Conflict nets: Efficient locally canonical MALL proof nets Dominic - PowerPoint PPT Presentation

Conflict nets: Efficient locally canonical MALL proof nets Dominic J. D. Hughes and Willem Heijltjes LICS, 6 July 2016

Conflict nets: Efficient locally canonical MALL proof nets

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ � NJ LJ = Intuitionistic sequent calculus NJ = Intuitionistic natural deduction

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ � NJ A � B � C � D ⊢ E A � B � C � D ⊢ E ≃ A � B � C ∧ D ⊢ E A ∧ B � C � D ⊢ E A ∧ B � C ∧ D ⊢ E A ∧ B � C ∧ D ⊢ E − − → ← A B C D A ∧ B C ∧ D E

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ � NJ A � B � C � D ⊢ E A � B � C � D ⊢ E ≃ A � B � C ∧ D ⊢ E A ∧ B � C � D ⊢ E A ∧ B � C ∧ D ⊢ E A ∧ B � C ∧ D ⊢ E − − → ← A B C D x : A ∧ B y : C ∧ D N E N [ � a , b � / x ] [ � c , d � / y ]

⊢ A B , C ⊢ D B , C ⊢ D ≃ A � B , C ⊢ D ⊢ A B ⊢ C � D A � B ⊢ C � D A � B ⊢ C � D − − → ← A � B A B [ C ] D C � D

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨

A ⊢ C D ⊢ E B ⊢ C D ⊢ E A ⊢ C B ⊢ C A , C � D ⊢ E B , C � D ⊢ E A ∨ B ⊢ C D ⊢ E ∼ A ∨ B , C � D ⊢ E A ∨ B , C � D ⊢ E − − → → [ A ] [ B ] [ A ] [ B ] A ∨ B C C C � D C C � D C C � D C D D D A ∨ B E E E E

A ⊢ C D ⊢ E B ⊢ C D ⊢ E A ⊢ C B ⊢ C A , C � D ⊢ E B , C � D ⊢ E A ∨ B ⊢ C D ⊢ E ∼ A ∨ B , C � D ⊢ E A ∨ B , C � D ⊢ E − − → → [ A ] [ B ] [ A ] [ B ] L R L R x : A ∨ B C C f : C � D C f : C � D C f : C � D C D D D N N N x : A ∨ B E E E E N [ fM / d ] where case v of inl a �→ N [ fL / d ] M = case v of inl a �→ L inr b �→ N [ fR / d ] inr b �→ R

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` )

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) P � P Q � Q Q � Q R � R ⊗ ⊗ ≃ � P ⊗ Q � Q Q � Q ⊗ R � R R � R ⊗ P P � P ⊗ � P ⊗ Q � Q ⊗ R � R � P ⊗ Q � Q ⊗ R � R P P − − → ← P ⊗ Q Q ⊗ R P R

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ canonical for MALL ( ⊗ ` ⊕ ) `

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ canonical

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ canonical • 2003 Hughes + Van Glabbeek: MALL Slice nets ✗ efficient � canonical

P � P P � P P � P Q � Q P � P Q � Q ≃ P � P ⊗ Q � Q P � P ⊗ Q � Q P ` P � P Q � Q P � P ⊗ Q � Q P � P ⊗ Q � Q P ` P ` − − → ← P ⊗ Q P ` P Q

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ canonical • 2003 Hughes + Van Glabbeek: MALL Slice nets ✗ efficient � canonical

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ canonical • 2003 Hughes + Van Glabbeek: MALL Slice nets ✗ efficient � canonical • This paper: MALL Conflict nets � efficient � locally canonical

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ locally canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ locally canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ locally canonical • 2003 Hughes + Van Glabbeek: MALL Slice nets ✗ efficient � strongly canonical • This paper: MALL Conflict nets � efficient � locally canonical

All MLL and ALL rule commutations are local R � R S S P P Q � Q R R S S � � � � ⊗ ⊗ ⊗ � R ⊗ S � P ⊗ Q � R ⊗ S � S ⊗ P P Q � Q R � S ` P � Q R ⊗ � → � P ⊗ Q R ⊗ S � P ⊗ Q � Q ⊗ ( R ⊗ S ) � R P � Q � R ` S ⊗ P � S ` � P ⊗ Q � Q ⊗ ( R ⊗ S ) � R ` S � P ⊗ Q � Q ⊗ ( R ⊗ S ) � R ` S P P ` ` P ` ( P ⊗ Q ) � Q ⊗ ( R ⊗ S ) � R ` S P ` ( P ⊗ Q ) � Q ⊗ ( R ⊗ S ) � R ` S

