Probabilistic Computability and Randomness in the Weihrauch Lattice - PowerPoint PPT Presentation

Probabilistic Computability and Randomness in the Weihrauch Lattice Vasco Brattka Universit¨ at der Bundeswehr M¨ unchen, Germany University of Cape Town, South Africa based on different joint work with Guido Gherardi Matthew Hendtlass Rupert H¨ olzl Alexander Kreuzer Arno Pauly Arbeitstreffen, Hiddensee, 8–12 August 2016

Outline 1 Algorithmic Randomness in the Weihrauch Lattice 2 Las Vegas Computability 3 Probabilistic Algorithms 4 Vitali Covering Theorem

Algorithmic Randomness in the Weihrauch Lattice

Mathematical Problems and Solutions Definition A problem is a partial multi-valued function f : ⊆ X ⇒ Y on represented spaces X , Y . ◮ There are a certain sets of potential inputs X and outputs Y . ◮ D = dom ( f ) contains the valid instances of the problem. ◮ f ( x ) is the set of solutions of the problem f for instance x . Definition g : ⊆ X ⇒ Y solves f : ⊆ X ⇒ Y , if dom ( f ) ⊆ dom ( g ) and g ( x ) ⊆ f ( x ) for all x ∈ dom ( f ). We write g ⊑ f in this situation.

Weihrauch Reducibility Let f : ⊆ X ⇒ Y and g : ⊆ Z ⇒ W be two mathematical problems. f g x f ( x ) K H ◮ f is Weihrauch reducible to g , f ≤ W g , if there are computable H : ⊆ X × W ⇒ Y , K : ⊆ X ⇒ Z such that H ( id X , gK ) ⊑ f . ◮ f is strongly Weihrauch reducible to g , f ≤ sW g , if there are computable H : ⊆ W ⇒ Y , K : ⊆ X ⇒ Z such that HgK ⊑ f . ◮ Equivalences f ≡ W g and f ≡ sW g are defined as usual. Theorem (Tavana and Weihrauch 2011) f ≤ W g ⇐ ⇒ there is a Turing machine that computes f and uses g as an oracle exactly once during its infinite computation.

Examples of Mathematical Problems ◮ The Limit Problem is the mathematical problem lim : ⊆ N N → N N , � p 0 , p 1 , ... � �→ lim i →∞ p i with dom (lim) := {� p 0 , p 1 , ... � : ( p i ) i is convergent } . ◮ Martin-L¨ of Randomness is the mathematical problem MLR : 2 N ⇒ 2 N with MLR( x ) := { y ∈ 2 N : y is Martin-L¨ of random relative to x } . ◮ Weak Weak K˝ onig’s Lemma is the mathematical problem WWKL : ⊆ Tr ⇒ 2 N , T �→ [ T ] with dom (WWKL) := { T ∈ Tr : µ ([ T ]) > 0 } . ◮ The Intermediate Value Theorem is the problem IVT : ⊆ Con[0 , 1] ⇒ [0 , 1] , f �→ f − 1 { 0 } with dom (IVT) := { f : f (0) · f (1) < 0 } . ◮ The Zero Problem Z X : ⊆ C ( X ) ⇒ X , f �→ f − 1 { 0 } . ◮ The Choice Problem C X : ⊆ A − ( X ) ⇒ X , A �→ A .

Algebraic Operations Definition For f : ⊆ X ⇒ Y and g : ⊆ W ⇒ Z we define: ◮ f × g : ⊆ X × W ⇒ Y × Z , ( x , w ) �→ f ( x ) × g ( w ) (Product) � f ( z ) if z ∈ X ◮ f ⊔ g : ⊆ X ⊔ W ⇒ Y ⊔ Z , z �→ (Coproduct) g ( z ) if z ∈ W ◮ f ⊓ g : ⊆ X × W ⇒ Y ⊔ Z , ( x , w ) �→ f ( x ) ⊔ g ( w ) (Sum) ◮ f ∗ : ⊆ X ∗ ⇒ Y ∗ , f ∗ = � ∞ i =0 f i (Star) ◮ � f : ⊆ X N ⇒ Y N , � f = X ∞ i =0 f (Parallelization) ◮ Weihrauch reducibility induces a lattice with the coproduct ⊔ as supremum and the sum ⊓ as infimum. ◮ Parallelization and star operation are closure operators in the Weihrauch lattice.

