Strong Normalization Akiyoshi and Terui Introduction Our Results Strong normalization for the parameter-free Strong polymorphic lambda calculus based on the Ω -rule Normalization Theorem Previous Results Buchholz’ Ω -Rule Ryota Akiyoshi 1 Kazushige Terui 2 1 Waseda Institute for Advanced Study, Waseda University 2 Research Institute for Mathematical Sciences, Kyoto University 17th., September 2016 1 / 21
Strong Motivation Normalization Akiyoshi and Terui Introduction Our Results Strong Normalization Theorem • Girard’s proof of the strong normalization of his system F requires Previous Results the third-order arithmetic on the meta-level. Buchholz’ Ω -Rule • Natural question: can we have a more predicative proof of the normalization for fragments of F ? • predicative proof = proof without circular reasoning. 2 / 21
Strong Aim of This Talk Normalization Akiyoshi and Terui • In this talk, we present a predicative proof of the strong Introduction normalization for F p n by studying Buchholz’ Ω -rule. Our Results Strong • F p Normalization n : a parameter-free polymorphic lambda calculus allowing Theorem n -times nested second-order quantifier. Previous Results Buchholz’ Ω -Rule • We transfer an important method in proof theory called the Ω -rule into computer science. • Moreover, we give a proof-theoretic bound of the strong normalization for it. Akiyoshi and Terui, “Strong normalization for the parameter-free polymorphic lambda calculus based on the Omega-rule”, First International Conference on Formal Structures for Computation and Deduction (FSCD) , 2016. 3 / 21
Strong Definition of Syntax Normalization Akiyoshi and Terui Definition (Cf. Aehlig08) Introduction Our Results For each n ∈ N ∪{− 1 } , we define Tp n as Strong Normalization Theorem A n , B n :: = α | A n ⇒ B n | ∀ α . A n − 1 . Previous Results Buchholz’ where FV ( A n − 1 ) ⊆ { α } in the last clause. Ω -Rule We write Tp simp = Tp − 1 . Types in this set are “parameter-free”. : = ∀ α . ( α ⇒ α ) ⇒ ( α ⇒ α ) ∈ Tp 0 N T : = ∀ α . ( α ⇒ α ⇒ α ) ⇒ ( α ⇒ α ) ∈ Tp 0 O : = ∀ α . (( N ⇒ α ) ⇒ α ) ⇒ ( α ⇒ α ) ⇒ ( α ⇒ α ) ∈ Tp 1 Remark An important property: A , B ∈ Tp n implies A [ B / α ] ∈ Tp n . 4 / 21
Strong Term Rules and Conversions Normalization Akiyoshi and Definition Terui Terms ( Tm ) and Conversions of F p are defined in the standard way: Introduction Our Results Strong M B ∈ Tm Normalization ( var ) ( con ) ( abs ) Theorem x A ∈ Tm c A ∈ Tm ( λ x A . M ) A ⇒ B ∈ Tm Previous Results Buchholz’ Ω -Rule M A ∈ Tm ∩ Ec ( α ) M A ⇒ B ∈ X N A ∈ Tm ( app ) ( Abs ) ( MN ) B ∈ Tm ( Λ α . M ) ∀ α . A ∈ Tm M ∀ α . A ∈ Tm ( App ) ( MB ) A [ B / α ] ∈ Tm ( λ x A . M ) N → M [ N / x A ] , ( Λ α . M ) B → M [ B / α ] . Definition F p n is obtained by restricting types to Tp n . 5 / 21
Strong Previous Results by Alternkirch, Coquand, and Aehlig Normalization Akiyoshi and • Girard’s proof of SN ( F ) requires the third-order arithmetic on the Terui meta-level. Introduction Our Results • Question: can we have a more predicative proof of the Strong normalization for fragments of system F? Normalization Theorem Previous Results Buchholz’ Ω -Rule 6 / 21
Strong Previous Results by Alternkirch, Coquand, and Aehlig Normalization Akiyoshi and • Girard’s proof of SN ( F ) requires the third-order arithmetic on the Terui meta-level. Introduction Our Results • Question: can we have a more predicative proof of the Strong normalization for fragments of system F? Normalization Theorem • Alternkirch and Coquand: a proof of weak normalization Previous Results (WN) of F p 0 for specific terms; Buchholz’ Ω -Rule Provably total in HA = representable in F p 0 . 6 / 21
Strong Previous Results by Alternkirch, Coquand, and Aehlig Normalization Akiyoshi and • Girard’s proof of SN ( F ) requires the third-order arithmetic on the Terui meta-level. Introduction Our Results • Question: can we have a more predicative proof of the Strong normalization for fragments of system F? Normalization Theorem • Alternkirch and Coquand: a proof of weak normalization Previous Results (WN) of F p 0 for specific terms; Buchholz’ Ω -Rule Provably total in HA = representable in F p 0 . • Aehlig: an indirect predicative proof of WN for F p n for a specific terms; Provably total in ID n = representable in F p n . (The problem of SN was left open in his Ph.D thesis) 6 / 21
