Proofs as Programs Revisited Ryota Akiyoshi Waseda Institute for - - PowerPoint PPT Presentation

“Proofs as Programs” Revisited

Ryota Akiyoshi

Waseda Institute for Advanced Study Keio University

July 27th., 2018

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Aim of This Talk

▶ The aim is to revisit Schwichtenberg’s works by focusing on parameter

subsystems of Girard’s F.

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Proofs as Programs by Schwichtenberg

▶ Proofs are regarded as programs (Curry=Howard isomorphism) ▶ Schwichtenberg measured the complexity as programs of proofs in

arithmetic.

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Proofs as Programs by Schwichtenberg

▶ Proofs are regarded as programs (Curry=Howard isomorphism) ▶ Schwichtenberg measured the complexity as programs of proofs in

arithmetic.

▶ Proofs as programs could contain such a “complicated” structures.

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Proofs as Programs by Schwichtenberg

▶ Proofs are regarded as programs (Curry=Howard isomorphism) ▶ Schwichtenberg measured the complexity as programs of proofs in

arithmetic.

▶ Proofs as programs could contain such a “complicated” structures.

Theorem (Schwichtenberg90)

Let r be a closed term of type N → N in arithmetic. Then, there is m such that all n ≥ m |rn| ≤ GD0Dm+2

1

0(n).

(D0,D1 are the collapsing functions, and G is a slow growing hierarchy.)

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Proofs as Programs by Schwichtenberg

▶ Proofs as programs could contain such a complicated structures.

|rn| ≤ GD0Dm+2

1

0(n).

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Proofs as Programs by Schwichtenberg

▶ Proofs as programs could contain such a complicated structures.

|rn| ≤ GD0Dm+2

1

0(n).

▶ Strategy for getting this result:

• 1. Normalize a given term rn and measure the size of it. (We need

D0Dm+2

1

0(n) here. )

• 2. To climb down the “big” tree ordinal by the slow growing

hierarchy using ideas by Wainer-Girard and Arai.

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Proofs as Programs by Schwichtenberg

▶ Proofs as programs could contain such a complicated structures.

|rn| ≤ GD0Dm+2

1

0(n).

▶ Strategy for getting this result:

• 1. Normalize a given term rn and measure the size of it. (We need

D0Dm+2

1

0(n) here. )

• 2. To climb down the “big” tree ordinal by the slow growing

hierarchy using ideas by Wainer-Girard and Arai.

▶ The bound is sharp. A specific program of ∀x∃yA(x,y) has such a

complexity.

▶ These arguments are implemented in Scheme.

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Some Literatures

▶ Arai, A slow growing analogue to Buchholz’ proof, 1991. ▶ Buchholz, An independence result for Π1

1-CA+BI, 1987.

▶ Girard, Proof Theory and Logical Complexity, Vol 1, 1987.

(Volume 2 is available: http://girard.perso.math.cnrs.fr/Archives4.html)

▶ Schwichtenberg, Proofs as Programs, 1990. ▶ Schwichtenberg and Wainer, Ordinal Bounds for Programs, 1994.

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Goal of This Talk

▶ The aim: to revisit S’s works by focusing on parameter subsystems of

Girard’s F.

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Goal of This Talk

▶ The aim: to revisit S’s works by focusing on parameter subsystems of

Girard’s F.

▶ Two advantages of our approach:

• 1. Our approach is simpler, smoother.

▶ The syntax of F is very simple.

• 2. This talk is about the weakest theory dealing with the type N:

N : ∀α.((α ⇒ α) ⇒ α ⇒ α)

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Goal of This Talk

▶ The aim: to revisit S’s works by focusing on parameter subsystems of

Girard’s F.

▶ Two advantages of our approach:

• 1. Our approach is simpler, smoother.

▶ The syntax of F is very simple.

• 2. This talk is about the weakest theory dealing with the type N:

N : ∀α.((α ⇒ α) ⇒ α ⇒ α)

• 3. It is possible to extend our result into stronger theories of

inductive definitions, uniformly.

▶ Typical example of the next level is Brouwer’s ordinals:

O : ∀α.((N ⇒ α) ⇒ α) ⇒ (α ⇒ α) ⇒ (α ⇒ α)

▶ This is more direct, too.

▶ Terms in F can be regarded as programs. 6 / 26

Another Motivation

▶ Another motivation:

▶ to connect a traditional method called the Ω-rule in proof-theory

with the context of the lambda calculus.

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Another Motivation

▶ Another motivation:

▶ to connect a traditional method called the Ω-rule in proof-theory

with the context of the lambda calculus.

