Weihrauch and constructive reducibility between existence statements - - PowerPoint PPT Presentation

Introduction Meta-theorems Parallelization Summary

Weihrauch and constructive reducibility between existence statements

Makoto Fujiwara

JSPS Research Fellow (PD), Meiji University/LMU Munich

CCA 2020 (virtual) 10 September 2020

This work is supported by JSPS KAKENHI Grant Numbers 18K13450 and 19J01239, as well as JSPS Core-to-Core Program (A. Advanced Research Networks).

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Introduction Meta-theorems Parallelization Summary

Introduction

Many existence statements in mathematics can be formalized as Π2 sentences of form ∀f (A(f ) → ∃gB(f , g)) . In general, f and g are possibly tuples of functions respectively, but in this talk, we present Π2 sentences as above for notational simplicity. Here f is called an instance and g is called a solution to f . During these decades, the interrelations between existence statements have been studied extensively in several contexts of reverse mathematics (RM).

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Introduction Meta-theorems Parallelization Summary

There are several corresponding results between the following two RM:

1 RM via Weihrauch reducibility from Computable Analysis; 2 RM over Constructive Mathematics.

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Introduction Meta-theorems Parallelization Summary

There are several corresponding results between the following two RM:

1 RM via Weihrauch reducibility from Computable Analysis; 2 RM over Constructive Mathematics.

• Fact. (Weihrauch RM)

DICHR is Weihrauch equivalent to LLPO.

• Fact. (Const. RM over BISH)

DICHR is constructively equivalent to LLPO. DICHR : ∀α ∈ R (α ≥ 0 ∨ α ≤ 0). LLPO : ∀f N→N ¬

• ∃nNf (2n) = 0 ∧ ∃nNf (2n + 1) = 0
• → ∀nNf (2n) = 0 ∨ ∀nNf (2n + 1) = 0
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Introduction Meta-theorems Parallelization Summary

Definition. The parallelization (or sequential version) P of P ≡ ∀f (A(f ) → ∃gB(f , g)) is ∀fnn∈N (∀n A(fn) → ∃gnn∈N∀n B(fn, gn)) .

• Remark. P ≤W Q =

⇒ P ≤W Q.

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Introduction Meta-theorems Parallelization Summary

Definition. The parallelization (or sequential version) P of P ≡ ∀f (A(f ) → ∃gB(f , g)) is ∀fnn∈N (∀n A(fn) → ∃gnn∈N∀n B(fn, gn)) .

• Remark. P ≤W Q =

⇒ P ≤W Q.

• Fact. (Weihrauch RM)

1 WKL is Weihrauch reducible to

IVT (and vice versa) but not so to IVT.

2 WKL is Weihrauch reducible to

LLPO (and vice versa) but not so to LLPO.

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Introduction Meta-theorems Parallelization Summary

Definition. The parallelization (or sequential version) P of P ≡ ∀f (A(f ) → ∃gB(f , g)) is ∀fnn∈N (∀n A(fn) → ∃gnn∈N∀n B(fn, gn)) .

• Remark. P ≤W Q =

⇒ P ≤W Q.

• Fact. (Weihrauch RM)

1 WKL is Weihrauch reducible to

IVT (and vice versa) but not so to IVT.

2 WKL is Weihrauch reducible to

LLPO (and vice versa) but not so to LLPO.

• Fact. (Const. RM over BISH which accepts AC0,ω)

IVT ⇔ WKL ⇔ LLPO.

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Introduction Meta-theorems Parallelization Summary

In this decade, there are several attempts to characterize Weihrauch RM from such a formalistic approach (Kuyper 2017, Hirst-Mummert 2019, Uftring 2020 etc.). Here we partially characterize the notions of P ≤W Q and P ≤W Q in Weihrauch RM by some derivability notions

• bserved in Constructive RM.

In these two decades, Constructive RM, as well as some

• f Weihrauch RM, have been developed over

(many-sorted) arithmetic as for Friedman-Simpson RM. Our approach: Weihrauch RM

• Const. RM over BISH

Classical RM (Formal)

• Const. RM (Formal)

We employ finite-type arithmetic as our framework.

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Introduction Meta-theorems Parallelization Summary

Hilbert-type system E-HAω (resp. E-PAω) is the finite type extension of HA (resp. PA), of which T is the terms.

• E-HA

ω↾ (resp.

E-PA

ω↾) is the restrictions of E-HAω (resp.