One MALL rule commutation is not local Π Π . . . . . . Π . B , ∆ , C B , ∆ , D Γ , A B , ∆ , C Γ , A B , ∆ , D ⊗ . . ⊗ Γ , A ⊗ B , ∆ , C ` → Γ , A ⊗ B , ∆ , D Γ , A B � ∆ � C ` D ⊗ ` Γ , A ⊗ B , ∆ , C Γ , A ⊗ B , ∆ , C D D ` `

Local/strong canonicity • Local canonicity Invariance under local rule commutations • Strong canonicity Invariance under all rule commutations

Conflict nets: Efficient locally canonical MALL proof nets • 1934 Gentzen: LJ NJ = Λ � efficient � canonical for → ∧ ✗ locally canonical for → ∧∨ • 1987 Girard: LL Box nets � efficient � canonical for MLL ( ⊗ ` ) ✗ locally canonical for MALL ( ⊗ ` ⊕ ) ` • 1996 Girard: MALL Monomial nets ✗ efficient ✗ locally canonical • 2003 Hughes + Van Glabbeek: MALL Slice nets ✗ efficient � strongly canonical • This paper: MALL Conflict nets � efficient � locally canonical

Conflict net 1 b c b a c a ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) > P ` �

Conflict net 2 c b c a b f g a d e P ⊗ R P ` P R ` R > � � � d f e g >

Conflict nets are locally canonical Q � Q Q � Q Q � Q Q � Q ⊕ 1 Q � Q ⊕ R ⊕ 1 ` Q � Q ⊕ R P � P Q ` Q � Q ⊗ ` P � P ⊗ ( Q ≃ Q � Q ⊕ R ⊗ Q ) � Q ` P � P Q ` ⊕ 1 P � P ⊗ ( Q Q ) � Q ⊕ R P � P ⊗ ( Q Q ) � Q ⊕ R ` ` ` ` � � � � P ⊗ ( Q Q ) ` ( Q ⊕ R ) P ⊗ ( Q Q ) ` ( Q ⊕ R ) P ` P � ` � − − → → b c b a c a ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) > P ` �

Conflict nets: efficient rather than strongly canonical a e a P � Pb Q � Qd P � P b P � P P � P Q � Q c ∼ P ` P � P Q � Q P � P ⊗ Q � Q P � P ⊗ Q � Q P � P ⊗ Q � Q P � P ⊗ Q � Q P ` P ` − − → → a a d a b a d b e c c � � P P P ⊗ Q Q P P P ⊗ Q Q ` ` > � b b e >

Coalescence correctness (generalizing MLL contractibility) b c b a a c > ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` � ` a bc a bc ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` � ⊕ 1 a a bc bc P ( P ⊗ ( Q ` Q )) ` ( Q ⊕ R ) � ⊗ a bc ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) a bc P ` ` a bc a bc P ( P ⊗ ( Q ` Q )) ` ( Q ⊕ R )

c bc Q � Q Q � Q Q Q � Q ` a Q � Q ⊕ R P � P Q ` P � P ⊗ ( Q Q ) � Q ⊕ R ` P � ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) ` b c b a c a P ( P ⊗ ( Q ` Q )) ` ( Q ⊕ R ) > �

c bc Q � Q Q � Q Q Q � Q ` a Q � Q ⊕ R P � P Q ` P � P ⊗ ( Q Q ) � Q ⊕ R ` P � ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) ` b c b a c a P ( P ⊗ ( Q ` Q )) ` ( Q ⊕ R ) > � ` a a bc bc P ( P ⊗ ( Q ` Q )) ` ( Q ⊕ R ) �

Q � Q Q � Q bc Q ` Q � Q a Q � Q ⊕ R P � P Q ` P � P ⊗ ( Q ` Q ) � Q ⊕ R P � ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) ` b c b a c a ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` > � ` a bc a bc ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` �

Q � Q Q � Q bc Q ` Q � Q a Q � Q ⊕ R P � P Q ` P � P ⊗ ( Q ` Q ) � Q ⊕ R P � ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) ` a bc a bc ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` �

Q � Q Q � Q bc Q ` Q � Q a Q � Q ⊕ R P � P Q ` P � P ⊗ ( Q ` Q ) � Q ⊕ R P � ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) ` a bc a bc ( P ⊗ ( Q Q )) ` ( Q ⊕ R ) P ` �