Effective computability and constructive provability for existence sentences Makoto Fujiwara Waseda Institute for Advanced Study (WIAS), Waseda University Workshop on Mathematical Logic and its Application, Kyoto University, 16 September 2016 This work is supported by JSPS KAKENHI Grant Number JP16H07289 and also by JSPS Core-to-Core Program (A. Advanced Research Networks). 1 / 16
Targets: Existence Statements Many theorems in ordinary mathematics (Analysis, Algebra, Combinatorics etc.) can be formalized as Π 1 2 sentences having a form ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) , where f and g are (possibly tuples of) functions on natural numbers. 2 / 16
In computable (recursive) mathematics, effective contents of classical theorems have been investigated. In particular, there are many “effectivized” results of classical theorems in combinatorics, e.g., Brooks’s theorem (Schmerl 1982, Carstens/P¨ appinghaus 1983, Tverberg 1984), Marriage theorem (Kierstead 1983), Dilworth’s theorem (Kierstead 1981). 3 / 16
In computable (recursive) mathematics, effective contents of classical theorems have been investigated. In particular, there are many “effectivized” results of classical theorems in combinatorics, e.g., Brooks’s theorem (Schmerl 1982, Carstens/P¨ appinghaus 1983, Tverberg 1984), Marriage theorem (Kierstead 1983), Dilworth’s theorem (Kierstead 1981). Two Kinds of Effectivization Non-uniform computability : For any computable f , there exists a computable g . Uniform computability : There exists a uniform algorithm to obtain a witness g for each (computable) f . 3 / 16
Correspondence in Reverse Mathematics Non-uniform computability (For any computable f , there exists a computable g .) Uniform computability (There exists a uniform algorithm to ob- tain a witness g for each f .) Constructive provability 4 / 16
Correspondence in Reverse Mathematics Non-uniform computability ≈ RCA 0 (For any computable f , there exists a computable g .) Uniform computability (There exists a uniform algorithm to ob- tain a witness g for each f .) ≈ EL 0 Constructive provability 4 / 16
Correspondence in Reverse Mathematics Non-uniform computability ≈ RCA 0 (For any computable f , there exists a computable g .) Uniform computability ≈ ?? (There exists a uniform algorithm to ob- tain a witness g for each f .) ≈ EL 0 Constructive provability Toward an axiomatization of uniform computability Investigate how uniform computability for existence statements can be captured by (semi-)intuitionistic provability in (many-sorted) arithmetic! 4 / 16
Formalization of Uniform Computability (F. 2015) Hilbert-type system E-HA ω (resp. E-PA ω ) is the finite type extension of HA (resp. PA), of which T is the terms. 5 / 16
Formalization of Uniform Computability (F. 2015) Hilbert-type system E-HA ω (resp. E-PA ω ) is the finite type extension of HA (resp. PA), of which T is the terms. � ω ↾ (resp. � ω ↾ ) is the restrictions of E-HA ω (resp. E-HA E-PA E-PA ω ) to primitive recursion of type 0 and quantifier-free induction, of which T 0 is the terms. 5 / 16
Formalization of Uniform Computability (F. 2015) Hilbert-type system E-HA ω (resp. E-PA ω ) is the finite type extension of HA (resp. PA), of which T is the terms. � ω ↾ (resp. � ω ↾ ) is the restrictions of E-HA ω (resp. E-HA E-PA E-PA ω ) to primitive recursion of type 0 and quantifier-free induction, of which T 0 is the terms. Intuitionistic Logic Classical Logic 0 HA PA 1 EL 0 EL RCA 0 RCA E-HA ω + QF - AC 1 , 0 � ω ↾ + QF - AC 1 , 0 E-HA RCA ω RCA ω ω 0 5 / 16
Formalization of Uniform Computability (F. 2015) Hilbert-type system E-HA ω (resp. E-PA ω ) is the finite type extension of HA (resp. PA), of which T is the terms. � ω ↾ (resp. � ω ↾ ) is the restrictions of E-HA ω (resp. E-HA E-PA E-PA ω ) to primitive recursion of type 0 and quantifier-free induction, of which T 0 is the terms. Intuitionistic Logic Classical Logic 0 HA PA 1 EL 0 EL RCA 0 RCA E-HA ω + QF - AC 1 , 0 � ω ↾ + QF - AC 1 , 0 E-HA RCA ω RCA ω ω 0 Fact. (Kohlenbach 2005) RCA ω 0 is a conservative extension of RCA 0 . 5 / 16
Formalization of Uniform Computability (F. 2015) Hilbert-type system E-HA ω (resp. E-PA ω ) is the finite type extension of HA (resp. PA), of which T is the terms. � ω ↾ (resp. � ω ↾ ) is the restrictions of E-HA ω (resp. E-HA E-PA E-PA ω ) to primitive recursion of type 0 and quantifier-free induction, of which T 0 is the terms. Intuitionistic Logic Classical Logic 0 HA PA 1 EL 0 EL RCA 0 RCA E-HA ω + QF - AC 1 , 0 � ω ↾ + QF - AC 1 , 0 E-HA RCA ω RCA ω ω 0 Fact. (Kohlenbach 2005) RCA ω 0 is a conservative extension of RCA 0 . That is also the case for WKL ω 0 (:= RCA ω 0 + WKL ) and ACA ω 0 (:= RCA ω 0 + ACA ). 5 / 16