E-PAω) to primitive recursion of type 0 and quantifier-free induction, of which T0 is the terms. Type-1 functions (functions of type NN) definable in T0 (resp. T) coincide with primitive (resp. PA-provably) recursive functions in the ordinary sense.

N HA NN EL0 EL ω

• E-HA

ω↾ + QF-AC0,0

E-HAω + QF-AC0,0

• Fact. (Kohlenbach 2005)

RCAω

0 :≡

E-PA

ω↾ + QF-AC1,0 is a conservative extension of

RCA0 in Friedman-Simpson RM. ACσ,τ : ∀xσ∃y τA(x, y) → ∃Y τ σ∀xσ A(x, Yx).

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Introduction Meta-theorems Parallelization Summary

Definition (Weihrauch reducibility for Π1

2 statements)

For Π1

2 statements P and Q of form ∀f (A(f ) → ∃gB(f , g)),

P is Weihrauch reducible to Q (denoted as P ≤W Q) if there exist Turing functionals Φ and Ψ such that whenever f is an instance of P, then f ′ := Φ(f ) is an instance of Q, and whenever g ′ is a solution to f ′, then g := Ψ(f ⊕ g ′) is a solution to f .

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Introduction Meta-theorems Parallelization Summary

Definition (Weihrauch reducibility for Π1

2 statements)

For Π1

2 statements P and Q of form ∀f (A(f ) → ∃gB(f , g)),

P is Weihrauch reducible to Q (denoted as P ≤W Q) if there exist Turing functionals Φ and Ψ such that whenever f is an instance of P, then f ′ := Φ(f ) is an instance of Q, and whenever g ′ is a solution to f ′, then g := Ψ(f ⊕ g ′) is a solution to f . In the following, we define the primitive recursive (in the sense of G¨

• del/Kleene) variants of Weihrauch

reducibility in which Turing functionals for the reduction are replaced by primitive recursive (total) functionals (in the sense of G¨

• del/Kleene).

The verification theory is also concerned.

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Introduction Meta-theorems Parallelization Summary

PR Variants of Weihrauch Reducibility in Sω

Definition

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)).

P is G¨

• del-primitive-recursive Weihrauch reducible

to Q in Sω if there exist closed terms s and t (of suitable types) in T such that Sω proves

∀f (A1(f ) → A2(sf ))∧∀f , g′ B2(sf , g′) ∧ A1(f ) → B1(f , tfg′)

• .

P is Kleene-primitive-recursive Weihrauch reducible to Q in Sω if there exist closed terms s and t (of suitable types) in T0 such that Sω proves the same sentence.

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Introduction Meta-theorems Parallelization Summary

• Proposition. (cf. Brattka/Gherardi 2011)

WKL is Kleene-primitive-recursive Weihrauch reducible to

• LLPO in

E-PA

ω↾ + QF-AC0,0 (which contains Π0 1-IND).

Remark.

• E-PA

ω↾ + QF-AC0,0 and

E-PA

ω↾ + Π0 1-AC0,0 are conservative

extensions of RCA0 and ACA0 in Friedman-Simpson RM respectively.

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Introduction Meta-theorems Parallelization Summary

Definition (Normal Reducibility in Sω)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)).

We say that P is normally reducible to Q in Sω if Sω proves

∀f

• A1(f ) → ∃f ′

A2(f ′) ∧ ∀g′ B2(f ′, g′) → ∃gB1(f , g)

• .

The normal reducibility, which requires a specific form of a proof of that Q implies P, is a stronger notion than just proving Q → P. Since intuitionistic finite-type arithmetic with a choice principle roughly corresponds to Bishop’s constructive mathematics, one may regard the normal reducibility in a nearly intuitionistic finite-type arithmetic as a sort of constructive reducibility.

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Introduction Meta-theorems Parallelization Summary

The normal reducibility in the context of a classical system is nothing but provability in the system: Proposition. Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)), and Sω be a classical finite-type arithmetic such that Sω ⊢ ∃f ′A2(f ′).∗ If Sω ⊢ Q → P, then P is normally reducible to Q in Sω.

∗Note that if Sω ⊢ ∀f ′¬A2(f ′), then Sω ⊢ Q → P just means Sω ⊢ P

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Introduction Meta-theorems Parallelization Summary

The normal reducibility in the context of a classical system is nothing but provability in the system: Proposition. Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)), and Sω be a classical finite-type arithmetic such that Sω ⊢ ∃f ′A2(f ′).∗ If Sω ⊢ Q → P, then P is normally reducible to Q in Sω.