▶ Examples of this direction:

▶ Terui, “MacNeille completion and Buchholz’ Omega rule for

parameter-free second order logics”, CSL, 2018.

▶ Akiyoshi and Terui, “Strong normalization for the parameter-free

polymorphic lambda calculus based on the Omega-rule”, FSCD, 2016.

▶ Maybe, we could apply this method to another type theories, but I

don’t know...

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Some Literatures

▶ Akiyoshi, “The Upperbound of the Length of the Reductions in a

Subsystem of Girard’s F”, preprint, 2018.

▶ Akiyoshi, ““Proofs as Programs” in Parameter-Free Fragments of

System F”, submitted, 2018.

▶ Akiyoshi, “A Formalization of Brouwer’s Argument for Bar

Induction”, WoLLIC, 2018.

▶ Terui, “MacNeille completion and Buchholz’ Omega rule for

parameter-free second order logics”, CSL, 2018.

▶ Akiyoshi, “An Ordinal-Free Proof of the Complete Cut-Elimination

Theorem for Π1

1-CA + BI with the ω-rule”, The Mints’ memorial issue

• f the IfCoLog Journal of Logics and their Applications, 2017.

▶ Akiyoshi and Terui, “Strong Normalization for the Parameter-Free

Polymorphic Lambda Calculus Based on the Ω-Rule”, FSCD 2016.

▶ Akiyoshi and Mints, “An Extension of the Omega-Rule”, AML, 2016.

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Definition of Syntax Definition

The types are defined by: A,B ::= α | A ⇒ B | ∀α.A where ∀α.A is closed and A is ∀-free. Types in this set are “parameter-free”.

Definition

Terms are defined as follows: xA (λxA.MB)A⇒B (MA⇒BNA)B (Λα.MA)∀α.A (M∀α.AB)A[α/B] with the standard proviso.

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Examples

Examples of types in this language: N := ∀α.(α ⇒ α) ⇒ (α ⇒ α) (natural numbers) T := ∀α.(α ⇒ α ⇒ α) ⇒ (α ⇒ α) (binary trees)

Remark

Girard’s maxim: Peano Arithmetic is (best viewed as) a theory of one inductive definition.

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Examples

Examples of types in this language: N := ∀α.(α ⇒ α) ⇒ (α ⇒ α) (natural numbers) T := ∀α.(α ⇒ α ⇒ α) ⇒ (α ⇒ α) (binary trees)

Remark

Girard’s maxim: Peano Arithmetic is (best viewed as) a theory of one inductive definition. But, we cannot express the following: L(N) := ∀α.(N ⇒ α ⇒ α) ⇒ (α ⇒ α) (lists over N) O := ∀α.((N ⇒ α) ⇒ α) ⇒ (α ⇒ α) ⇒ (α ⇒ α) (Brouwer ordinals)

Remark

This kind of restriction originally goes back to Gaisi Takeuti’s works in 1950’s. Cf. his “On the fundamental conjecture of GLC I-VI”.

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Tree Ordinals Definition (Buchholz87)

The tree classes Tσ (σ ≤ 2) are defined as follows:

▶ If α : I → Tσ is a function with I : /

0,{0}, or Tρ for some ρ < σ, then α ∈ Tσ. Some notations.

• 1. 0 for α : /

0 → Tσ,

• 2. β + for α : {0} → Tσ with α(0) = β.

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Tree Ordinals Definition (Buchholz87)

The tree classes Tσ (σ ≤ 2) are defined as follows:

▶ If α : I → Tσ is a function with I : /

0,{0}, or Tρ for some ρ < σ, then α ∈ Tσ. Some notations.

• 1. 0 for α : /

0 → Tσ,

• 2. β + for α : {0} → Tσ with α(0) = β.

Remark

• 1. T0 is identified with N (the set of natural numbers) ,
• 2. T1 is the set of countable trees.

The operations of addition, multiplication, and exponentiation of trees are defined in the standard way. For example, (α +β)+γ = α +(β +γ),(α ×β)×γ = α ×(β ×γ),etc...

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Collapsing Functions on Tree Ordinals

Let Ω0 := N, Ω1 := the set of countable tree ordinals.

Definition (Buchholz87, Arai91)

The collapsing functions Dσ : Tv → Tσ+1 for σ < v ≤ 2 are defined as follows:

• 1. Dσ0 := Ωσ,
• 2. Dσ(α +1) := (Dσ(α)×(n+1))n∈ω,
• 3. If ρ ≤ σ, then Dσ((αξ)ξ∈Tρ) := (Dσαξ)ξ∈Tρ,
• 4. If σ < µ +1, then Dσ((αξ)ξ∈Tµ+1) := (Dσαξn)n∈ω where

ξ0 := Ωµ,ξn+1 := Dµαξn.

