Weihrauch-completeness for layerwise computability 1 Arno Pauly - - PowerPoint PPT Presentation

Weihrauch-completeness for layerwise computability1

Arno Pauly

Clare College University of Cambridge

CCR 2015, Heidelberg

1Joint work with George Davie & Willem Fouché (UNISA).

Outline

Definitions The main result Examples A non-example

Layerwise computability

Fix a universal Martin-Löf test U = (Un)n∈N.

Definition

A (multivalued) function f : MLR ⇒ X is layerwise computable w.r.t. U, iff there exists a computable partial function F :⊆ N × MLR → X such that whenever p / ∈ Un then F(n, p) ∈ f(p).

Theorem (Hölzl & Shafer)

Layerwise computability does depend on the choice of U in general, but all optimal Martin-Löf tests yield the same class.

More extended computability notions

Definition

A finitely-revising machine is a Type-2 machine with the extra capability to erase its output and restart writing it, to be used finitely many times during the computation. A function is computable with finitely many mindchanges, if there this a finitely-revising machine computing it.

Definition

A non-deterministic Type-2 machine with advice space Z computes a multivalued function f : X ⇒ Y as follows:

• 1. On input x ∈ X, guess some z ∈ Z.
• 2. Either: Halt and reject the guess.
• 3. Or: Run indefinitely, and output some y ∈ f(x).

Such that for any x ∈ X there is some z ∈ Z leading to case 3.

Connections

Observation (Brattka, de Brecht & P .)

Finitely revising machines and non-deterministic machines with advice space N are equivalent.

Observation

Any layerwise computable function is computable by non-deterministic machine with advice space N.

Represented spaces and computability

Definition

A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.

Definition

F :⊆ NN → NN is a realizer of f : X ⇒ Y, iff δY(F(p)) ∈ f(δX(p)) for all p ∈ δ−1

X (dom(F)).

NN

F

− − − − → NN   δX   δY X

f

− − − − → Y

Definition

f : X ⇒ Y is called computable (continuous), iff it has a computable (continuous) realizer.

Weihrauch-reducibility

Definition

For f :⊆ X ⇒ Y, g :⊆ V ⇒ W say f ≤W g iff there are computable H, K :⊆ NN → NN, such that KidNN, GH is a realizer of f for every realizer G of g.

Theorem (Brattka & Gherardi 2011, P . 2010)

W is a distributive lattice. The cartesian product × is an

• peration on W.

Theorem (Higuchi & P . 2013)

For A ⊆ NN, let dA : A → {0}. Then d· : Mop → W is a lattice embedding.

The motivation

• 1. Identify a theorem

∀x ∈ X ∃y ∈ Y . D(x) ⇒ T(x, y) with the multi-valued function T :⊆ X ⇒ Y, dom(T) = D

• btained by Skolemization.
• 2. Then compare theorems via Weihrauch-reducibility to

learn about their constructive content. Similar spirit as (constructive) reverse mathematics, but:

Theorem (Higuchi & P . 2013)

W is not a Brouwer algebra.

The degree of CN

Lemma

The following are Weihrauch equivalent:

• 1. CN :⊆ A(N) ⇒ N be defined via n ∈ CN(A) iff n ∈ A
• 2. UCN, defined via UCN = (CN) |{A∈A(N)||A|=1}
• 3. minA :⊆ A(N) → N
• 4. maxO :⊆ O(N) → N
• 5. Bound :⊆ O(N) ⇒ N, where n ∈ Bound(U) iff

∀m ∈ U n ≥ m.

Weihrauch-completeness for layerwise-computability

Definition

Let LAYU : MLR ⇒ N be defined via n ∈ LAYU(p) iff p / ∈ Un. Let rdU : MLR → N be defined via rdU(p) = min{n ∈ N | p / ∈ Un}.

Observation

LAYU is layerwise computable w.r.t. U. Whenver f : MLR ⇒ X is layerwise computable w.r.t. U, then f ≤W LAYU.

◮ If f is layerwise-computable and f ≡W LAYU, call f

Weihrauch-complete for layerwise computability.

◮ The problems that are Weihrauch-complete for layerwise

computability are the most non-computable layerwise-computable problems.

The main theorem

Theorem

LAYU ≡W rdU ≡W CN × dMLR

Proof.

LAYU ≤W rdU Trivial. rdU ≤W minA ×dMLR We have a random sequence available as input for dMLR, and the presence of this degree does not matter further. Note that given p we can compute {n | p / ∈ Un} ∈ A(N).

Proof continued

Proof.

Bound ×dMLR ≤W LAYU The input is an enumeration of some finite set I ⊂ N (which we may safely assume to be an interval) and a random sequence p. Let w be the current prefix of the output (i.e. the input to LAYU). If we learn that n ∈ I, we consider w0N. As this is not random and U is universal, we know that w0N ∈ Un. As Un is open, there is some – effectively findable – k ∈ N such that w0k{0, 1}N ⊆ Un. We proceed to amend the current output to w0k, and then start outputting p (until we potentially learn n + 1 ∈ I. As I is finite, the output q will have some tail identical to p, and thus is Martin Löf random. By construction, whenever n ∈ I, then q ∈ Un, thus if b ∈ LAYU(q) then b ∈ Bound(p).