• Remark. The above proposition does not hold for intuitionistic

finite-type arithmetic. Thus, in an intuitionistic context, the notion of normal reducibility is a strictly stronger notion than provability for existence statements.

∗Note that if Sω ⊢ ∀f ′¬A2(f ′), then Sω ⊢ Q → P just means Sω ⊢ P

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Introduction Meta-theorems Parallelization Summary

Characterization of a Weakening of P ≤W Q

Proposition.

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)) with ∃-free (containing neither ∃ nor ∨) formulas A1, A2, B1, B2.

P is G¨

• del-primitive-recursive Weihrauch reducible to Q

in E-PAω ⇐ ⇒ P is normally reducible to Q in E-HAω. P is Kleene-primitive-recursive Weihrauch reducible to Q in E-PA

ω↾ ⇐

⇒ P is normally reducible to Q in E-HA

ω↾.

Idea of the Proof. (⇐) is shown by using the modified realizability interpretation; (⇒) is shown by using the negative translation.

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Introduction Meta-theorems Parallelization Summary

Meta-theorems with respect to RCA0 and ACA0

AC0,0 : ∀xN∃yNA(x, y) → ∃Y NN∀xN A(x, Yx). QF-AC0,0 : AC0,0 where A is restricted to Aqf. Π0

1-AC0,0 :

AC0,0 where A is restricted to ∀zNAqf DNS0 : ∀xN¬¬A → ¬¬∀xNA. Σ0

1-DNS0 :

DNS0 where A is restricted to ∃yNAqf. Σ0

2-DNS0 :

DNS0 where A is restricted to ∃yN∀zNAqf.

• Lemma. (Kohlenbach/F. 2018, F. 2020)

The negative translation of an instance of QF-AC0,0 (resp. Π0

1-AC0,0) is derived from QF-AC0,0 + Σ0 1-DNS0 (resp.

Π0

1-AC0,0 + Σ0 2-DNS0).

The modified realizability interpretation of an instance of Σ0

1-DNS0 (resp. Σ0 2-DNS0) is derived from

Σ0

1-DNS0 + QF-AC0,0 (resp. Σ0 2-DNS0 + Π0 1-AC0,0).

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Introduction Meta-theorems Parallelization Summary

• Theorem. (F. 2020)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g))

with ∃-free formulas A1, A2, B1, and B2. P is G¨

• del-primitive-recursive Weihrauch reducible to Q

in E-PAω (resp. E-PAω + QF-AC0,0, E-PAω + Π0

1-AC0,0)

if and only if P is normally reducible to Q in E-HAω (resp. E-HAω + QF-AC0,0 + Σ0

1-DNS0, E-HAω +

Π0

1-AC0,0 + Σ0 2-DNS0).

P is Kleene-primitive-recursive Weihrauch reducible to Q

in E-PA

ω↾ (resp.

E-PA

ω↾ + QF-AC0,0,

E-PA

ω↾ + Π0 1-AC0,0)

if and only if P is normally reducible to Q in E-HA

ω↾

(resp. E-HA

ω↾ + QF-AC0,0 + Σ0 1-DNS0,

E-HA

ω↾ +

Π0

1-AC0,0 + Σ0 2-DNS0).

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Introduction Meta-theorems Parallelization Summary

Observation: IVT, WKLc, and WKL2

A lot of existing proofs in constructive reverse mathematics show not only provability but rather normal

• reducibility. However, this is not always the case.

Here, as an example, we deal with some derivability results in constructive reverse mathematics on the intermediate value theorem IVT, the convex weak K¨

• nig’s

lemma WKLc, and the weak K¨

• nig’s lemma for trees

having exactly 2 branches for each non-0-height WKL2.

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Introduction Meta-theorems Parallelization Summary

Observation: IVT, WKLc, and WKL2

A lot of existing proofs in constructive reverse mathematics show not only provability but rather normal

• reducibility. However, this is not always the case.

Here, as an example, we deal with some derivability results in constructive reverse mathematics on the intermediate value theorem IVT, the convex weak K¨

• nig’s

lemma WKLc, and the weak K¨

• nig’s lemma for trees

having exactly 2 branches for each non-0-height WKL2. Theorem.(Berger/Ishihara/Kihara/Nemoto 2019)

1 EL0 ⊢ IVT → WKL2. 2 EL0 ⊢ IVT ↔ WKLc.

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Introduction Meta-theorems Parallelization Summary

• Remark. All of IVT, WKLc and WKL2 are formalized as Π2

sentences in the applicable form of our meta-theorems.