Uniform Provability in Γ : 1 There exists a term t 1 s.t. Γ ⊢ ∀ f ( ϕ ( f ) → t | f ↓ ∧ ψ ( f , t | f )) , where { β n ) − 1 where n is the least n ′ s.t. α (¯ α (¯ β n ′ ) ̸ = 0 . α ( β ) := ↑ if there is no such n ′ . α | β := λ n . α ( ⟨ n ⟩ ⌢ β ) . odel prim. rec.) term t 1 → 1 ∈ T s.t. 2 There exists a (G¨ Γ ⊢ ∀ f ( ϕ ( f ) → ψ ( f , tf )) . 3 There exists a (Kleene prim. rec.) term t 1 → 1 ∈ T 0 s.t. Γ ⊢ ∀ f ( ϕ ( f ) → ψ ( f , tf )) . 6 / 16
Proposition 1. (F. 2015) Let ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) be a L (EL 0 )-formula such that ϕ ( f ) is purely universal and ψ ( f , g ) is equivalent to some formula ∀ w ρ ∃ s 0 ψ qf ( f , g , w , s ) over EL 0 . There exists a term t 1 such that RCA 0 (+ WKL ) ⊢ ∀ f ( ϕ ( f ) → t | f ↓ ∧ ψ ( f , t | f )) if and only if EL 0 ⊢ ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) . 7 / 16
Proposition 1. (F. 2015) Let ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) be a L (EL 0 )-formula such that ϕ ( f ) is purely universal and ψ ( f , g ) is equivalent to some formula ∀ w ρ ∃ s 0 ψ qf ( f , g , w , s ) over EL 0 . There exists a term t 1 such that RCA 0 (+ WKL ) ⊢ ∀ f ( ϕ ( f ) → t | f ↓ ∧ ψ ( f , t | f )) if and only if EL 0 ⊢ ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) . On the Proof. IF direction is by the function realizability (Dorais 2014). ONLY IF direction is by Kuroda’s negative translation and the (monotone) Dialectica interpretation. 7 / 16
Application. For example, Kierstead’s effective marriage theorem EMT has the required syntactical form in Proposition 1 and uniformly provable in RCA 0 , then it follows that EMT is provable in EL 0 . Remark. It is known that many existence theorems are formalized as a Π 1 2 formula of the syntactical form in Proposition 1. The analogous results for EL, RCA instead of EL 0 , RCA 0 also hold. 8 / 16
The Next Step Most of discussion in ordinary mathematics is carried out in the presence of Arithmetical Comprehension Axiom. 9 / 16
The Next Step Most of discussion in ordinary mathematics is carried out in the presence of Arithmetical Comprehension Axiom. 1 - AC 0 , 0 : Π 0 ∀ α 1 ( ∀ x 0 ∃ y 0 ∀ z 0 α ( x , y , z ) = 0 → ∃ β 1 ∀ x , z α ( x , β ( x ) , z ) = 0) classically derives ACA . 9 / 16
The Next Step Most of discussion in ordinary mathematics is carried out in the presence of Arithmetical Comprehension Axiom. 1 - AC 0 , 0 : Π 0 ∀ α 1 ( ∀ x 0 ∃ y 0 ∀ z 0 α ( x , y , z ) = 0 → ∃ β 1 ∀ x , z α ( x , β ( x ) , z ) = 0) classically derives ACA . Question. How is uniform provability in classical systems with Π 0 1 - AC 0 , 0 characterized? 9 / 16
The Next Step Most of discussion in ordinary mathematics is carried out in the presence of Arithmetical Comprehension Axiom. 1 - AC 0 , 0 : Π 0 ∀ α 1 ( ∀ x 0 ∃ y 0 ∀ z 0 α ( x , y , z ) = 0 → ∃ β 1 ∀ x , z α ( x , β ( x ) , z ) = 0) classically derives ACA . Question. How is uniform provability in classical systems with Π 0 1 - AC 0 , 0 characterized? Fact. 1 - AC 0 , 0 is intuitionistically The negative translation of Π 0 1 - AC 0 , 0 and Σ 0 derived from Π 0 2 - DNS 0 : ∀ α 1 ( ∀ x 0 ¬¬∃ y 0 ∀ z 0 α ( x , y , z ) = 0 → ¬¬∀ x ∃ y ∀ z α ( x , y , z ) = 0) . 9 / 16
Uniform ⇒ Intuitionistic Proposition 2. Let ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) be a L (EL 0 )-formula such that ϕ ( f ) ∈ A and ψ ( f , g ) ∈ B . If there exists a term t 1 → 1 ∈ T 0 such that 1 - AC 0 , 0 ⊢ ∀ f ( ϕ ( f ) → ψ ( f , tf )) , � ω ↾ + Π 0 E-PA 1 - AC 0 , 0 + Σ 0 2 - DNS 0 ⊢ ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) . EL 0 + Π 0 then 10 / 16
Uniform ⇒ Intuitionistic Proposition 2. Let ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) be a L (EL 0 )-formula such that ϕ ( f ) ∈ A and ψ ( f , g ) ∈ B . If there exists a term t 1 → 1 ∈ T 0 such that 1 - AC 0 , 0 ⊢ ∀ f ( ϕ ( f ) → ψ ( f , tf )) , � ω ↾ + Π 0 E-PA 1 - AC 0 , 0 + Σ 0 2 - DNS 0 ⊢ ∀ f ( ϕ ( f ) → ∃ g ψ ( f , g )) . EL 0 + Π 0 then The classes A , B of formulas are defined simultaneously by P , A 1 ∧ A 2 , A 1 ∨ A 2 , ∀ xA 1 , ∃ xA 1 , B 1 → A 1 are in A ; P , B 1 ∧ B 2 , ∀ xB 1 , A 1 → B 1 are in B ; where P , A i , B i range over prime formulas, formulas in A , B respectively. 10 / 16
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