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Collapsing Functions on Tree Ordinals

Let Ω0 := N, Ω1 := the set of countable tree ordinals.

Definition (Buchholz87, Arai91)

The collapsing functions Dσ : Tv → Tσ+1 for σ < v ≤ 2 are defined as follows:

• 1. Dσ0 := Ωσ,
• 2. Dσ(α +1) := (Dσ(α)×(n+1))n∈ω,
• 3. If ρ ≤ σ, then Dσ((αξ)ξ∈Tρ) := (Dσαξ)ξ∈Tρ,
• 4. If σ < µ +1, then Dσ((αξ)ξ∈Tµ+1) := (Dσαξn)n∈ω where

ξ0 := Ωµ,ξn+1 := Dµαξn.

Remark

• 1. In the last clause, the point is that the indexes ξn are tree ordinals.
• 2. D00 = ω, D01 = ω2,D0ω = ωω,

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Collapsing Functions on Tree Ordinals

Let Ω0 := N, Ω1 := the set of countable tree ordinals.

Definition (Buchholz87, Arai91)

The collapsing functions Dσ : Tv → Tσ+1 for σ < v ≤ 2 are defined as follows:

• 1. Dσ0 := Ωσ,
• 2. Dσ(α +1) := (Dσ(α)×(n+1))n∈ω,
• 3. If ρ ≤ σ, then Dσ((αξ)ξ∈Tρ) := (Dσαξ)ξ∈Tρ,
• 4. If σ < µ +1, then Dσ((αξ)ξ∈Tµ+1) := (Dσαξn)n∈ω where

ξ0 := Ωµ,ξn+1 := Dµαξn.

Remark

• 1. In the last clause, the point is that the indexes ξn are tree ordinals.
• 2. D00 = ω, D01 = ω2,D0ω = ωω, ...,D0Ω1 = ε0.

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Collapsing Functions on Tree Ordinals

Let Ω0 := N, Ω1 := the set of countable tree ordinals.

Definition (Buchholz87, Arai91)

The collapsing functions Dσ : Tv → Tσ+1 for σ < v ≤ 2 are defined as follows:

• 1. Dσ0 := Ωσ,
• 2. Dσ(α +1) := (Dσ(α)×(n+1))n∈ω,
• 3. If ρ ≤ σ, then Dσ((αξ)ξ∈Tρ) := (Dσαξ)ξ∈Tρ,
• 4. If σ < µ +1, then Dσ((αξ)ξ∈Tµ+1) := (Dσαξn)n∈ω where

ξ0 := Ωµ,ξn+1 := Dµαξn.

Remark

• 1. In the last clause, the point is that the indexes ξn are tree ordinals.
• 2. D00 = ω, D01 = ω2,D0ω = ωω, ...,D0Ω1 = ε0.
• 3. In this talk, we identify ordinals with its notations.

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The Relation |=a M : A

▶ |=a M : A means “a term M has the size or complexity α”.

Definition

The relation |=a M : A for a ∈ T is defined inductively as follows: (Var) If |=a Ni : Bi for i = 1,...,m with 0 ≤ m, then |=a+1 x⃗ N : A, (<1) If |=b M : A and b <1 a, then |=a M : A, (abs) If |=a N : C, then |=a+1 λx.N : B → C, (Abs) If |=a N : B, then |=a+1 λα.N : ∀α.B.

Remark

▶ Idea is to define the “logical” domain over which we can quantify. ▶ In this definition, α could be just a natural number.

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The Relation ⊢a

m M : A

Definition

The relation ⊢a

m M : A for a ∈ T and m < ω is defined by adding the

following to |=a: (ω+) If the following conditions are satisfied

• i. tp(a) = Ω1, ⊢a−

m N : ∀α.B,

• ii. ∀z ∈ T1∀K ∈ Π1
• 1. |=z K : B(α) implies ⊢a[z]

m Hz : A,

then ⊢a

m ND : A,

(Cut) If ⊢a

m N with lev(N) ≤ m and and there is a sequence of terms⃗

L such that ⊢a

m Li for i = 1,...,n, then ⊢a+1 m

N⃗ L.

Remark

▶ Hz in the formulation of (ω+) could depend on z. ▶ a is a limit of a[0],a[1],..., hence could be infinite.