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Introduction Meta-theorems Parallelization Summary

• Remark. All of IVT, WKLc and WKL2 are formalized as Π2

sentences in the applicable form of our meta-theorems. Proposition. WKL2 is normally reducible to IVT in E-HA

ω↾ + QF-AC0,0.

• Proof. By inspecting the proof of EL0 ⊢ IVT → WKL2.

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Introduction Meta-theorems Parallelization Summary

• Remark. All of IVT, WKLc and WKL2 are formalized as Π2

sentences in the applicable form of our meta-theorems. Proposition. WKL2 is normally reducible to IVT in E-HA

ω↾ + QF-AC0,0.

• Proof. By inspecting the proof of EL0 ⊢ IVT → WKL2.

Proposition. IVT is normally reducible to WKLc in E-HA

ω↾ + QF-AC0,0.

• Proof. By inspecting the proof of EL0 ⊢ WKLc → IVT.

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Introduction Meta-theorems Parallelization Summary

• Remark. All of IVT, WKLc and WKL2 are formalized as Π2

sentences in the applicable form of our meta-theorems. Proposition. WKL2 is normally reducible to IVT in E-HA

ω↾ + QF-AC0,0.

• Proof. By inspecting the proof of EL0 ⊢ IVT → WKL2.

Proposition. IVT is normally reducible to WKLc in E-HA

ω↾ + QF-AC0,0.

• Proof. By inspecting the proof of EL0 ⊢ WKLc → IVT.

Corollary. WKL2 (resp. IVT) is Kleene-primitive-recursive Weihrauch reducible to IVT (resp. WKLc) in E-PA

ω↾ + QF-AC0,0 (even

in E-HA

ω↾ by the proof of our meta-theorem).

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Introduction Meta-theorems Parallelization Summary

Remark. On the other hand, in the proof of IVT → WKLc in Berger/Ishihara/Kihara/Nemoto 2019, for a given infinite convex tree T, IVT is first used to construct an infinite convex subtree T ′ having at most 2 branches for each height, and then it is used again for taking an infinite path through T ′.

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Introduction Meta-theorems Parallelization Summary

• Definition. (2-copies Generalizations)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)). For a finite-type arithmetic Sω containing E-HAω, P is G¨

• del/Kleene-primitive-recursive Weihrauch reducible to

the 2-copies of Q in Sω if there exist closed terms s, t, and u in T/T0 such that Sω proves ∀f (A1(f ) → A2(sf ))∧ ∀f , g′ (B2(sf , g′) ∧ A1(f ) → A2(tfg′)) ∧ ∀f , g′, g′′ (B2(tfg′, g′′) ∧ B2(sf , g′) ∧ A1(f ) → B1(f , ufg′g′′)) . For a finite-type arithmetic Sω containing E-HA

ω↾, P is

normally reducible to the 2-copies of Q in Sω if Sω proves ∀f     A1(f ) → ∃f ′  A2(f ′) ∧ ∀g′   B2(f ′, g′) → ∃f ′′

• A2(f ′′) ∧ ∀g′′

B2(f ′′, g′′) → ∃gB1(f , g)

• 

       .

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Introduction Meta-theorems Parallelization Summary

• Corollary. (k-copies Generalizations)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g))

with ∃-free formulas A1, A2, B1, and B2.

P is G¨

• del-primitive-recursive Weihrauch reducible to the

k-copies of Q in E-PAω (resp. E-PAω + QF-AC0,0, E-PAω + Π0

1-AC0,0) if and only if P is

normally reducible to the k-copies of Q in E-HAω (resp. E-HAω + QF-AC0,0 + Σ0

1-DNS0, E-HAω + Π0 1-AC0,0 +

Σ0

2-DNS0).

P is Kleene-primitive-recursive Weihrauch reducible to the k-copies of Q in E-PA

ω↾ (resp.

• E-PA

ω↾ + QF-AC0,0,

E-PA

ω↾ + Π0 1-AC0,0) if and only if P is

normally reducible to the k-copies of Q in E-HA

ω↾ (resp.

• E-HA

ω↾ + QF-AC0,0 + Σ0 1-DNS0,

E-HA

ω↾ + Π0 1-AC0,0 +

Σ0

2-DNS0).

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Introduction Meta-theorems Parallelization Summary

Applying the meta-theorem for k = 2 to the proof of IVT → WKLc in Berger/Ishihara/Kihara/Nemoto 2019, one can obtain a non-trivial result in the style of computable analysis: Proposition. WKLc is Kleene-primitive-recursive Weihrauch reducible to the 2-copies of IVT in E-PA

ω↾ + QF-AC0,0.