Basic Complexity Classes and Reverse Mathematics C N N ATR 0 lim ≡ sW � ACA 0 C N WKL 0 + IΣ 0 C R ≡ sW C N × C 2 N 1 WKL ≡ sW C 2 N ≡ sW � WKL 0 C 2 WWKL ≡ sW PC 2 N WWKL 0 IΣ 0 lim N ≡ sW C N 1 BΣ 0 K N ≡ sW C ∗ 2 1 C 1 RCA 0

Compositional Product and Implication The Weihrauch lattice is not complete and infinite suprema and infima do not always exist. There are some known existent ones. Definition For two mathematical problem f , g we define ◮ f ∗ g := max { f 0 ◦ g 0 : f 0 ≤ W f , g 0 ≤ W g } compos. product ◮ g → f := min { h : f ≤ W g ∗ h } implication Theorem (B. and Pauly 2016) The compositional product f ∗ g and the implication g → f exist for all problems f , g.

Martin-L¨ of Randomness Proposition (B., Gherardi and H¨ olzl 2015) MLR ∗ MLR ≤ W MLR Proof. This is a consequence of van Lambalgen’s Theorem. � Corollary The class of functions f ≤ W MLR is closed under composition. Theorem (B. and Pauly 2016) MLR ≡ W (C N → WWKL) . Proof. (C N → WWKL) ≤ W MLR: It suffices to prove WWKL ≤ W C N ∗ MLR, which follows from Kuˇ cera’s Lemma. MLR ≤ W (C N → WWKL): Given some h with WWKL ≤ W C N ∗ h we need to prove that MLR ≤ W h . Given some universal of test ( U i ) i , we use A 0 := 2 N \ U 0 and the fact that Martin-L¨ Martin-L¨ of randoms are stable under finite changes. �

Further Notions of Randomness Theorem (H¨ olzl and Miyabe 2015) WR < W SR ≤ W CR < W MLR < W W2R < W 2 - RAN . Proof. The strictness has been proved using hyperimmune degrees, high degrees and minimal degrees. � ◮ WR: Kurtz random ◮ SR: Schnorr random ◮ CR: computable random ◮ W2R: weakly 2-random ◮ n -RAN: n -random Proposition (Bienvenu and Kuyper 2016) n- RAN ∗ n- RAN ≤ W n- RAN . Proof. The proof is based on van Lambalgen’s Theorem and generalized lowness properties. �

Quantitative Versions of WWKL Definition (Dorais, Dzhafarov, Hirst, Mileti and Shafer 2016) By ε -WWKL : ⊆ Tr ⇒ 2 N we denote the restriction of WKL to dom ( ε -WWKL) := { T : µ ([ T ]) > ε } for ε ∈ R . Theorem (DDHMS 2016 and B., Gherardi and H¨ olzl 2015) ε - WWKL ≤ W δ - WWKL ⇐ ⇒ ε ≥ δ for all ε, δ ∈ [0 , 1] . Proof. (Idea) “= ⇒ ” Assume ε < δ . Then there are positive integers a , b with ε < a b ≤ δ . We consider ◮ C a , b which is C b restricted to sets A ⊆ { 0 , ..., b − 1 } with | A | ≥ a . Then C a , b ≤ W ε -WWKL and C a , b �≤ W δ -WWKL. Hence ε -WWKL �≤ W δ -WWKL �

Joint Results with Hendtlass and Kreuzer 2015 lim ≡ sW � C N DNC 2 ≡ sW WKL WWKL C N DNC 3 ≡ sW WKL 3 1 2 -WWKL ACC 2 ≡ sW LLPO n − 1 DNC n +1 ≡ sW WKL n +1 ACC 3 ≡ sW LLPO 3 n -WWKL ACC n +1 ≡ sW LLPO n +1 (1 − ∗ )-WWKL DNC N ACC N MLR ≡ W (C N → WWKL) PA ≡ (C ′ N → WKL) NON � ∞ ◮ (1 −∗ )-WWKL : ⊆ Tr N ⇒ 2 N , ( T i ) i �→ (1 − 2 − i )-WWKL( T i ) i =0