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Intuition of the ω+-Rule

▶ Picture of the ω+-rule:

. . . . ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

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Intuition of the ω+-Rule

▶ Picture of the ω+-rule:

. . . . ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ The right deduction is based on BHK-reading of ⇒:

∀α.B ⇒ B[α/D]

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Intuition of the ω+-Rule

▶ Picture of the ω+-rule:

. . . . ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ The right deduction is based on BHK-reading of ⇒:

∀α.B ⇒ B[α/D]

▶ We can derive Comprehension using this:

[∀α.B]1 ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+ ∀α.B ⇒ B[α/D] → I,1

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Intuition of the ω+-Rule

▶ We can compare the picture with the definition:

. . . . ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+ (ω+) If the following conditions are satisfied

• i. tp(a) = Ω1, ⊢a−

m N : ∀α.B,

• ii. ∀z ∈ T1∀K ∈ Π1
• 1. |=z K : B(α) implies ⊢a[z]

m Hz : B[α/D],

then ⊢a

m ND : B[α/D].

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Predicative Normalization

The following corresponds to predicative c.e. in infinitary proof-theory.

Lemma

There is an operation D1 on terms such that If ⊢α

m+1 M : A, then ⊢D1a m

D1(M) : A.

• Proof. The argument is more or less the same as the standard one. □

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Idea of Impredicative Normalization

. . . . B ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ Impredicative Normalization (Collapsing): Elimination of ω+

impredicaticve normalization = taking a subtree

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Idea of Impredicative Normalization

. . . . B ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ Impredicative Normalization (Collapsing): Elimination of ω+

impredicaticve normalization = taking a subtree

▶ Strategy of normalization by induction on a given derivation:

• 1. Normalize the proof of B. Let d be the result.

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Idea of Impredicative Normalization

. . . . B ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ Impredicative Normalization (Collapsing): Elimination of ω+

impredicaticve normalization = taking a subtree

▶ Strategy of normalization by induction on a given derivation:

• 1. Normalize the proof of B. Let d be the result.
• 2. Take the subproof of ω+ indexed by d, say hd.

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Idea of Impredicative Normalization

. . . . B ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ Impredicative Normalization (Collapsing): Elimination of ω+

impredicaticve normalization = taking a subtree

▶ Strategy of normalization by induction on a given derivation:

• 1. Normalize the proof of B. Let d be the result.
• 2. Take the subproof of ω+ indexed by d, say hd.
• 3. (If necessary) normalize hd.

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Idea of Impredicative Normalization

. . . . B ∀α.B ... [K : B(α)] B[α/D] S... ∀α.B ⇒ B[α/D] B[α/D] ω+

▶ Impredicative Normalization (Collapsing): Elimination of ω+

impredicaticve normalization = taking a subtree

▶ Strategy of normalization by induction on a given derivation:

• 1. Normalize the proof of B. Let d be the result.
• 2. Take the subproof of ω+ indexed by d, say hd.
• 3. (If necessary) normalize hd.

▶ The following is the result:

. . . . hd B(α) B[α/D] S

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Impredicative Normalization Lemma

There is an operation D0 on terms such that if ⊢a

0 M : A with A ∈ Π1 1, then

|=D0a D0(M) : A.

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Embedding Theorem Theorem

Let M be a term such that all subterms of it have levels ≤ m. Also, let⃗ y be any sequence of variables such that M ⃗ y is well-typed. Then, there exists k such that ⊢Ω1×k

m

M ⃗ y.

Corollary

If M : A and⃗ y is any sequence of variables such that A is Π1

1 and M

⃗ y is well-typed, then |=D0Dm

1 Ω×n D0Dm

1 (M

⃗ y) : C for some n,C.

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Slow Growing Hierarchy on Ordinals

Next, we introduce the slow growing hierarchy by which we climb down the set of countable trees (T1).

Definition

Ga : N → N for a ∈ T1 is defined by induction on a:

• 1. G0(n) := 0,
• 2. Ga+1(n) := Ga(n)+1,
• 3. Ga(n) := Ga[n](n) if tp(a) := ω. (when a is limit)

Note that ω[n] = n holds (by the definition).

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Slow Growing Hierarchy on Ordinals

Next, we introduce the slow growing hierarchy by which we climb down the set of countable trees (T1).

Definition

Ga : N → N for a ∈ T1 is defined by induction on a:

• 1. G0(n) := 0,
• 2. Ga+1(n) := Ga(n)+1,
• 3. Ga(n) := Ga[n](n) if tp(a) := ω. (when a is limit)

Note that ω[n] = n holds (by the definition).