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Introduction Meta-theorems Parallelization Summary

Remark In general, for an intuitionistic finite-type arithmetic iSω which satisfies the deduction theorem and proves the existence of an instance of Q, iSω proves Q → P if and

• nly if P is normally reducible to Q in iSω + Q.

In the context of computable analysis (or higher order reverse mathematics), this corresponds to the notion that there exists a G¨

• del/Kleene-primitive-recursive functional

Φ (verifiably in Sω) which transforms a functional providing a solution to an instance of Q to a functional providing a solution to an instance of P.

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Introduction Meta-theorems Parallelization Summary

Recap

P is normally reducible to Q in iSω. ↓ P is normally reducible to the 2-copies of Q in iSω. ↓ P is normally reducible to the 3-copies of Q in iSω. ↓ . . . ↓ P is normally reducible to Q in iSω + Q

• iSω ⊢ Q → P

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Introduction Meta-theorems Parallelization Summary

Definition.(Normal Derivability)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)). P is normally T-derivable from Q in iSω if there exists a closed term s in T such that iSω proves the following two:

1 ∀f

• A1(f ) → ∀mNA2(smf )
• ;

2 ∀f

• A1(f ) ∧ ∀mN (A2(smf ) → ∃g′B2 (smf , g′))

→ ∃gB1(f , g)

• .

The normal T0-derivability is defined with using T0 in stead of T. The fact that P is normally derivable from Q (in iSω) demands some proof of that Q implies P with the following structure:

1 Fix f such that A1(f ); 2 Assuming A1(f ), derive ∃gB1(f , g) by using Q for the

countably many instances which are provided primitive recursively (in the sense of G¨

• del/Kleene) in f .

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Introduction Meta-theorems Parallelization Summary

Definition.(Normal Derivability)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)). P is normally T-derivable from Q in iSω if there exists a closed term s in T such that iSω proves the following two:

1 ∀f

• A1(f ) → ∀mNA2(smf )
• ;

2 ∀f

• A1(f ) ∧ ∀mN (A2(smf ) → ∃g′B2 (smf , g′))

→ ∃gB1(f , g)

• .

The normal T0-derivability is defined with using T0 in stead of T.

• Remark. The normal derivability is (properly) weaker than the

normal reducibility: Omitting ∀mN from the definition of the normal derivability makes it (intuitionistically) equivalent to the normal reducibility.

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Introduction Meta-theorems Parallelization Summary

• Proposition. (cf. Berger/Ishihara/Schuster 2012)

WKL is normally T0-derivable from LLPO in

• E-HA

ω↾ + Π0 1-AC0,0 ∨ + QF-AC0,0 + Σ0 1-DNS0, where

Π0

1-AC0,0 ∨ :

∀nN ∀xNAqf(n, x) ∨ ∀y NBqf(n, y)

• → ∃hN→N∀nN

h(n) = 0 → ∀xNAqf(n, x)

• ∧
• h(n) = 0 → ∀y NBqf(n, y)
• .

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Introduction Meta-theorems Parallelization Summary

• Theorem. (Parallelization)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)) with ∃-free formulas A1, A2, B1, and B2.

1 P is G¨

• del-primitive-recursive Weihrauch reducible to

Q in E-PAω ⇐ ⇒ P is normally T-derivable from Q in E-HAω + AC0,ω

2 P is Kleene-primitive-recursive Weihrauch reducible to

Q in

• E-PA

ω↾

⇐ ⇒ P is normally T0-derivable from Q in E-HA

ω↾ + AC0,ω

AC0,ω(countable choice) : ∀xN∃f τA(x, f ) → ∃F N→τ∀xNA(x, Fx).

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Introduction Meta-theorems Parallelization Summary

• Theorem. (Parallelization)

Let P : ∀f (A1(f ) → ∃g B1(f , g)) , Q : ∀f (A2(f ) → ∃gB2(f , g)) with ∃-free formulas A1, A2, B1, and B2.

1 P is G¨

• del-primitive-recursive Weihrauch reducible to

Q in E-PAω (resp. E-PAω + QF-AC0,0, E-PAω + Π0

1-AC0,0)

⇐ ⇒ P is normally T-derivable from Q in E-HAω + AC0,ω (resp. E-HAω + AC0,ω + Σ0

1-DNS0, E-HAω + AC0,ω + Σ0 2-DNS0). 2 P is Kleene-primitive-recursive Weihrauch reducible to

Q in

• E-PA

ω↾ (resp.