Jumps ◮ For every representation δ : ⊆ N N → X we define the jump δ ′ : ⊆ N N → X by δ ′ := δ ◦ lim. ◮ X ′ = ( X , δ ′ ) denotes the corresponding represented space. ◮ For f : ⊆ X ⇒ Y we define its jump by f ′ : ⊆ X ′ ⇒ Y , x �→ f ( x ). ◮ For instance id ′ ≡ sW lim, WKL ′ ≡ sW KL ≡ sW BWT R , etc. ◮ n -RAN ≡ sW MLR ( n − 1) . Proposition (B., Gherardi and Marcone 2012) ⇒ f ′ ≤ sW g ′ and f ≤ sW f ′ . f ≤ sW g = ◮ f < W f ′ does not hold in general: f ≡ sW f ′ for a constant f . ◮ f < W g is compatible with: f ′ ≡ W g ′ , f ′ < W g ′ , g ′ < W f ′ , f ′ | W g ′ . Theorem (B., H¨ olzl and Kuyper 2016) f ′ ≤ W g ′ = ⇒ f ≤ W g with respect to the halting problem.

Uniform Theorem of Kurtz Theorem of Kurtz. Every 2–random computes a 1–generic. Theorem (B., Hendtlass and Kreuzer 2015) 1 - GEN < W (1 − ∗ ) - WWKL ′ . Proof. (Idea) We apply the “fireworks technique” of Rumyantsev and Shen to get a uniform reduction. � Theorem (B., Hendtlass and Kreuzer 2015) 0 �≤ W WWKL ( n ) for all n ∈ N . BCT ′ Proof. (Idea) There exists a co-c.e. comeager set A ⊆ 2 N such that no point of A is low for Ω. WWKL ( n ) has a realizer that maps computable inputs to outputs that are low for Ω for n ≥ 1. � Corollary BCT ′ 0 �≤ W 1 - GEN .

Las Vegas Computability

Turing Machines with Advice input advice x ∈ X r ∈ R Turing Machine computes f : ⊆ X ⇒ Y or y ∈ f ( x ) failure! correct output Condition: ( ∀ x ∈ dom ( f )) { r ∈ R : r does not fail with x } � = ∅

Las Vegas Turing Machines input advice x ∈ X r ∈ R Las Vegas Turing Machine computes f : ⊆ X ⇒ Y or y ∈ f ( x ) failure! correct output Condition: ( ∀ x ∈ dom ( f )) µ { r ∈ R : r does not fail with x } > 0

Calibrating Computability with Choice Theorem (B., de Brecht and Pauly 2012) For R ⊆ N N and f : ⊆ X ⇒ Y the following are equivalent: ◮ f ≤ W C R , ◮ f is computable on a Turing machine with advice from R. Corollary ◮ f ≤ C 1 ⇐ ⇒ f is computable, ◮ f ≤ W C N ⇐ ⇒ f comp. with finitely many mind changes, ◮ f ≤ W C 2 N ⇐ ⇒ f is non-deterministically computable, ◮ f ≤ W PC 2 N ⇐ ⇒ f is Las Vegas computable, ◮ f ≤ W � C N ⇐ ⇒ f is limit computable, ◮ f ≤ W C N N ⇐ ⇒ f is effectively Borel measurable. In the last case f is single-valued on computable Polish spaces.

Independent Choice Theorem Theorem (B., de Brecht and Pauly 2012) C R ∗ C S ≤ W C R × S for all R , S ⊆ N N . Proof. Run a Turing machine that simulates upon advice ( r , s ) two consecutive machines with advice r and s , respectively. � Proposition If s : R → S is a computable surjection, then C S ≤ W C R . Corollary C R is closed under composition for R ∈ { N , 2 N , N × 2 N , N N } . Corollary (Gherardi and Marcone 2009, B. and Gherardi 2011) WKL is closed under composition.