On MALL proof nets Willem Heijltjes University of Bath LL2016, - PowerPoint PPT Presentation

ALL: Coalescence A & ( B & C ) A & ( B & C ) A & ( B & C ) � � ( A ⊕ B ) ⊕ C ( A ⊕ B ) ⊕ C ( A ⊕ B ) ⊕ C � A & ( B & C ) A & ( B & C ) A & ( B & C ) � � ( A ⊕ B ) ⊕ C ( A ⊕ B ) ⊕ C ( A ⊕ B ) ⊕ C � A & ( B & C ) A & ( B & C ) � ( A ⊕ B ) ⊕ C ( A ⊕ B ) ⊕ C

Remark The set of subsets of a set X ordered by inclusion ( ⊆ ) § Is a free distributive lattice: A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) § Models ALL: A ⊢ B A ⊆ B ⇒ A & B A ∩ B ⇒ A ⊕ B A ∪ B ⇒ § Correctness: every resolution contains at least one link P Q P P ∪ Q P ∪ Q P ∩ P

Remark The set of subsets of a set X ordered by inclusion ( ⊆ ) § Is a free distributive lattice: A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) § Models ALL: A ⊢ B A ⊆ B ⇒ A & B A ∩ B ⇒ A ⊕ B A ∪ B ⇒ § Correctness: every resolution contains at least one link P Q P P ∪ Q P ∪ Q P ∩ P But distributivity destroys coalescence: ( A ∪ B ) ∩ ( A ∪ C ) A ∪ ( B ∩ C )

Properties of ALL proof nets De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N 1 ◦ N 2 ✓ P-time ? P 1 P 2 ∼ Proof equivalence / canonicity = = ✓ ⇒ ⇒ N 1 N 2 =

ALLU A ⊤ Additive units: 0, ⊤ A , ⊤ A ⊤ Equivalence: ⊤ ⊤ ⊤ ⊤ ⊤ ∼ ∼ ∼ A ⊕ B A ⊕ B A ⊕ B A & B A & B Non-confluence: ⊤ ⊕ A ⊤ ⊕ A ⊤ ⊕ A ⊤ ⊕ A ⊤ ⊕ A � � � � ⊤ ⊕ B ⊤ ⊕ B ⊤ ⊕ B ⊤ ⊕ B ⊤ ⊕ B

Saturation ⊤ ⊤ ⊤ ∼ ∼ A ⊕ B A ⊕ B A ⊕ B ⊤ ⊤ � � � � A ⊕ B A ⊕ B ⊤ � � A ⊕ B ⊤ ⊤ ∼ A & B A & B ⊤ � � A & B

Properties of ALLU proof nets De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N 1 ◦ N 2 ✓ P-time ? P 1 P 2 ∼ Proof equivalence / canonicity = = ✓ ⇒ ⇒ N 1 N 2 =

MLLU A , B , C · ·· · = 1 | ⊥ | A ⊗ B | A & B Γ , A B , ∆ Γ , A , B Γ 1 Γ , A ⊗ B , ∆ Γ , A & B Γ , ⊥ 1 ⊥ & 1 ⊥ ⊥ ⊥ ⊥ 1 & ⊗ ⊗

Equivalence A B A B A B ∼ ∼ ⊗ ⊗ ⊗ ⊥ ⊥ ⊥ A A A ∼ ∼ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ A B A B A B and ∼ & & & ⊥ ⊥ ⊥

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗ ⊥ ⊗ ⊥, 1 , 1 , 1 , ⊥ ⊗ ⊥

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

⊗ ⊥ ⊥ 1 1 1 ⊥ ⊥ ⊗

Properties of MLLU proof nets De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N 1 ◦ N 2 ✓ P-time ? P 1 P 2 ∼ Proof equivalence / canonicity = = ✖ PSPACE ⇒ ⇒ N 1 N 2 =

Proof nets and complexity MLL MLLU ALL ALLU De-sequentialization ✓ ✓ ✓ ✓ ✓ 2 ✓ 6 ✓ 6 Sequentialization / correctness ✓ Composition / cut-elimination ✓ ✓ ✓ ✓ ✓ 1 ✖ 5 ✓ 3 ✓ 4 Proof equivalence / canonicity 1 [Girard 1987]; 2 [Guerrini 1999]; 3 [Hu 1999]; 4 [H 2011]; 5 [H & Houston 2014]; 6 [H & Hughes 2015]

