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

Weihrauch-completeness for layerwise computability 1 Arno Pauly Clare College University of Cambridge CCR 2015, Heidelberg 1 Joint 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 = ( U n ) 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 / ∈ U n 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 , δ X ) where X is a set and δ X : ⊆ N N → X a surjective partial function. Definition F : ⊆ N N → N N is a realizer of f : X ⇒ Y , iff δ Y ( F ( p )) ∈ f ( δ X ( p )) for all p ∈ δ − 1 X ( dom ( F )) . F N N → N N − − − −     � δ X � δ Y f − − − − → X 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 : ⊆ N N → N N , such that K � id N N , 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 operation on W . Theorem (Higuchi & P . 2013) For A ⊆ N N , let d A : A → { 0 } . Then d · : M op → 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 obtained 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 C N Lemma The following are Weihrauch equivalent: 1. C N : ⊆ A ( N ) ⇒ N be defined via n ∈ C N ( A ) iff n ∈ A 2. UC N , defined via UC N = ( C N ) | { A ∈A ( N ) || A | = 1 } 3. min A : ⊆ A ( N ) → N 4. max O : ⊆ O ( N ) → N 5. Bound : ⊆ O ( N ) ⇒ N , where n ∈ Bound ( U ) iff ∀ m ∈ U n ≥ m.

Weihrauch-completeness for layerwise-computability Definition Let LAY U : MLR ⇒ N be defined via n ∈ LAY U ( p ) iff p / ∈ U n . Let rd U : MLR → N be defined via rd U ( p ) = min { n ∈ N | p / ∈ U n } . Observation LAY U is layerwise computable w.r.t. U . Whenver f : MLR ⇒ X is layerwise computable w.r.t. U , then f ≤ W LAY U . ◮ If f is layerwise-computable and f ≡ W LAY U , 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 LAY U ≡ W rd U ≡ W C N × d MLR Proof. LAY U ≤ W rd U Trivial. rd U ≤ W min A × d MLR We have a random sequence available as input for d MLR , and the presence of this degree does not matter further. Note that given p we can compute { n | p / ∈ U n } ∈ A ( N ) .

Proof continued Proof. Bound × d MLR ≤ W LAY U 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 LAY U ). If we learn that n ∈ I , we consider w 0 N . As this is not random and U is universal, we know that w 0 N ∈ U n . As U n is open, there is some – effectively findable – k ∈ N such that w 0 k { 0 , 1 } N ⊆ U n . We proceed to amend the current output to w 0 k , 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 ∈ U n , thus if b ∈ LAY U ( q ) then b ∈ Bound ( p ) .

Corollaries ◮ LAY < W C N ◮ LAY × LAY ≡ W LAY and LAY ⋆ LAY ≡ W LAY ◮ LAY ⋆ C N ≡ W C N ⋆ LAY ≡ W LAY ◮ LAY < W � LAY ≡ W lim × d MLR ◮ LAY < W LAY ∗ ≡ W id N N + LAY < W C N ◮ If f ≤ W C N 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 Π 0 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 C 0 ([ 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 = � α 0 , α 1 , . . . , α jn , . . . � , where n ≤ 2 j . Then we define: 1. Φ( α )( 1 ) := η ( α 0 ) 2. Φ( α )( 1 2 ) := 1 2 ( η ( α 0 ) + η ( α 1 )) � � 3. Φ( α )( 2 n + 1 2 j + 1 ) := 1 2 − j / 2 η ( α jn ) + Φ( α )( n + 1 2 j ) + Φ( α )( n 2 j ) 2 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 < sup t ∈ [ 0 , 1 ] Φ( vw α )( t ) .

Law of the iterated logarithm Definition Let LIL : MLR ⇒ N be defined via N ∈ LIL ( α ) iff: n − 1 � � ∀ n ≥ N | ( 2 α ( i ) − 1 ) | < 2 n log log n i = 0 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 ( 2 ( uv )( i ) − 1 ) | > 2 | uv | log log | uv | . i = 0

Birkhoff’s theorem Definition Let S : { 0 , 1 } N → { 0 , 1 } N be the usual shift-operator, and π 1 : { 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: � � � n 1 − 1 π 1 ( S i ( p )) 2 | < 2 − k | n + 1 i = 0 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:   | uv |− 1 �  1  − 1 π 1 ( S i ( uv )) 2 | > 2 − k | | uv | i = 0

Hitting times Definition Let A λ> 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 HittingTime A : MLR × A λ> 0 ( { 0 , 1 } N ) → N be defined via HittingTime A ( p , A ) = min { n ∈ N | T n ( p ) ∈ A } . Theorem (Kuˇ cera) HittingTime A is well-defined. Theorem HittingTime A ≡ W LAY , but not even HittingTime A ( · , U C 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.