Cut elimination for infinitary proofs Amina Doumane IRIF-Universit - PowerPoint PPT Presentation

Cut elimination for infinitary proofs Amina Doumane IRIF-Université Paris Diderot August 2016 - CSL Joint work with: David Baelde Alexis Saurin � LSV-ENS Cachan IRIF-Université Paris 7

Introduction

Introduction Inductive and coinductive definitions A natural number is either 0 or the successor of a natural number.

Introduction Inductive and coinductive definitions N = 1 ⊕ N

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X A stream is made of a natural number (head) and a stream (tail).

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X S = N ⊗ S

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X S = ν X . N ⊗ X

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X S = ν X . N ⊗ X

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X S = ν X . N ⊗ X Proofs-programs over these data types double ( n ) = 0 if n = 0 = succ ( succ ( double ( m ))) if n = succ ( m )

Introduction Inductive and coinductive definitions N = µ X . 1 ⊕ X S = ν X . N ⊗ X Proofs-programs over these data types double ( n ) = 0 if n = 0 = succ ( succ ( double ( m ))) if n = succ ( m ) Π double N ⊢ N ( ⊕ 2 ) N ⊢ 1 ⊕ N (1) ( µ r ) 1 ⊢ 1 N ⊢ N Π double = ( ⊕ 1 ) ( ⊕ 2 ) 1 ⊢ 1 ⊕ N N ⊢ 1 ⊕ N ( µ l ) ( µ r ) 1 ⊢ N N ⊢ N ( ⊕ l ) 1 ⊕ N ⊢ N ( µ l ) N ⊢ N

Infinitary (circular) proofs in the litterature Verification device : Complete deduction sytem giving algorithms for checking validity (Tableaux, sequent calculi) Success → Validity µ -calulus formula → Proof search ր ց Failure → Invalidity

Infinitary (circular) proofs in the litterature Verification device : Complete deduction sytem giving algorithms for checking validity (Tableaux, sequent calculi) Success → Validity µ -calulus formula → Proof search ր ց Failure → Invalidity Completeness arguments : Intermediate objects between syntax and semantics (Kozen, Kaivola, Walukiewicz) µ -calulus formula → Circular proof → Finite axiomatization

Infinitary (circular) proofs in the litterature Verification device : Complete deduction sytem giving algorithms for checking validity (Tableaux, sequent calculi) Success → Validity µ -calulus formula → Proof search ր ց Failure → Invalidity Completeness arguments : Intermediate objects between syntax and semantics (Kozen, Kaivola, Walukiewicz) µ -calulus formula → Circular proof → Finite axiomatization But rarely as proof/programm objects in themselves

Structural proof theory Two main properties: Syntactic cut-elimination

Structural proof theory Two main properties: Syntactic cut-elimination Motivation: At the heart of proofs-as-programms viewpoint Focalization Motivation: Proof search strategy based on the notion of polarity State of art de la focalization: le nothing peut être provoquant.

Structural proof theory Two main properties: Syntactic cut-elimination Motivation: At the heart of proofs-as-programms viewpoint State of art: Semantical cut elimination (Brotherstone), Additive fragment (Fortier-Santocanale) Focalization Motivation: Proof search strategy based on the notion of polarity State of art: Nothing State of art de la focalization: le nothing peut être provoquant.

Structural proof theory Two main properties: Syntactic cut-elimination Motivation: At the heart of proofs-as-programms viewpoint State of art: Semantical cut elimination (Brotherstone), Additive fragment (Fortier-Santocanale) Focalization Motivation: Proof search strategy based on the notion of polarity State of art: Nothing State of art de la focalization: le nothing peut être provoquant.

Infinitary proof system µ MALL ∞

Formulas µ MALL ∞ formulas X | ⊤ |⊥| 0 | 1 | F ⊗ F | F � F | F � F | F ⊕ F F ::= MALL | µ X . F least fixed point | ν X . F greatest fixed point µ and ν are dual. Example: ¬ ( ν X . X ⊗ X ) = µ X . X � X . Data types encoding Nat := µ X . 1 ⊕ X Stream( A ) := ν X . A ⊗ X

Sequent calculus µ MALL ∞ pre-proofs are the trees coinductively generated by: Usual logical rules ⊢ Γ , F ⊢ ∆ , G ⊢ Γ , F , G ⊢ Γ , F ⊢ Γ , G ⊢ Γ , F i ( ⊗ ) ( ⊕ i ) ( � ) ( � ) ⊢ Γ , ∆ , F ⊗ G ⊢ Γ , F � G ⊢ Γ , F � G ⊢ Γ , F 1 ⊕ F 2 Identity rules ⊢ Γ , F ⊢ ∆ , ¬ F (ax) ⊢ F , ¬ F (cut) ⊢ Γ , ∆ Rules for µ and ν ⊢ Γ , F [ µ X . F / X ] ⊢ Γ , F [ ν X . F / X ] ( µ ) ( ν ) ⊢ Γ , µ X . F ⊢ Γ , ν X . F

Sequent calculus . . . . . . ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F (cut) ⊢ F

Sequent calculus . . . . . . ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F (cut) ⊢ F Pre-proofs are unsound, hence the need for a validity condition.

Sequent calculus . . . . . . ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F ( µ ) ( ν ) ⊢ µ X . X ⊢ ν X . X , F (cut) ⊢ F Pre-proofs are unsound, hence the need for a validity condition. Validity condition A proof is a pre-proof such that every infinite branch must unfold a ν formula infinitly often.

