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 1 / 26

Aim of This Talk ▶ The aim is to revisit Schwichtenberg’s works by focusing on parameter subsystems of Girard’s F. 2 / 26

Proofs as Programs by Schwichtenberg ▶ Proofs are regarded as programs (Curry=Howard isomorphism) ▶ Schwichtenberg measured the complexity as programs of proofs in arithmetic. 3 / 26

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. 3 / 26

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 | ≤ G D 0 D m + 2 0 ( n ) . 1 ( D 0 , D 1 are the collapsing functions, and G is a slow growing hierarchy.) 3 / 26

Proofs as Programs by Schwichtenberg ▶ Proofs as programs could contain such a complicated structures. | rn | ≤ G D 0 D m + 2 0 ( n ) . 1 4 / 26

Proofs as Programs by Schwichtenberg ▶ Proofs as programs could contain such a complicated structures. | rn | ≤ G D 0 D m + 2 0 ( n ) . 1 ▶ Strategy for getting this result: 1. Normalize a given term rn and measure the size of it. (We need D 0 D m + 2 0 ( n ) here. ) 1 2. To climb down the “big” tree ordinal by the slow growing hierarchy using ideas by Wainer-Girard and Arai. 4 / 26

Proofs as Programs by Schwichtenberg ▶ Proofs as programs could contain such a complicated structures. | rn | ≤ G D 0 D m + 2 0 ( n ) . 1 ▶ Strategy for getting this result: 1. Normalize a given term rn and measure the size of it. (We need D 0 D m + 2 0 ( n ) here. ) 1 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. 4 / 26

Some Literatures ▶ Arai, A slow growing analogue to Buchholz’ proof, 1991. ▶ Buchholz, An independence result for Π 1 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. 5 / 26

Goal of This Talk ▶ The aim: to revisit S’s works by focusing on parameter subsystems of Girard’s F. 6 / 26

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 ) 6 / 26

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. 7 / 26

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... 7 / 26

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 Π 1 1 - CA + BI with the ω -rule”, The Mints’ memorial issue of 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. 8 / 26

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: ( λ x A . M B ) A ⇒ B ( M A ⇒ B N A ) B x A ( Λ α . M A ) ∀ α . A ( M ∀ α . A B ) A [ α / B ] with the standard proviso. 9 / 26

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. 10 / 26

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”. 10 / 26

Tree Ordinals Definition (Buchholz87) The tree classes T σ ( σ ≤ 2 ) are defined as follows: ▶ If α : I → T σ is a function with I : / 0 , { 0 } , or T ρ for some ρ < σ , then α ∈ T σ . Some notations. 1. 0 for α : / 0 → T σ , 2. β + for α : { 0 } → T σ with α ( 0 ) = β . 11 / 26

Tree Ordinals Definition (Buchholz87) The tree classes T σ ( σ ≤ 2 ) are defined as follows: ▶ If α : I → T σ is a function with I : / 0 , { 0 } , or T ρ for some ρ < σ , then α ∈ T σ . Some notations. 1. 0 for α : / 0 → T σ , 2. β + for α : { 0 } → T σ with α ( 0 ) = β . Remark 1. T 0 is identified with N (the set of natural numbers) , 2. T 1 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 ... 11 / 26

Collapsing Functions on Tree Ordinals Let Ω 0 : = N , Ω 1 : = the set of countable tree ordinals. Definition (Buchholz87, Arai91) The collapsing functions D σ : T v → T σ + 1 for σ < v ≤ 2 are defined as follows: 1. D σ 0 : = Ω σ , 2. D σ ( α + 1 ) : = ( D σ ( α ) × ( n + 1 )) n ∈ ω , 3. If ρ ≤ σ , then D σ (( α ξ ) ξ ∈ T ρ ) : = ( D σ α ξ ) ξ ∈ T ρ , 4. If σ < µ + 1 , then D σ (( α ξ ) ξ ∈ T µ + 1 ) : = ( D σ α ξ n ) n ∈ ω where ξ 0 : = Ω µ , ξ n + 1 : = D µ α ξ n . 12 / 26

Collapsing Functions on Tree Ordinals Let Ω 0 : = N , Ω 1 : = the set of countable tree ordinals. Definition (Buchholz87, Arai91) The collapsing functions D σ : T v → T σ + 1 for σ < v ≤ 2 are defined as follows: 1. D σ 0 : = Ω σ , 2. D σ ( α + 1 ) : = ( D σ ( α ) × ( n + 1 )) n ∈ ω , 3. If ρ ≤ σ , then D σ (( α ξ ) ξ ∈ T ρ ) : = ( D σ α ξ ) ξ ∈ T ρ , 4. If σ < µ + 1 , then D σ (( α ξ ) ξ ∈ T µ + 1 ) : = ( D σ α ξ n ) n ∈ ω where ξ 0 : = Ω µ , ξ n + 1 : = D µ α ξ n . Remark 1. In the last clause, the point is that the indexes ξ n are tree ordinals. 2. D 0 0 = ω , D 0 1 = ω 2 , D 0 ω = ω ω , 12 / 26

Collapsing Functions on Tree Ordinals Let Ω 0 : = N , Ω 1 : = the set of countable tree ordinals. Definition (Buchholz87, Arai91) The collapsing functions D σ : T v → T σ + 1 for σ < v ≤ 2 are defined as follows: 1. D σ 0 : = Ω σ , 2. D σ ( α + 1 ) : = ( D σ ( α ) × ( n + 1 )) n ∈ ω , 3. If ρ ≤ σ , then D σ (( α ξ ) ξ ∈ T ρ ) : = ( D σ α ξ ) ξ ∈ T ρ , 4. If σ < µ + 1 , then D σ (( α ξ ) ξ ∈ T µ + 1 ) : = ( D σ α ξ n ) n ∈ ω where ξ 0 : = Ω µ , ξ n + 1 : = D µ α ξ n . Remark 1. In the last clause, the point is that the indexes ξ n are tree ordinals. 2. D 0 0 = ω , D 0 1 = ω 2 , D 0 ω = ω ω , ..., D 0 Ω 1 = ε 0 . 12 / 26