MALL

Monomial nets Γ , A Γ , B Γ , A i Γ , A & B Γ , A 1 ⊕ A 2 Γ p A p A i Γ p B p Γ & p ⊕ i A ⊗ B Γ A ⊗ B Γ Links are indexed by monomial weights: elements p 1 · p 2 · p 3 . . . p n · q 1 · q 2 · q 3 . . . q m from a boolean algebra ( P , 0 , 1 , + , ·, ) whose atomic elements p , p ∈ P indicate the two branches of a subformula A & p B

w · p w · p w · p w · p ax ax ax ax w · p w · p w ∼ w w & p & p ⊕ 1 ⊕ 1 ⊕ 1 A & A A ⊕ A A & A A ⊕ A Distributivity: ( w · p ) + ( w · p ) = w · ( p + p ) = w · 1 = w

MALL proof nets M : Monomial nets [Girard 1996, Laurent & Maieli 2008] S : Slice nets [Hughes & Van Glabbeek 2005] C : Conflict nets [Hughes & H 2016] M S C De-sequentialization ✓ Sequentialization / correctness ✖ Composition / cut-elimination ? Proof equivalence / canonicity ✖

Sequent + linking A , A B , B A , A B , B A , A ⊗ B , B ⊕ B A , A ⊗ B , B ⊕ B A & A , A ⊗ B , B ⊕ B A & A A ⊗ B B ⊕ B A , A B , B A , A B , B A , A ⊗ B , B ⊕ B A , A ⊗ B , B ⊕ B A & A , A ⊗ B , B ⊕ B

Slice nets A & A A ⊗ B B & B A set of links for each slice

Slice nets A & A A ⊗ B B & B A set of links for each slice B , B B , B A A ⊗ ( B & B ) B A , A B & B , B A , A ⊗ ( B & B ) , B But there may be 2 n slices, for n the number of &-occurrences

MALL proof nets M : Monomial nets [Girard 1996, Laurent & Maieli 2008] S : Slice nets [Hughes & Van Glabbeek 2005] C : Conflict nets [Hughes & H 2016] M S C De-sequentialization ✓ ✖ Sequentialization / correctness ✖ ✓ Composition / cut-elimination ? ✓ Proof equivalence / canonicity ✖ ✓

The problem: size v canonicity Π 1 Π 3 Π 2 Π 3 Π 1 Π 2 A , C D B , C D A , C B , C Π 3 ∼ A , C ⊗ D B , C ⊗ D A & B , C D A & B , C ⊗ D A & B , C ⊗ D Distributivity: ( a · x ) + ( a · b ) + ( y · b ) ( a · ( x + b )) + ( y · b ) ( a · x ) + (( a + y ) · b )

, A , B , C , D , A , B , C , D , A , C , A , D & , B , C , B , D & , A , C , B , C , A , D , B , D Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ & & & Γ , A & B , C & Γ , A & B , D & , A , C & D Γ , B , C & D Γ , A , B , C & D Γ , A & B , C , D Γ ↔ ↔ & & & , A & B , C & D , A & B , C & D Γ , A & B , C & D Γ , A & B , C & D Γ Γ B , ∆ , C D , Σ ⊗ , A B , ∆ , C , A i , B j , A i , B j Γ Γ Γ ⊗ Γ ⊕ j ⊕ i Γ , A B , ∆ , C ⊗ D , Σ , A ⊗ B , ∆ , C D , Σ ⊗ , A 1 ⊕ A 2 , B , C j Γ ↔ Γ , A 1 , B 1 ⊕ B 2 ↔ ⊗ Γ ⊕ i Γ ⊕ j , A ⊗ B , ∆ , C ⊗ D , Σ , A ⊗ B , ∆ , C ⊗ D , Σ Γ , A 1 ⊕ A 2 , B 1 ⊕ B 2 , A 1 ⊕ A 2 , B 1 ⊕ B 2 Γ B , ∆ , C , D , A B , ∆ , C , D , A , C i , B , C i & , A , C i , B , C i Γ Γ Γ Γ Γ & ⊗ Γ ⊕ i Γ ⊕ i Γ , A B , ∆ , C & D , A ⊗ B , ∆ , C , D , A & B , C i , A , C 1 ⊕ C 2 , B , C 1 ⊕ C 2 & Γ ↔ Γ ↔ & ⊗ Γ ⊕ i Γ , A ⊗ B , ∆ , C & D , A ⊗ B , ∆ , C & D , A & B , C 1 ⊕ C 2 , A & B , C 1 ⊕ C 2 Γ Γ B , ∆ , C i , A B , ∆ , C i Γ ⊕ i ⊗ Γ , A B , ∆ , C 1 ⊕ C 2 , A ⊗ B , ∆ , C i Γ ↔ ⊗ Γ ⊕ i , A ⊗ B , ∆ , C 1 ⊕ C 2 , A ⊗ B , ∆ , C 1 ⊕ C 2 Γ , A i , B , C , A i , B , C Γ Γ & ⊕ i Γ , A 1 , B & C , A 1 ⊕ A 2 , B , C Γ ↔ & ⊕ i Γ , A 1 ⊕ A 2 , B & C , A 1 ⊕ A 2 , B & C Γ , A , B , C , A , B , D & , A , B , C , A , B , D Γ Γ Γ Γ & & , A , B , C & D , A & B , C , A & B , D & Γ ↔ Γ Γ & , A & B , C & D , A & B , C & D Γ Γ