Remark

It holds that Ga+b(n) = Ga(n)+Gb(n). Gk(n) = k,

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Slow Growing Hierarchy on Ordinals

Next, we introduce the slow growing hierarchy by which we climb down the set of countable trees (T1).

Definition

Ga : N → N for a ∈ T1 is defined by induction on a:

• 1. G0(n) := 0,
• 2. Ga+1(n) := Ga(n)+1,
• 3. Ga(n) := Ga[n](n) if tp(a) := ω. (when a is limit)

Note that ω[n] = n holds (by the definition).

Remark

It holds that Ga+b(n) = Ga(n)+Gb(n). Gk(n) = k, Gω(n) = Gω[n](n) = Gn(n) = n,

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Slow Growing Hierarchy on Ordinals

Next, we introduce the slow growing hierarchy by which we climb down the set of countable trees (T1).

Definition

Ga : N → N for a ∈ T1 is defined by induction on a:

• 1. G0(n) := 0,
• 2. Ga+1(n) := Ga(n)+1,
• 3. Ga(n) := Ga[n](n) if tp(a) := ω. (when a is limit)

Note that ω[n] = n holds (by the definition).

Remark

It holds that Ga+b(n) = Ga(n)+Gb(n). Gk(n) = k, Gω(n) = Gω[n](n) = Gn(n) = n, GD01(n) = Gω×(n+1)(n) = Gω(n)×(n+1) = n×(n+1).

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Climbing Down Tree Ordinals

Recall that Ga(4) is a natural number even if a ∈ T1 is infinite.

Lemma

If ⊢a

0 Sm0 with 4 ≤ a ∈ T1, then Sm0 < Ga(4).

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The Upper Bound Theorem Definition

A function f : N → N is representable in our system if there is a term M : N ⇒ N such that M(Sn0) →∗

β Sk0 iff f(n) = k, where Sk0 is the

Church numeral corresponding to k.

Theorem

Let f be a representable function in our system with M : N ⇒ N. Then, |=D0(d×(n+1)) D0Dm

1 (MSn) : N with d = Dm 1 (Ω×m) for some m. Therefore,

there is m such that for all n ≥ m |D0Dm

1 (MSn)| < GD0Dm+2

1

0(n).

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Lower Bound Theorem Theorem (Schwichtenberg90)

For any m, we can formally prove in arithmetic: ∀x∃y(D0Dm

1 0)[x]y = 0.

Theorem (Aehlig05, 08)

• 1. The following are equivalent:

1.1 IDc

0 ⊢ ∀x∃yR(x,y),

1.2 HA2

1 ⊢ ∀x.N → ¬∀y(Ny → ¬R(x,y)).

• 2. If HA2

1 ⊢ ∀x.N → ¬∀y(Ny → ¬R(x,y)), then there is a term in our

system computing this function on Church numerals, that is, for every n the term tcn reduced to a Church numeral cl and R(n,l) holds.

Theorem

The function expressed by ∀x∃y(D0Dm

1 0)[x]y = 0 is representable in our

system.

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Summary

▶ As expected, the complexity of a term of type N ⇒ N is bounded:

|D0Dm

1 (MSn)| < GD0Dm+2

1

0(n).

▶ We can iterate our approach to handle with Brouwer’s ordinals (and

more...) O : ∀α.((N ⇒ α) ⇒ α) ⇒ (α ⇒ α) ⇒ (α ⇒ α)

▶ If we formalized tree ordinals and the slow growing hierarchy, then we

could see the bound in a visible way by considering concrete examples.

▶ We computed the upperbound of the length of β-reductions in our

system, too.

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References

▶ Aehlig, Induction and Inductive Definitions in Fragments of Second

Order Arithmetic, 2005.

▶ Aehlig, Parameter-Free Polymorphic Types, 2008. ▶ Akiyoshi, “Proofs as Programs” in Parameter-Free Fragments of

System F, 2018, draft, 16 pages.

▶ Akiyoshi, The Upperbound of the Length of the Reductions in a

Parameter-Free Fragment of System F, 2018, draft, 17 pages.

▶ Arai, A slow growing analogue to Buchholz’ proof, 1991. ▶ Buchholz, An independence result for Π1

1-CA+BI, 1987.

▶ Girard, Proof Theory and Logical Complexity, Vol. 1, 1987.

(For Vol. 2, see http://girard.perso.math.cnrs.fr/Archives4.html)

▶ Schwichtenberg, Proofs as Programs, 1990. ▶ Schwichtenberg and Wainer, Ordinal Bounds for Programs, 1994.

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