The characterization of Weihrauch reducibility in systems containing E-PA ω + QF-AC 0 , 0 Patrick Uftring September 8, 2020 1/30
Motivation We represent problems as formulas: P := ∀ x ( A ( x ) → ∃ y B ( x , y ) ). ���� � �� � domain matrix Can we find a system of (at least) second-order arithmetic A and a calculus C such that the following holds for two problems P and Q ? A ⊢ ” Q ≤ W P ” ⇔ C ⊢ P ′ → Q ′ . 2/30
Results in this direction Theorem (Hirst and Mummert 2019) Suppose P and Q are nice problems of the form P := ∀ x ( A ( x ) → ∃ yB ( x , y )) Q := ∀ u ( C ( u ) → ∃ vD ( u , v )) . then the following are equivalent: a) i RCA ω 0 proves Q with one typical use of P, b) i RCA ω 0 ⊢ Q ≤ W P. Theorem (Fujiwara 2020) Several characterization results of Weihrauch reducibility in ω ↾ + AC ω / Π 0 E-PA ω / � 1 -AC 0 , 0 / QF-AC 0 , 0 . E-PA Both results rely on a special proof structure 3/30
A first approach Theorem 6.4 (Kuyper JSL 2017) Characterizes compositional Weihrauch reducibility in RCA 0 using EL 0 (elementary intuitionistic analysis)+ MP (Markov’s principle). Theorem 7.1 (Kuyper JSL 2017) Characterizes Weihrauch reducibility in RCA 0 using (EL 0 + MP) ∃ α a that is defined like EL 0 + MP but ◮ contraction is only allowed for formulas without function quantifiers and ◮ weakening is only allowed for subformulas of ∃ α A where A does not contain function quantifiers. Counterexamples (Uftring M.Sc. thesis 2018) But the general idea seems to be correct. 4/30
The goal Consider P : ≡ ∀ x 1 ( A ( x ) → ∃ y 1 B ( x , y )) Q : ≡ ∀ u 1 ( C ( u ) → ∃ v 1 D ( u , v )) Theorem (Simplified) The following are equivalent: ℓ +Γ • proves P ′ ⊸ Q ′ a) E-LPA ω b) E-PA ω + QF-AC 0 , 0 +Γ proves Q ≤ W P 5/30
Linear Logic Every formula is a resource Symbols of linear logic ◮ Conjunctions: A ⊗ B , A & B & ◮ Disjunctions: A B , A ⊕ B ◮ Modal: ! A , ? A ◮ Involution: A ⊥ ◮ Abbreviation: ( A ⊸ B ) : ≡ A ⊥ & B Embedding of classical logic into linear logic A • : ≡ A where A is atomic, ( ¬ A ) • : ≡ ( A • ) ⊥ , ( A ∧ B ) • : ≡ A • ⊗ B • , ( A ∨ B ) • : ≡ A • & B • , ( A → B ) • : ≡ A • ⊸ B • . 6/30
Linear Logic (Intuition) Every argument must be used exactly once: Examples ⊢ A ⊗ B ⊸ B ⊗ A ⊢ A ⊸ ( B ⊸ A ⊗ B ) � A ⊸ A ⊗ A We cannot simply multiply A . ⊢ ! A ⊸ A ⊗ A We may use ! A as often as we like. � A ⊗ B ⊸ A We must use B . ⊢ A ⊗ ! B ⊸ A We may choose to use ! B not at all. Dualities ( A ⊗ B ) ⊥ ≡ A ⊥ & B ⊥ (! A ) ⊥ ≡ ? A ⊥ & Connectives and “?” do not have a simple intuition. 7/30
Motivating a linear predicate Problem: Quantifiers in problems cause problems Solution: Proof theory on nonstandard arithmetic (van den Berg, Briseid, Safarik 2012) Standard Predicate ◮ st( x ) ∧ x = y → st( y ) ◮ st( t c ) where t c is closed ◮ st( f ) ∧ st( x ) → st( fx ) ◮ Φ(0) ∧ ∀ st n 0 (Φ( n ) → Φ( n + 1)) → ∀ st n 0 Φ( n ) Nonstandard Dialectica only extracts information about standard values. Idea: Adapt this predicate to linear logic ◮ Only extract information about the Weihrauch reduction ◮ Uniform extraction that works with problems involving quantifiers 8/30
