Weihrauch-completeness for layerwisecomputability1

Arno Pauly

Clare CollegeUniversity 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.

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

Theorem (Hölzl & Shafer)Layerwise computability does depend on the choice of U ingeneral, but all optimal Martin-Löf tests yield the same class.

More extended computability notions

DefinitionA finitely-revising machine is a Type-2 machine with the extracapability to erase its output and restart writing it, to be usedfinitely many times during the computation. A function iscomputable with finitely many mindchanges, if there this afinitely-revising machine computing it.

DefinitionA non-deterministic Type-2 machine with advice space Zcomputes 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 withadvice space N are equivalent.

ObservationAny layerwise computable function is computable bynon-deterministic machine with advice space N.

Represented spaces and computability

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

DefinitionF :⊆ 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−−−−→ NNyδX

yδY

X f−−−−→ Y

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

Weihrauch-reducibility

DefinitionFor f :⊆ X⇒ Y, g :⊆ V⇒W say

f ≤W g

iff there are computable H,K :⊆ NN → NN, such thatK 〈idNN ,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 anoperation on W.

Theorem (Higuchi & P. 2013)For A ⊆ NN, let dA : A→ {0}. Then d· : Mop →W is a latticeembedding.

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 ) = Dobtained by Skolemization.

2. Then compare theorems via Weihrauch-reducibility tolearn 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

LemmaThe following are Weihrauch equivalent:

1. CN :⊆ A(N)⇒ N be defined via n ∈ CN(A) iff n ∈ A2. UCN, defined via UCN = (CN) |{A∈A(N)||A|=1}

3. minA :⊆ A(N)→ N4. maxO :⊆ O(N)→ N5. Bound :⊆ O(N)⇒ N, where n ∈ Bound(U) iff∀m ∈ U n ≥ m.

Weihrauch-completeness for layerwise-computability

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

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

I If f is layerwise-computable and f ≡W LAYU , call fWeihrauch-complete for layerwise computability.

I The problems that are Weihrauch-complete for layerwisecomputability are the most non-computablelayerwise-computable problems.

The main theorem

TheoremLAYU ≡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 degreedoes not matter further. Note that given p we cancompute {n | p /∈ Un} ∈ A(N).

Proof continuedProof.

Bound×dMLR ≤W LAYU The input is an enumeration of somefinite set I ⊂ N (which we may safely assume to bean interval) and a random sequence p. Let w bethe current prefix of the output (i.e. the input toLAYU ). If we learn that n ∈ I, we consider w0N. Asthis is not random and U is universal, we know thatw0N ∈ Un. As Un is open, there is some –effectively findable – k ∈ N such thatw0k{0,1}N ⊆ Un. We proceed to amend thecurrent 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 tailidentical to p, and thus is Martin Löf random. Byconstruction, whenever n ∈ I, then q ∈ Un, thus ifb ∈ LAYU (q) then b ∈ Bound(p).

Corollaries

I LAY <W CN

I LAY× LAY ≡W LAY and LAY ? LAY ≡W LAYI LAY ? CN ≡W CN ? LAY ≡W LAYI LAY <W LAY ≡W lim×dMLR

I LAY <W LAY∗ ≡W idNN + LAY <W CN

I If f ≤W CN for f :⊆ MLR⇒ Y, then f ≤W LAY.

More consequences

CorollaryThe following are equivalent for f :⊆ MLR→ Y for a computablemetric space Y:

1. f is effectively ∆02-measurable.

2. f is Π01-piecewise computable.

3. f ≤W LAY.

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

Complex oscillationsDefinitionThe complex oscillations CO are the Martin-Löf randomelements of C0([0,1],R) equipped with the Wiener measure.Let computable η : MLR→ R induce the normal distributionN (0,1) on R.

DefinitionWe define the function Φ : MLR→ CO by recursively providingthe values Φ(α) takes on dyadic rationals, and extending itcontinuously to the interval. Let α = 〈α0, α1, . . . , αjn, . . .〉, wheren ≤ 2j . Then we define:

1. Φ(α)(1) := η(α0)

2. Φ(α)(12) := 1

2 (η(α0) + η(α1))

3. Φ(α)(2n+12j+1 ) := 1

2

(2−j/2η(αjn) + Φ(α)(n+1

2j ) + Φ(α)( n2j ))

Theorem (Davie & Fouché)Φ is a layerwise computable bijection with computable inverse.

The completeness result

TheoremΦ ≡W LAY

LemmaGiven 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

DefinitionLet LIL : MLR⇒ N be defined via N ∈ LIL(α) iff:

∀n ≥ N |n−1∑i=0

(2α(i)− 1)| <√

2n log log n

TheoremLIL ≡W LAY.

LemmaGiven 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

DefinitionLet S : {0,1}N → {0,1}N be the usual shift-operator, andπ1 : {0,1}N → {0,1} be the projection to the first bit. LetBirkhoff : 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

TheoremBirkhoff ≡W LAY

Proof ingredient

LemmaGiven u ∈ {0,1}∗ and k ,N ∈ N, k > 0, we can compute somev ∈ {0,1}∗ such that |uv | ≥ N and:

|

1|uv |

|uv |−1∑i=0

π1(Si(uv))

− 12| > 2−k

Hitting times

DefinitionLet Aλ>0({0,1}N) be the restriction of A({0,1}N) to sets ofpositive Lebesgue measure. Let T : {0,1}N → {0,1}N be theusual shift-operator. DefineHittingTimeA : MLR×Aλ>0({0,1}N)→ N be defined viaHittingTimeA(p,A) = min{n ∈ N | T n(p) ∈ A}.

Theorem (Kucera)HittingTimeA is well-defined.

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

100) islayerwise computable.

Some last minute-additions

I Finding the suitable n from the multiple recurrencetheorem for Martin-Löf randoms is Weihrauch-equivalent toLAY (but not layerwise computable).

I Computing the time-reversal of a Brownian motion on[0,∞) should be Weihrauch-reducible to LAY (but whatabout the other direction)?

Some open questions

I Investigate further layerwise-computable problems.I 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 computabilityarXiv, 1505.02091, 2015.

R. Hölzl and P. Shafer.Universality, optimality, and randomness deficiencyAnnals of Pure and Applied Logic, 2015.