.e. continuous semimeasure if and only ifdν is auniversal left r.e. supermartinagle, in the sense that ifdwere another left r.e. supermartingale then there is ac∈Nsuch thatd(σ) ≤c⋅dν(σ) for all σ∈h^∗.

Proof.

(a) This follows from the fact thathis recursive.

(b) Givenσ∈h^∗, we have

∑

i<h(∣σ∣)

dν(σ^⌢⟨i⟩) = ∑

i<h(∣σ∣)

∣h^∣σ

⌢⟨i⟩∣∣ ⋅ν(σ^⌢⟨i⟩) = ∣h^∣σ∣+1∣ ⋅ ∑

i<h(∣σ∣)

ν(σ^⌢⟨i⟩) =h(∣σ∣) ⋅⎛

⎝

∣h^∣σ∣∣ ⋅ ∑

i<h(∣σ∣)

ν(σ^⌢⟨i⟩)⎞

⎠ . By comparing the first and last expressions with the definitions of what it means for d_ν to be a supermartingale or for ν to be a continuous semimeasure shows that d_ν is a supermartinagle if and only ifν is a continuous semimeasure.

(c) Straight-forward.

Lemma IV.1.25. SupposeS andT are subsets of h^∗.

(a) IfS⊆T, then dwtf(S) ≤dwtf(T)andpwt_f(S) ≤pwt_f(T).

(b) dwt_f(S∪T) =dwt_f(S) +dwt_f(T) −dwt_f(S∩T).

(c) pwt_f(S∪T) ≤pwt_f(S) +pwt_f(T), with equality if the strings inS andT are pairwise incompatible.

Proof.

(a) Straight-forward.

(b) Straight-forward.

(c) IfP ⊆S∪T is prefix-free, thenP∩S andP∩T are prefix-free subsets ofS andT, respectively, so dwt_f(P) ≤dwt_f(P∩S) +dwt_f(P∩T) ≤pwt_f(S) +pwt_f(T).

Taking the supremum among all prefix-freeP ⊆S∪T yields pwt_f(S∪T) ≤pwt_f(S) +pwt_f(T).

If the strings in S and T are pairwise incompatible, then given prefix-free subsets A⊆S and B⊆T, A∩B= ∅and A∪B is a prefix-free subset ofS∪T, so

dwtf(A) +dwtf(B) =dwtf(A∪B) ≤pwt_f(S∪T).

Taking the supremum among all prefix-freeA⊆S andB⊆T yields pwt_f(S) +pwt_f(T) ≤pwt_f(S∪T).

Theorem IV.1.26. SupposeX∈h^N. The following are equivalent.

(i) X is stronglyf-random.

(ii) X is stronglyf-complex.

(iii) d^h does notf-succeed onX, whered^his the universal left r.e. supermartingale corresponding to Mas in Lemma IV.1.24.

(iv) No left r.e. supermartingale f-succeeds on X.

Proof.

(i) ⇐⇒ (ii) SupposeX is stronglyf-random. LetS_i= {σ∈h^∗∣KA(σ) < (log_1/2γ(σ)) ⋅f(σ) −i}. IfP⊆S_i is prefix-free, then

dwtf(P) = ∑

σ∈P

γ(σ)^f(σ)≤ ∑

σ∈P

γ(σ)^{(KA(σ)+i)⋅(logγ(σ)1/2)}≤ 1 2ⁱ ∑

σ∈P

M(σ) ≤ 1

2ⁱM(⟨⟩) ≤ 1 2ⁱ.

Thus, ⟨Si⟩i∈N forms a weakf-ML test. BecauseX is strongly f-random,X is not covered by ⟨Si⟩i∈N

and so there is ani∈Nsuch thatX∉JSiK, i.e., for everyn∈Nwe have KA(X↾n) ≥ (log_1/2γ(X↾n)) ⋅ f(X↾n) −i, so X is stronglyf-complex.