E-LPA ω ℓ Extensional Linear Peano Arithmetic in all finite types with linear predicate consists of the following three parts: ◮ The axioms and rules of linear logic, ◮ The axioms of E-PA ω translated to linear logic, ◮ Additional axioms for the new linear predicate ℓ : ⊢ A ⊥ ⊢ ℓ ( t c ) ⊢ ℓ ( t ) ⊸ ℓ ( t ) ⊗ ℓ ( t ) nl , ! A nl ⊢ ℓ ⊥ ( t ) , ℓ ⊥ ( r ) , ℓ ( tr ) ⊢ ( ∀ x 0 ∃ y 0 α xy = 0 0) ⊥ , ∃ Y 1 ( ∀ x 0 ( α x ( Yx ) = 0 0) ⊗ !( ℓ ( α ) ⊸ ℓ ( Y ))) Abbreviations: ∀ ℓ xA : ≡ ∀ x ( ℓ ( x ) ⊸ A ) ∃ ℓ xA : ≡ ∃ x ( ℓ ( x ) ⊗ A ) ∃ ℓ ǫ xA : ≡ ∃ x ( ℓ ( x ) ⊗ ǫ = 0 0 ⊗ A ) For ǫ := 0 and ǫ := 1, ∃ ℓ ǫ xA behaves like ∃ ℓ xA and ⊥ , respectively. 9/30
Formalization of Weihrauch reducibility Problems P : ≡ ∀ x 1 ( A ( x ) → ∃ y 1 B ( x , y )) Q : ≡ ∀ u 1 ( C ( u ) → ∃ v 1 D ( u , v )) In E-LPA ω P ′ : ≡ ∀ ℓ x 1 ( A • ( x ) ⊸ ∃ ℓ ℓ ǫ y 1 B • ( x , y )) Q ′ : ≡ ∀ ℓ u 1 ( C • ( u ) ⊸ ∃ ℓ ǫ v 1 D • ( u , v )) Weihrauch reducibility formalized using associates There are closed terms t and s such that the formulas ∀ u 1 ( C ( u ) → t · u ↓ ∧ A ( t · u )) ∀ u 1 , y 1 ( C ( u ) ∧ B ( t · u , y ) → s · j ( u , y ) ↓ ∧ D ( u , s · j ( u , y ))) and hold. 10/30
The Characterization of Weihrauch reducibility Theorem (Uftring 2018, 2020) Let A ( x 1 ) , B ( x , y 1 ) , C ( u 1 ) , and D ( u , v 1 ) be formulas of E-PA ω . Let Γ be a set of formulas of the same language. Consider: ⊢ ∀ ℓ x 1 ( A • ( x ) ⊸ ∃ ℓ ǫ y 1 B • ( x , y )) ⊸ ∀ ℓ u 1 ( C • ( u ) ⊸ ∃ ℓ ǫ v 1 D • ( u , v )) . The following are equivalent: ℓ +Γ • proves the sequent. a) E-LPA ω ℓ +Γ • proves the sequent. b) E-APA ω c) E-PA ω + QF-AC 0 , 0 +Γ proves both C ( u ) → t · u ↓ ∧ A ( t · u ) and C ( u ) ∧ B ( t · u , y ) → s · j ( u , y ) ↓ ∧ D ( u , s · j ( u , y )) for some closed terms t 1 and s 1 of L (E-PA ω ) . 11/30
G¨ odel’s Dialectica interpretation for linear logic Inspired by work due to de Paiva (1991), Shirahata (2006), and Oliva (2008-2011): | A | : ≡ A for unnegated + nonlinear atomic A , u ) ⊥ for unnegated atomic A , | A ⊥ | u : ≡ ( | A | v v | A ⊕ B | x , u , k 0 : ≡ (! k = 0 0 ⊗ | A | x y ) ⊕ (! k � = 0 0 ⊗ | B | u v ), y , v | A & B | x , u y , v , k 0 : ≡ (! k = 0 0 ⊸ | A | x y ) & (! k � = 0 0 ⊸ | B | u v ), & B | f , g : ≡ | A | fu & | B | gx | A u , x , u x | A ⊗ B | x , u : ≡ | A | x fu ⊗ | B | u gx , f , g |∃ zA | x : ≡ ∃ z | A | x y , y |∀ zA | x : ≡ ∀ z | A | x y , y : ≡ ? ∃ x | A | x | ? A | y y , | ! A | x : ≡ ! ∀ y | A | x y . Biggest modification: Quantified values are not interpreted 12/30