Focalization

Focalization in MALL Idea: classify the connectives into 2 categories Negative connectives: Invertible connectives ie. we don’t lose provability by applying these rules ( � , � ). If ⊢ Γ , A � B is provable then ⊢ Γ , A , B is also provable. Positive connectives: Non Invertible connectives ie. there is a choice to make, a bad choice may lead to a loss of provability ( ⊕ , ⊗ ). ⊢ 1 , X ⊥ ⊢ X ⊢ ⊥ ( ⊕ ) ( ⊗ ) ⊢ ⊤⊕⊥ ⊢ X ⊗ 1 , X ⊥

Focalization in MALL To prove a sequent Γ , apply the following: Γ contains a negative formula Γ contains no negative formula choose a negative choose some positive formula and formula and apply the unique decompose it hereditarily until negative rule available. negative subformulas are reached. ( ax ) ( ax ) ⊢ B , B ⊥ ⊢ C , C ⊥ ( ⊕ ) ( ⊕ ) ⊢ B , D ⊕ B ⊥ ⊢ C , D ⊕ C ⊥ ( ⊗ ) ⊢ B ⊗ C , D ⊕ B ⊥ , D ⊕ C ⊥ ( ⊕ ) ⊢ A ⊕ ( B ⊗ C ) , D ⊕ B ⊥ , D ⊕ C ⊥ ( � ) ⊢ A ⊕ ( B ⊗ C ) , ( D ⊕ B ⊥ ) � ( D ⊕ C ⊥ )

Focalization in MALL To prove a sequent Γ , apply the following: Γ contains a negative formula Γ contains no negative formula choose a negative choose some positive formula and formula and apply the unique decompose it hereditarily until negative rule available. negative subformulas are reached. ( ax ) ( ax ) ⊢ B , B ⊥ ⊢ C , C ⊥ ( ⊕ ) ( ⊕ ) ⊢ B , D ⊕ B ⊥ ⊢ C , D ⊕ C ⊥ ( ⊗ ) ⊢ B ⊗ C , D ⊕ B ⊥ , D ⊕ C ⊥ ( � ) ⊢ B ⊗ C , ( D ⊕ B ⊥ ) � ( D ⊕ C ⊥ ) ( ⊕ ) ⊢ A ⊕ ( B ⊗ C ) , ( D ⊕ B ⊥ ) � ( D ⊕ C ⊥ )

Focalization in MALL To prove a sequent Γ , apply the following: Γ contains a negative formula Γ contains no negative formula choose a negative choose some positive formula and formula and apply the unique decompose it hereditarily until negative rule available. negative subformulas are reached. ( ax ) ⊢ C , C ⊥ ( ⊕ ) ( ax ) ⊢ B , B ⊥ ⊢ C , D ⊕ C ⊥ ( ⊗ ) ⊢ B ⊗ C , B ⊥ , D ⊕ C ⊥ ( ⊕ ) ⊢ B ⊗ C , B ⊥ , D ⊕ C ⊥ ( ⊕ ) ⊢ A ⊕ ( B ⊗ C ) , D ⊕ B ⊥ , D ⊕ C ⊥ ( � ) ⊢ A ⊕ ( B ⊗ C ) , ( D ⊕ B ⊥ ) � ( D ⊕ C ⊥ )

Focalization for µ MALL Classification of connectives ν is classified negative and µ is classified positive , even though both are invertible. If µ is classified negative, we would have . . ⊢ . ( µ ) ⊢ ⊤⊗⊤ , µ X . X ( µ ) ⊢ ⊤⊗⊤ , µ X . X ... which is not a valid proof.

Proof of completeness of Focalization for MALL Transforms a MALL proof into a focused proof by using: Strong commutation of Negatives: negative connectives commute down with all other connectives. Exemple: ( � / � ) ⊢ F , P , Q ⊢ G , P , Q ⊢ F , P , Q ⊢ G , P , Q ( � ) ( � ) ( � ) ⊢ F , P � Q ⊢ G , P � Q → ⊢ F � G , P , Q ( � ) ( � ) ⊢ F � G , P � Q ⊢ F � G , P � Q Weak commutation of positives: positive connectives commute with each others only. Exemple: ( ⊕ / ⊕ ) ⊢ G , P , Γ ⊢ G , P , Γ ( ⊕ ) ( ⊕ ) ⊢ G , P ⊕ Q , Γ → ⊢ F ⊕ G , P , Γ ( ⊕ ) ( ⊕ ) ⊢ F ⊕ G , P ⊕ Q , Γ ⊢ F ⊕ G , P ⊕ Q , Γ

Proof of completeness of Focalization for µ MALL Works in the same way, under some adaptations. Rules commutations cannot be performed locally: π ( ⋆ ) ⊢ F , P , Q ( � ) ⊢ F , P � Q ⊢ F , P � Q π ( � ) ⊢ F � F , P � Q ⊢ F , P , Q ( � ) ⊢ F , P � Q ⊢ F , P � Q ( � ) ⊢ F � F , P � Q ( ν ) ( ⋆ ) ⊢ F , P � Q The commutation process is productive. The commutation process preserves validity.

Cut elimination

Cut elimination procedure Strategy: “push” the cuts away from the root. Cut-Cut: ⊢ Γ , F ⊢ ¬ F , ∆ , G (cut) ⊢ Γ , ∆ , G ⊢ ¬ G , Σ (cut) ⊢ Γ , ∆ , Σ � ⊢ ¬ F , ∆ , G ⊢ ¬ G , Σ (cut) ⊢ Γ , F ⊢ ¬ F , ∆ , Σ (cut) ⊢ Γ , ∆ , Σ