MacNeille completion and Buchholz Omega rule Kazushige Terui RIMS, - PowerPoint PPT Presentation

MacNeille completion and Buchholz’ Omega rule Kazushige Terui RIMS, Kyoto University 08/03/18, Kanazawa 1 / 45

Summary of this talk MacNeille Buchholz’ Ω -rule (1981) completion Parameter-free 2nd { ∆ ⇒ Π θ } ∆ ⇒ LI order intuitionistic Y ϕ θ ( Y ) logics Ω -rule ∀ X.ϕ ( X ) ⇒ Π Ω -valuation For the lambda calculus audience where ∆ is 1st order and Π is 2nd order, For the nonclassical logics audience is similar to a characteristic property of MacNeille completion A ⊆ A : { a ≤ y } a ≤ x x ≤ y where a ∈ A and x, y ∈ A . 2 / 45

Cut elimination proofs for higher order logics/arithmetic Syntactic cut elimination MacNeille completion Parameter-free 2nd 1. Ordinal assignment order intuitionistic logics 2. Ω -rule technique (Buchholz, Aehlig, Mints, Akiyoshi, Ω -rule . . . ). Works only for fragments of higher order Ω -valuation logic/arithmetic (so far). For the lambda calculus audience Semantic cut elimination For the nonclassical logics audience 1. Semi-valuation (Sch¨ utte, Takahashi, Prawitz). 3-valued semantics (Girard 74) = Kleene’s semantics. Employs reductio ad absurdum and WKL. Destroys the proof structure. 2. MacNeille completion and reducibility candidates (Maehara 91, Okada 96, after Girard 71). Fully constructive. Extends to strong normalization. 3 / 45

Cut elimination proofs for higher order logics/arithmetic Syntactic cut elimination MacNeille completion Parameter-free 2nd 1. Ordinal assignment order intuitionistic logics 2. Ω -rule technique (Buchholz, Aehlig, Mints, Akiyoshi, Ω -rule . . . ). Works only for fragments of higher order Ω -valuation logic/arithmetic (so far). For the lambda calculus audience Semantic cut elimination For the nonclassical logics audience 1. Semi-valuation (Sch¨ utte, Takahashi, Prawitz). 3-valued semantics (Girard 74) = Kleene’s semantics. Employs reductio ad absurdum and WKL. Destroys the proof structure. 2. MacNeille completion and reducibility candidates (Maehara 91, Okada 96, after Girard 71). Fully constructive. Extends to strong normalization. What is the relationship? (Mints’ question) 3 / 45

Cut elimination proofs for higher order logics/arithmetic MacNeille Target system Fragments Full higher-order logics completion Algebraic proof ??? MacNeille completion Parameter-free 2nd order intuitionistic + reducibility candidates logics Syntactic proof Ω -rule Takeuti’s Conjecture Ω -rule Ω -valuation For the lambda calculus audience In this talk we fill in the ??? slot by introducing the concept For the nonclassical of Ω -valuation. The target systems are parameter-free 2nd logics audience order intuitionistic logics. 4 / 45

Cut elimination proofs for higher order logics/arithmetic MacNeille Target system Fragments Full higher-order logics completion Algebraic proof ??? MacNeille completion Parameter-free 2nd order intuitionistic + reducibility candidates logics Syntactic proof Ω -rule Takeuti’s Conjecture Ω -rule Ω -valuation For the lambda calculus audience In this talk we fill in the ??? slot by introducing the concept For the nonclassical of Ω -valuation. The target systems are parameter-free 2nd logics audience order intuitionistic logics. Notice: It is mostly a reworking of known results (especially those of Klaus Aehlig). Our purpose is just to provide an algebraic perspective on them. 4 / 45

Outline MacNeille completion MacNeille � completion Parameter-free 2nd order intuitionistic logics � Parameter-free 2nd order intuitionistic Ω -rule technique (syntactic) � logics Ω -valuation technique (semantic) � Ω -rule For the lambda calculus audience � Ω -valuation For the lambda For the nonclassical logics audience � calculus audience For the nonclassical logics audience 5 / 45

MacNeille ⊲ completion Parameter-free 2nd order intuitionistic logics Ω -rule Ω -valuation For the lambda calculus audience MacNeille completion For the nonclassical logics audience 6 / 45

MacNeille completion A : a lattice. MacNeille completion A completion of A is an embedding e : A − → B into a Parameter-free 2nd order intuitionistic complete lattice B (we often assume A ⊆ B ). logics Ω -rule Examples: Ω -valuation For the lambda Q ⊆ R ∪ {±∞} calculus audience � For the nonclassical e : A − → ℘ ( uf ( A )) � logics audience ( A : Boolean algebra, uf = ultrafilters). A ⊆ B is a MacNeille completion if for any x ∈ B , � � x = { a ∈ A : x ≤ B a } = { a ∈ A : a ≤ B x } . Theorem (Banachewski 56, Schmidt 56) Every lattice A has a unique MacNeille completion A . MacNeille completion is regular, i.e., preserves � and � that already exist in A . 7 / 45

MacNeille completion MacNeille (Recap) A ⊆ B is a MacNeille completion if for any x ∈ B , completion Parameter-free 2nd order intuitionistic � � { a ∈ A : a ≤ B x } = { a ∈ A : x ≤ B a } . x = logics Ω -rule Ω -valuation Q ⊆ R ∪ {±∞} is MacNeille, since � For the lambda calculus audience For the nonclassical x = inf { a ∈ Q : x ≤ a } = sup { a ∈ Q : a ≤ x } logics audience for any x ∈ R . It is regular, e.g., 1 1 0 = lim n ( in Q ) = lim n ( in R ) . n →∞ n →∞ e : A − → ℘ ( uf ( A )) is not regular, hence not MacNeille � (actually a canonical extension). 8 / 45