Interpretation of the linear predicate Interpreting the standard predicate (simplified) | st( t ) | x : ≡ x = t Constructing a term � 0 : ≡ 1, � ( τρ ) : ≡ � τ� ρ . Hereditary version of associates (Kleene, Kreisel 1959) con 0 ( s 1 , t 0 ) : ≡ ∃ x 0 ( sx � = 0 0) ∧ ∀ x 0 ( sx � = 0 0 → sx = 0 t + 1), con τρ ( s � τρ , t τρ ) : ≡ ∀ x � ρ , y ρ (con ρ ( x , y ) → con τ ( sx , ty )). Theorem: For each closed term t there is some ˜ t with con(˜ t , t ). Interpreting the linear predicate | ℓ ( t ) | x : ≡ con • ( x , t ) 13/30
“History” of our functional interpretation G¨ odel’s Dialectica Linear Dialectica Nonstandard Dialectica (de Paiva 91 / Shirahata 06) (van den Berg, Briseid, Safarik 12) + Oliva 08–11 Linear Dialectica + linear predicate Linear Dialectica + linear predicate + computability 14/30
Soundness Theorem of Dialectica for E-LPA ω ℓ Theorem Let A 1 , . . . , A n be formulas of L (E-LPA ω ℓ ) , and Γ a set of formulas ℓ +Γ • (or E-APA ω in L (E-PA ω ) , and assume that E-LPA ω ℓ +Γ • ) proves ⊢ A 1 , . . . , A n . ℓ +Γ • (or E-APA ω then E-LPA ω ℓ +Γ • ) proves ⊢ | A 1 | a 0 x 0 , . . . , | A n | a n x n for tuples of terms a 0 , . . . , a n where the free variables of each a i are among those in the sequence of terms x 0 , . . . , x i − 1 , x i +1 , . . . , x n . In particular, the variables x i are not free in a i . Proof. Induction on the proof length, i.e., for all rules. 15/30
Proof sketch for the Characterization Theorem ℓ +Γ • + QF-AC 0 , 0 : Given a proof of the following in E-LPA ω ⊢ ∀ ℓ x 1 ( A • ( x ) ⊸ ∃ ℓ ǫ y 1 B • ( x , y )) ⊸ ∀ ℓ u 1 ( C • ( u ) ⊸ ∃ ℓ ǫ v 1 D • ( u , v )). “ ǫ := 1” ⊢ ∀ ℓ x 1 ( A • ( x ) ⊸ ⊥ )) ⊸ ∀ ℓ u 1 ( C • ( u ) ⊸ ⊥ ). Extract term t ′ mapping each ˜ u with C ( u ) to an ˜ x with A ( x ) ⇒ Associate t computing for each u with C ( u ) an x with A ( x ) “ ǫ := 0” + previous result ⊢ ∀ ℓ u 1 ( ∃ ℓ y 1 B • ( t · u , y ) ⊸ C • ( u ) ⊸ ∃ ℓ v 1 D • ( u , v )). Extract term s ′ mapping each ˜ u , ˜ y with B ( t · u , y ) and C ( u ) to ˜ v with D ( u , v ). ⇒ Associate s computing for each u and y with B ( t · u , y ) and C ( u ) a v with D ( u , v ). Associates t and s compute the Weihrauch reduction in E-PA ω +Γ 16/30
The Characterization of Weihrauch reducibility (pretty) Theorem (Uftring 2020) Let A ( x 1 ) , B ( x , y 1 ) , C ( u 1 ) , and D ( u , v 1 ) be formulas of E-PA ω . Let Γ be a set of formulas of the same language. Consider: ⊢ ∀ ℓ x 1 ( A • ( x ) ⊸ ∃ ℓ y 1 B • ( x , y )) ⊸ ∀ ℓ u 1 ( C • ( u ) ⊸ ∃ ℓ v 1 D • ( u , v )) . The following are equivalent: ℓ +Γ • proves the sequent. a) E-LPA ω ℓ +Γ • proves the sequent. b) E-APA ω c) E-PA ω + QF-AC 0 , 0 +Γ proves both C ( u ) → t · u ↓ ∧ A ( t · u ) and C ( u ) ∧ B ( t · u , y ) → s · j ( u , y ) ↓ ∧ D ( u , s · j ( u , y )) for some closed terms t 1 and s 1 of L (E-PA ω ) . 17/30
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