Corollaries

◮ LAY <W CN ◮ LAY × LAY ≡W LAY and LAY ⋆ LAY ≡W LAY ◮ LAY ⋆ CN ≡W CN ⋆ LAY ≡W LAY ◮ LAY <W

LAY ≡W lim ×dMLR

◮ LAY <W LAY∗ ≡W idNN + LAY <W CN ◮ If f ≤W CN for f :⊆ MLR ⇒ Y, then f ≤W LAY.

More consequences

Corollary

The following are equivalent for f :⊆ MLR → Y for a computable metric space Y:

• 1. f is effectively ∆0

2-measurable.

• 2. f is Π0

1-piecewise computable.

• 3. f ≤W LAY.

Proof.

By combining the computable Jayne-Rogers theorem (P . & de Brecht 2014) with the main theorem.

Complex oscillations

Definition

The complex oscillations CO are the Martin-Löf random elements of C0([0, 1], R) equipped with the Wiener measure. Let computable η : MLR → R induce the normal distribution N(0, 1) on R.

Definition

We define the function Φ : MLR → CO by recursively providing the values Φ(α) takes on dyadic rationals, and extending it continuously to the interval. Let α = α0, α1, . . . , αjn, . . ., where n ≤ 2j. Then we define:

• 1. Φ(α)(1) := η(α0)
• 2. Φ(α)( 1

2) := 1 2 (η(α0) + η(α1))

• 3. Φ(α)( 2n+1

2j+1 ) := 1 2

• 2−j/2η(αjn) + Φ(α)( n+1

2j ) + Φ(α)( n 2j )

• Theorem (Davie & Fouché)

Φ is a layerwise computable bijection with computable inverse.

The completeness result

Theorem

Φ ≡W LAY

Lemma

Given k ∈ N and v ∈ {0, 1}∗ we can compute some w ∈ {0, 1}∗ such that for all α ∈ MLR we find that k < supt∈[0,1] Φ(vwα)(t).

Law of the iterated logarithm

Definition

Let LIL : MLR ⇒ N be defined via N ∈ LIL(α) iff: ∀n ≥ N |

n−1

• i=0

(2α(i) − 1)| <

• 2n log log n

Theorem

LIL ≡W LAY.

Lemma

Given N ∈ N and u ∈ {0, 1}∗ we can compute some v ∈ {0, 1}∗ such that |uv| > N and | |uv|−1

i=0

(2(uv)(i) − 1)| >

• 2|uv| log log |uv|.

Birkhoff’s theorem

Definition

Let S : {0, 1}N → {0, 1}N be the usual shift-operator, and π1 : {0, 1}N → {0, 1} be the projection to the first bit. Let Birkhoff : MLR × N ⇒ N be defined via N ∈ Birkhoff(p, k) iff ∀n ≥ N we find that: |

• 1

n + 1

n

• i=0

π1(Si(p))

• − 1

2| < 2−k

Theorem

Birkhoff ≡W LAY

Proof ingredient

Lemma

Given u ∈ {0, 1}∗ and k, N ∈ N, k > 0, we can compute some v ∈ {0, 1}∗ such that |uv| ≥ N and: |   1 |uv|

|uv|−1

• i=0

π1(Si(uv))   − 1 2| > 2−k

Hitting times

Definition

Let Aλ>0({0, 1}N) be the restriction of A({0, 1}N) to sets of positive Lebesgue measure. Let T : {0, 1}N → {0, 1}N be the usual shift-operator. Define HittingTimeA : MLR × Aλ>0({0, 1}N) → N be defined via HittingTimeA(p, A) = min{n ∈ N | T n(p) ∈ A}.

Theorem (Kuˇ cera)

HittingTimeA is well-defined.

Theorem

HittingTimeA ≡W LAY, but not even HittingTimeA(·, UC

100) is

layerwise computable.

Some last minute-additions

◮ Finding the suitable n from the multiple recurrence

theorem for Martin-Löf randoms is Weihrauch-equivalent to LAY (but not layerwise computable).

◮ Computing the time-reversal of a Brownian motion on

[0, ∞) should be Weihrauch-reducible to LAY (but what about the other direction)?

Some open questions

◮ Investigate further layerwise-computable problems. ◮ Is there a (natural) problem which is non-computable,

layerwise computable and strictly below LAY?

Reference

• A. Pauly, G. Davie and W. Fouché.

Weihrauch-completeness for layerwise computability arXiv, 1505.02091, 2015.

• R. Hölzl and P

. Shafer. Universality, optimality, and randomness deficiency Annals of Pure and Applied Logic, 2015.