If X is not stronglyf-random, then there is a weakf-ML test⟨Si⟩i∈N which coversX. Uniformly in i∈ N, we letνi be defined by νi(σ) =pwt_f({τ ∈Si ∣τ ⊇ σ}). νi is a continuous semimeasure; using Lemma IV.1.25 we have

νi(σ) =pwt_f({τ∈Si∣τ⊇σ})

≥pwt ({τ∈Si∣τ⊃σ})

=pwt_f⎛

⎝

⋃

j<h(∣σ∣)

{τ∈Si∣τ⊇σ^⌢⟨j⟩}⎞

⎠

= ∑

j<h(∣σ∣)

pwt_f({τ∈S_i∣τ⊇σ^⌢⟨j⟩})

= ∑

j<h(∣σ∣)

νi(σ^⌢⟨j⟩).

Thatνiis left r.e. follows from the fact thatSi is r.e. Observe that forτ∈Si we have dwtf(τ) ≤νi(τ).

Because νi(⟨⟩) =pwt_f(Si) ≤2⁻ⁱ for eachi, the mapν∶h^∗→ [0,1]defined by ν(σ) ∶=

∞

∑

i=0

2ⁱν2i(σ)

forσ∈h^∗ is a left r.e. semimeasure, and hence there is ac∈Nsuch thatν(σ) <c⋅M(σ)for allσ∈h^∗. Then forσ∈S_2i, we have

exp₂(i− (log_1/2γ(σ)) ⋅f(σ)) =2ⁱdwt_f(σ) ≤2ⁱν_2i(σ) ≤ν(σ) <c⋅M(σ) =exp₂(−(KA(σ) +log_1/2c)) and hence

KA(σ) +i+log_1/2c< (log_1/2γ(σ)) ⋅f(σ).

Being covered by ⟨Si⟩i∈Nand hence by⟨S2i⟩i∈N as well,X is not strongly f-complex.

(ii) ⇐⇒ (iii) Letd^h be the universal left r.e. supermartingale corresponding toM, as in Lemma IV.1.24.

Now observe that for anyX∈h^Nandn∈N,

d^h(X↾n) ⋅µ(X↾n)^{1−f(X↾n)/n}=M(X↾n) ⋅µ(X↾n)⁻¹⋅µ(X↾n)^{1−f(X↾n)/n}

=exp₂(−(KA(X↾n) − (log_1/2γ(X↾n)) ⋅f(X↾n))).

Thus, lim sup

n

d0(X↾n) ⋅µ(X↾n)^{1−f(X↾n)/n}= ∞ ⇐⇒ ∀c∃n(KA(X↾n) < (log_1/2γ(X↾n)) ⋅f(X↾n) −c.

In other words,d^h f-succeeds onX if and only ifX is not stronglyf-complex.

(iii) ⇐⇒ (iv) If no left r.e. supermartingale f-succeeds on X, then in particular d^h does not f-succeed on X. Conversely, if d^h does not f-succeed on X, then the universality of d^h shows that no left r.e.

supermartingale f-succeeds onX.

Corollary IV.1.27. There exists a universal weak f-ML test, i.e., a weak f-ML test ⟨Si⟩i∈N such that X∈h^N is stronglyf-random inh^N if and only ifX is not covered by⟨Si⟩i∈N.

Proof. The proof of Theorem IV.1.26 shows that lettingSi= {σ∈h^∗∣KA(σ) < (log_1/2γ(σ)) ⋅f(σ) −i}yields a universal weakf-ML test.

Corollary IV.1.28. Suppose f(σ) =f˜(σ) for almost all σ. Then strong f-randomness is equivalent to strongf˜-randomness.

Proof. Supposef(σ) =f˜(σ)for all σ∈h^∗ for which∣σ∣ >N. Letc=max_σ∈h^∗_{,∣σ∣≤N}KA(σ). Then for every i∈ N, KA(σ) ≥ (log_1/2γ(σ)) ⋅f(σ) −c−i if and only if KA(σ) ≥ (log_1/2γ(σ)) ⋅f˜(σ) −c−i for allσ ∈h^∗. It follows that strongf-complexity and strong ˜f-complexity are equivalent, and so Theorem IV.1.26 shows strongf-randomness and strong ˜f-randomness are equivalent.

Remark IV.1.29. All of the above results hold withµreplaced by any computable measure onh^N for which µ(σ) >0 for allσ∈h^∗, with one adjustment – a supermartingaled f-succeeds on X with respect toµif and only if

lim sup

n

(d(X↾n) ⋅γ(X↾n)^−f(X↾n)⋅ ∣hⁿ∣⁻¹) = ∞.