MacNeille completion: its limitation DL : the class of distributive lattices. MacNeille completion HA : the class of Heyting algebras. Parameter-free 2nd order intuitionistic BA : the class of Boolean algebras. logics Ω -rule Theorem Ω -valuation For the lambda DL is not closed under MacNeille (Funayama 44). calculus audience � For the nonclassical logics audience HA and BA are closed under MacNeille completions. � These are the only nontrivial subvarieties of HA closed � under MacNeille (Harding-Bezhanishvili 04). Conservative extension by MacNeille completion does not work for proper intermediate logics. 9 / 45

MacNeille completion: link to Ω -rule MacNeille Fact completion Parameter-free 2nd A completion A ⊆ B is MacNeille iff the inferences below are order intuitionistic logics valid: Ω -rule { x ≤ a } y ≤ a { a ≤ y } a ≤ x Ω -valuation x ≤ y x ≤ y For the lambda calculus audience For the nonclassical where x, y range over B and a over A . logics audience “If a ≤ x implies a ≤ y for any a ∈ A , then x ≤ y .” This looks similar to the Ω -rule. 10 / 45

MacNeille completion Parameter-free 2nd order intuitionistic ⊲ logics Ω -rule Ω -valuation For the lambda Parameter-free 2nd order intuitionistic logics calculus audience For the nonclassical logics audience 11 / 45

Background: Theories of iterated inductive definitions ID 0 := PA . MacNeille completion ID 1 : Let ϕ ( X, x ) be a formula of PA ( X ) in which X Parameter-free 2nd order intuitionistic occurs positively and FV ( ϕ ) ⊆ { X, x } . logics It can be seen as a monotone function Ω -rule Ω -valuation ϕ ( Y ) := { n ∈ N : ϕ ( Y, n ) holds } : ℘ ( N ) − → ℘ ( N ) . For the lambda calculus audience For the nonclassical For each such ϕ , add to PA a new constant I ϕ and axioms logics audience ϕ ( I ϕ ) ⊆ I ϕ , ϕ ( T ) ⊆ T ⇒ I ϕ ⊆ T. for every T = λx.ψ ( x ) . This defines the theory ID 1 . ID n +1 := ID n + least fixpoints definable in ID n . . . ID <ω := � n ID n . 12 / 45

Parameter-free fragments of 2nd order intuitionistic logic Tm: the set of terms MacNeille completion X, Y, Z, . . . : 2nd order variables Parameter-free 2nd order intuitionistic Fm : the formulas of 1st-order intuitionistic logic logics Ω -rule ϕ, ψ ::= p ( t ) | t ∈ X | ⊥ | ϕ ∧ ψ | ϕ ∨ ψ | ∀ x.ϕ | ∃ x.ϕ Ω -valuation For the lambda calculus audience FM − 1 := Fm. For the nonclassical FM n +1 : logics audience ϕ n +1 ::= p ( t ) | t ∈ X | · · · | ∀ X.ϕ n | ∃ X.ϕ n where ϕ n ∈ FM n doesn’t contain 2nd order variables except X . 13 / 45

Parameter-free fragment of 2nd order intuitionistic logic MacNeille (Recap) FM n +1 : completion Parameter-free 2nd ϕ n +1 ::= · · · | ∀ X.ϕ n | ∃ X.ϕ n order intuitionistic logics Ω -rule where ϕ n ∈ FM n doesn’t contain 2nd order variables except Ω -valuation X . For the lambda calculus audience For the nonclassical Examples (over L PA ) logics audience N ( t ) := ∀ X. [ ∀ x ( x ∈ X → x +1 ∈ X ) ∧ 0 ∈ X → t ∈ X ] ∈ FM 0 Any arithmetical formula ϕ translates to ϕ N ∈ FM 0 . If ϕ ( X, x ) is an arithmetical formula, I ϕ ( t ) := ∀ X. [ ∀ x ( ϕ N ( X, x ) → x ∈ X ) → t ∈ X ] ∈ FM 1 Any formula ϕ of ID 1 translates to ϕ I ∈ FM 1 . 14 / 45

Digression: full 2nd order logic FM: the set of all 2nd-order formulas. MacNeille completion G 1 LI : sequent calculus for 2nd order intuitionistic logic with Parameter-free 2nd order intuitionistic full comprehension logics Ω -rule ϕ ( λx.ψ ) , Γ ⇒ Π Γ ⇒ Y ϕ ( Y ) Ω -valuation ∀ X.ϕ ( X ) , Γ ⇒ Π Γ ⊢ ∀ X.ϕ ( X ) For the lambda calculus audience For the nonclassical logics audience where Γ ⇒ Y ϕ ( Y ) means Y �∈ FV (Γ) (eigenvariable). � ϕ ( λx.ψ ) obtained by replacing t ∈ X �→ ψ ( t ) . � Theorem (Takeuti 53) If Z 2 ⊢ ϕ , then G 1 LC ⊢ Γ 0 ⇒ ϕ N for some universal Γ 0 . Cut elimination for G 1 LC implies 1-consistency of Z 2 , i.e., provable Σ 0 1 -sentences are true. 15 / 45