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). 1 / 31
Introduction Meta-theorems Parallelization Summary Introduction Many existence statements in mathematics can be formalized as Π 2 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 Π 2 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). 2 / 31
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. 3 / 31
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) Fact. (Const. RM over BISH) DICH R is Weihrauch DICH R is constructively equivalent to LLPO . equivalent to LLPO . DICH R : ∀ α ∈ R ( α ≥ 0 ∨ α ≤ 0). LLPO : � ¬ � � � ∃ n N f (2 n ) = 0 ∧ ∃ n N f (2 n + 1) = 0 ∀ f N → N → ∀ n N f (2 n ) � = 0 ∨ ∀ n N f (2 n + 1) � = 0 3 / 31
Introduction Meta-theorems Parallelization Summary Definition. The parallelization (or sequential version) � P of P ≡ ∀ f ( A ( f ) → ∃ gB ( f , g )) is ∀� f n � n ∈ N ( ∀ n A ( f n ) → ∃� g n � n ∈ N ∀ n B ( f n , g n )) . ⇒ P ≤ W � Remark. P ≤ W Q = Q . 4 / 31
Introduction Meta-theorems Parallelization Summary Definition. The parallelization (or sequential version) � P of P ≡ ∀ f ( A ( f ) → ∃ gB ( f , g )) is ∀� f n � n ∈ N ( ∀ n A ( f n ) → ∃� g n � n ∈ N ∀ n B ( f n , g n )) . ⇒ P ≤ W � Remark. P ≤ W Q = 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 . 4 / 31
Introduction Meta-theorems Parallelization Summary Definition. The parallelization (or sequential version) � P of P ≡ ∀ f ( A ( f ) → ∃ gB ( f , g )) is ∀� f n � n ∈ N ( ∀ n A ( f n ) → ∃� g n � n ∈ N ∀ n B ( f n , g n )) . ⇒ P ≤ W � Remark. P ≤ W Q = 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 AC 0 ,ω ) IVT ⇔ WKL ⇔ LLPO . 4 / 31
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 observed in Constructive RM. In these two decades, Constructive RM, as well as some of Weihrauch RM, have been developed over (many-sorted) arithmetic as for Friedman-Simpson RM. Our approach: Weihrauch RM Const. RM over BISH � Const. RM (Formal) Classical RM (Formal) � We employ finite-type arithmetic as our framework. 5 / 31
Introduction Meta-theorems Parallelization Summary 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 ↾ ) is the restrictions of E-HA ω (resp. � 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. Type-1 functions (functions of type N N ) definable in T 0 (resp. T ) coincide with primitive (resp. PA-provably) recursive functions in the ordinary sense. N HA N N EL 0 EL ω ↾ + QF - AC 0 , 0 E-HA ω + QF - AC 0 , 0 � ω E-HA Fact. (Kohlenbach 2005) ω ↾ + QF - AC 1 , 0 is a conservative extension of 0 : ≡ � RCA ω E-PA RCA 0 in Friedman-Simpson RM. AC σ,τ : ∀ x σ ∃ y τ A ( x , y ) → ∃ Y τ σ ∀ x σ A ( x , Yx ). 6 / 31
Introduction Meta-theorems Parallelization Summary Definition (Weihrauch reducibility for Π 1 2 statements) For Π 1 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 . 7 / 31
Introduction Meta-theorems Parallelization Summary Definition (Weihrauch reducibility for Π 1 2 statements) For Π 1 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¨ odel/Kleene) variants of Weihrauch reducibility in which Turing functionals for the reduction are replaced by primitive recursive (total) functionals (in the sense of G¨ odel/Kleene). The verification theory is also concerned. 7 / 31
Introduction Meta-theorems Parallelization Summary PR Variants of Weihrauch Reducibility in S ω Definition Let P : ∀ f ( A 1 ( f ) → ∃ g B 1 ( f , g )) , Q : ∀ f ( A 2 ( f ) → ∃ gB 2 ( f , g )). P is G¨ odel-primitive-recursive Weihrauch reducible to Q in S ω if there exist closed terms s and t (of suitable types) in T such that S ω proves ∀ f ( A 1 ( f ) → A 2 ( sf )) ∧∀ f , g ′ � � B 2 ( sf , g ′ ) ∧ A 1 ( f ) → B 1 ( f , tfg ′ ) . P is Kleene-primitive-recursive Weihrauch reducible to Q in S ω if there exist closed terms s and t (of suitable types) in T 0 such that S ω proves the same sentence. 8 / 31
Introduction Meta-theorems Parallelization Summary Proposition. (cf. Brattka/Gherardi 2011) WKL is Kleene-primitive-recursive Weihrauch reducible to ω ↾ + QF - AC 0 , 0 (which contains Π 0 LLPO in � � E-PA 1 -IND). Remark. ω ↾ + QF - AC 0 , 0 and � 1 - AC 0 , 0 are conservative ω ↾ + Π 0 � E-PA E-PA extensions of RCA 0 and ACA 0 in Friedman-Simpson RM respectively. 9 / 31
Introduction Meta-theorems Parallelization Summary Definition (Normal Reducibility in S ω ) Let P : ∀ f ( A 1 ( f ) → ∃ g B 1 ( f , g )) , Q : ∀ f ( A 2 ( f ) → ∃ gB 2 ( f , g )). We say that P is normally reducible to Q in S ω if S ω proves � A 1 ( f ) → ∃ f ′ � A 2 ( f ′ ) ∧ ∀ g ′ � ��� B 2 ( f ′ , g ′ ) → ∃ gB 1 ( f , g ) ∀ f . 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. 10 / 31
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 ( A 1 ( f ) → ∃ g B 1 ( f , g )) , Q : ∀ f ( A 2 ( f ) → ∃ gB 2 ( f , g )) , and S ω be a classical finite-type arithmetic such that S ω ⊢ ∃ f ′ A 2 ( f ′ ). ∗ If S ω ⊢ Q → P , then P is normally reducible to Q in S ω . ∗ Note that if S ω ⊢ ∀ f ′ ¬ A 2 ( f ′ ), then S ω ⊢ Q → P just means S ω ⊢ P 11 / 31
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 ( A 1 ( f ) → ∃ g B 1 ( f , g )) , Q : ∀ f ( A 2 ( f ) → ∃ gB 2 ( f , g )) , and S ω be a classical finite-type arithmetic such that S ω ⊢ ∃ f ′ A 2 ( f ′ ). ∗ If S ω ⊢ Q → P , then P is normally reducible to Q in S ω . 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 S ω ⊢ ∀ f ′ ¬ A 2 ( f ′ ), then S ω ⊢ Q → P just means S ω ⊢ P 11 / 31
Introduction Meta-theorems Parallelization Summary Characterization of a Weakening of P ≤ W Q Proposition. Let P : ∀ f ( A 1 ( f ) → ∃ g B 1 ( f , g )) , Q : ∀ f ( A 2 ( f ) → ∃ gB 2 ( f , g )) with ∃ -free (containing neither ∃ nor ∨ ) formulas A 1 , A 2 , B 1 , B 2 . P is G¨ odel-primitive-recursive Weihrauch reducible to Q in E-PA ω ⇐ ⇒ P is normally reducible to Q in E-HA ω . P is Kleene-primitive-recursive Weihrauch reducible to Q ω ↾ ⇐ ω ↾ . in � ⇒ P is normally reducible to Q in � E-PA E-HA Idea of the Proof. ( ⇐ ) is shown by using the modified realizability interpretation; ( ⇒ ) is shown by using the negative translation . 12 / 31
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