E-PA

ω↾ + QF-AC0,0,

E-PA

ω↾ + Π0 1-AC0,0)

⇐ ⇒ P is normally T0-derivable from Q in E-HA

ω↾ + AC0,ω (resp.

• E-HA

ω↾ + AC0,ω + Σ0 1-DNS0,

E-HA

ω↾ + AC0,ω + Σ0 2-DNS0).

AC0,ω(countable choice) : ∀xN∃f τA(x, f ) → ∃F N→τ∀xNA(x, Fx).

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Introduction Meta-theorems Parallelization Summary

A Example: WKL and LLPO

There seem to be many results (proofs) in computable analysis and constructive reverse mathematics to which our meta-theorems are applicable. By applying (the proof of) our meta-theorem, the following two results are shown to be equivalent:

• Proposition. (cf. Berger/Ishihara/Schuster 2012)

WKL is normally T0-derivable from LLPO in

• E-HA

ω↾ + Π0 1-AC0,0 ∨ + QF-AC0,0 + Σ0 1-DNS0.

• Proposition. (cf. Brattka/Gherardi 2011)

WKL is Kleene-primitive-recursive Weihrauch reducible to

• LLPO in

E-PA

ω↾ + QF-AC0,0.

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Introduction Meta-theorems Parallelization Summary

Observation

The direct proofs of the previous two propositions are somewhat similar (using Π0

1-IND for the verifications).

Nevertheless, the primitive recursive witnesses for the latter is not obvious from the proof of the former. On the other hand, the proof of the latter heavily uses classical logic for the verification, and hence, the former is also not an immediate consequence from the latter. Thus our meta-theorems should give rise to new results in

• ne of the contexts from the results (proofs) in the other

context.

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Introduction Meta-theorems Parallelization Summary

Remark. In our meta-theorems for parallelizations, the countable choice AC0,ω is crucial (it cannot be replaced by QF-AC0,0): If AC0,ω be replaced by QF-AC0,0, since WKL is Kleene-primitive-recursive Weihrauch reducible to LLPO in

• E-PA

ω↾ + QF-AC0,0, we have that WKL is normally

T0-derivable from LLPO in E-HA

ω↾ + QF-AC0,0 + Σ0 1-DNS0,

and hence, WKL is provable in E-PA

ω↾ + QF-AC0,0. This is a

contradiction.

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Introduction Meta-theorems Parallelization Summary

Recap

P is normally reducible to Q in iSω. = ⇒ P is normally derivable from Q in iSω + AC0,ω ↓ P is normally reducible to the 2-copies of Q in iSω. ↓ ↓ P is normally reducible to the 3-copies of Q in iSω. ↓ ↓ . . . . . . P is normally reducible to Q in iSω + Q

• iSω ⊢ Q → P

= ⇒ iSω + AC0,ω ⊢ Q → P

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Introduction Meta-theorems Parallelization Summary

Syntactical Restriction

Remark. Our meta-theorems are applicable only for existence statements formalized in finite-type arithmetic with ∃-free

• formulas. Of course, there are many mathematical

statements to which our meta-theorems are not applicable (e.g. the Bolzano–Weierstrass theorem). There is a counterexample which shows that our meta-theorems does not hold already for ∀f NN ∃xN∀y NAqf(f , x, y) → ∃g NN∀zNBqf(f , g, z)

• .

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Introduction Meta-theorems Parallelization Summary

References

1 J. Berger, H. Ishihara, T. Kihara and T. Nemoto, The binary

expansion and the intermediate value theorem in constructive reverse mathematics, Arch. Math. Logic 58, pp. 203–217, 2019.

2 J. Berger, H. Ishihara and P. Schuster, The weak K¨

• nig

lemma, Brouwer’s fan theorem, de Morgan’s law, and dependent choice, Rep. Math. Logic 47, pp. 63–86, 2012.

3 V. Brattka and Gherardi, Weihrauch degrees, omniscience

principles and weak computability, J. Symbolic Logic 76(1),

• pp. 143–176, 2011.

4 M. Fujiwara, Weihrauch and constructive reducibility between

existence statements, Computability, Pre-press, 2020.

5 M. Fujiwara, Parallelizations in Weihrauch reducibility and

constructive reverse mathematics, LNCS 12098, pp. 38-49, 2020.

Thank you!

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