Strong Previous Results by Alternkirch, Coquand, and Aehlig Normalization Akiyoshi and • Girard’s proof of SN ( F ) requires the third-order arithmetic on the Terui meta-level. Introduction Our Results • Question: can we have a more predicative proof of the Strong normalization for fragments of system F? Normalization Theorem • Alternkirch and Coquand: a proof of weak normalization Previous Results (WN) of F p 0 for specific terms; Buchholz’ Ω -Rule Provably total in HA = representable in F p 0 . • Aehlig: an indirect predicative proof of WN for F p n for a specific terms; Provably total in ID n = representable in F p n . (The problem of SN was left open in his Ph.D thesis) • Our aim is to improve the situation by giving a direct predicative proof of the strong normalization of such fragments for all terms. Altenkirch and Coquand, “A Finitary Subsystem of the Polymorphic λ -calculus”, TLCA 2001. Aehlig, “Parameter-free polymorphic types”, APAL , 2008. 6 / 21
Strong Our Results Normalization Akiyoshi and Terui • Systems of inductive definitions: Introduction 1 ID 1 = PA + the least fixed points for PA -definable monotone Our Results Strong operators. Normalization Theorem Previous Results Buchholz’ Ω -Rule 7 / 21
Strong Our Results Normalization Akiyoshi and Terui • Systems of inductive definitions: Introduction 1 ID 1 = PA + the least fixed points for PA -definable monotone Our Results Strong operators. Normalization 2 ID n + 1 = ID n + the least fixed points for ID n -definable Theorem Previous Results monotone operators with 1 ≤ n . Buchholz’ Ω -Rule 7 / 21
Strong Our Results Normalization Akiyoshi and Terui • Systems of inductive definitions: Introduction 1 ID 1 = PA + the least fixed points for PA -definable monotone Our Results Strong operators. Normalization 2 ID n + 1 = ID n + the least fixed points for ID n -definable Theorem Previous Results monotone operators with 1 ≤ n . Buchholz’ 3 ID < ω : = ∪ n ∈ ω ID n . Ω -Rule 4 ID ω : a proper extension of ID < ω . Theorem ID n + 1 ⊢ SN ( F p n ) for all n < ω . Theorem ID ω ⊢ SN ( F p ) with F p : = ∪ n ∈ ω F p n . Theorem (Aehlig 08) Every representable function in F p n is provably total in ID n . 7 / 21
Strong What is the Ω -Rule? Normalization Akiyoshi and Terui • The Ω -rule: infinitary rule introduced by Buchholz (1977) for Introduction ordinal analysis of iterated inductive definitions. Our Results Strong • Sch¨ utte’s ω -rule: branching over natural numbers. Normalization Theorem • The Ω -rule: branching over arithmetical cut-free proofs. Previous Results Buchholz’ • Main theorems by Buchholz: Ω -Rule 1 - CA ) is embedded to BI Ω . Embedding: BI (parameter free Π 1 Collapsing: weak normalization for arithmetical formulas for BI Ω . 8 / 21
Strong What is the Ω -Rule? Normalization Akiyoshi and Terui • The Ω -rule: infinitary rule introduced by Buchholz (1977) for Introduction ordinal analysis of iterated inductive definitions. Our Results Strong • Sch¨ utte’s ω -rule: branching over natural numbers. Normalization Theorem • The Ω -rule: branching over arithmetical cut-free proofs. Previous Results Buchholz’ • Main theorems by Buchholz: Ω -Rule 1 - CA ) is embedded to BI Ω . Embedding: BI (parameter free Π 1 Collapsing: weak normalization for arithmetical formulas for BI Ω . • Recent developments: 1. For a stronger system ( µ -calculus): H.Towsner (2008). 2. modal µ -calculus like ID 1 : G. J¨ ager and T. Studer (2010). 3. Complete cut-elimination theorem: R.Akiyoshi and G.Mints (2016, AML). 8 / 21
Strong Buchholz’ Ω -Rule Normalization Akiyoshi and Terui • Idea of the Ω -rule: BHK-reading of ∀ XA → B . Introduction Our Results • Meaning of ∀ XA → B : some transformation f (function) from Strong Normalization any (cut-free) proof of ∀ XA to a proof of B (BHK-reading). Theorem Previous Results Buchholz’ Ω -Rule 9 / 21
Strong Buchholz’ Ω -Rule Normalization Akiyoshi and Terui • Idea of the Ω -rule: BHK-reading of ∀ XA → B . Introduction Our Results • Meaning of ∀ XA → B : some transformation f (function) from Strong Normalization any (cut-free) proof of ∀ XA to a proof of B (BHK-reading). Theorem Previous Results • So, if we have a proof f ( d ) of B for any (cut-free) proof d of ∀ XA , Buchholz’ Ω -Rule then we have a proof of ∀ XA → B . { d : ∀ XA ( X ) } . . . . ... B ... ∀ XA ( X ) → B Ω Remark The Ω -rule works well not only for a formal system based on intuitionistic logic, but for one based on classical logic as well. 9 / 21
Recommend
More recommend