Conflict nets: idea (strong) canonicity: invariance under all commutations local canonicity: invariance under local commutations

A , A A , A ⊕ 1 ⊕ 2 A , A B , B B , B A , ( A ⊕ A ) B , B A , ( A ⊕ A ) B , B ⊕ 2 & ⊗ ⊗ A , ( A ⊕ A ) B , B & B A , ( A ⊕ A ) ⊗ B , B A , ( A ⊕ A ) ⊗ B , B ⊗ & A , ( A ⊕ A ) ⊗ B , B & B A , ( A ⊕ A ) ⊗ B , B & B & A & A , ( A ⊕ A ) ⊗ B , B & B & ( A & A ) & (( A ⊕ A ) ⊗ B ) , B & B & (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) ax ax ax ax ax ⊕ 1 ax ⊕ 2 ax ⊕ 2 & ⊗ ⊗ & ⊗ & & &

A , A A , A ⊕ 1 ⊕ 2 A , A B , B B , B A , ( A ⊕ A ) B , B A , ( A ⊕ A ) B , B ⊕ 2 & ⊗ ⊗ A , ( A ⊕ A ) B , B & B A , ( A ⊕ A ) ⊗ B , B A , ( A ⊕ A ) ⊗ B , B ⊗ & A , ( A ⊕ A ) ⊗ B , B & B A , ( A ⊕ A ) ⊗ B , B & B & A & A , ( A ⊕ A ) ⊗ B , B & B & ( A & A ) & (( A ⊕ A ) ⊗ B ) , B & B & (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) ax ax ax ax ax ax ax & ⊗ ⊗ & ⊗ &

A , A A , A ⊕ 1 ⊕ 2 A , A B , B B , B A , ( A ⊕ A ) B , B A , ( A ⊕ A ) B , B ⊕ 2 & ⊗ ⊗ A , ( A ⊕ A ) B , B & B A , ( A ⊕ A ) ⊗ B , B A , ( A ⊕ A ) ⊗ B , B ⊗ & A , ( A ⊕ A ) ⊗ B , B & B A , ( A ⊕ A ) ⊗ B , B & B & A & A , ( A ⊕ A ) ⊗ B , B & B & ( A & A ) & (( A ⊕ A ) ⊗ B ) , B & B & (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) ax ax ax ax ax ax ax & ⊗ ⊗ ⊗ &

A , A A , A ⊕ 1 ⊕ 2 A , A B , B B , B A , ( A ⊕ A ) B , B A , ( A ⊕ A ) B , B ⊕ 2 & ⊗ ⊗ A , ( A ⊕ A ) B , B & B A , ( A ⊕ A ) ⊗ B , B A , ( A ⊕ A ) ⊗ B , B ⊗ & A , ( A ⊕ A ) ⊗ B , B & B A , ( A ⊕ A ) ⊗ B , B & B & A & A , ( A ⊕ A ) ⊗ B , B & B & ( A & A ) & (( A ⊕ A ) ⊗ B ) , B & B & (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) ax ax ax ax ax ax ax & ⊗ ⊗ ⊗ &

A , A A , A ⊕ 1 ⊕ 2 A , A B , B B , B A , ( A ⊕ A ) B , B A , ( A ⊕ A ) B , B ⊕ 2 & ⊗ ⊗ A , ( A ⊕ A ) B , B & B A , ( A ⊕ A ) ⊗ B , B A , ( A ⊕ A ) ⊗ B , B ⊗ & A , ( A ⊕ A ) ⊗ B , B & B A , ( A ⊕ A ) ⊗ B , B & B & A & A , ( A ⊕ A ) ⊗ B , B & B & ( A & A ) & (( A ⊕ A ) ⊗ B ) , B & B & (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) ax ax ax ax ax ax ax # ⌢ ⌢ ⌢ #

c a b (( A & A ) & (( A ⊕ A ) ⊗ B )) & ( B & B ) d e f g c b g a e d f # ⌢ ⌢ ⌢ #

Conflict nets Data: ( # /⌢ ⌢ ) alternating, n-ary conflict tree T T · ·· · = ∆ ⊆ Γ | ( T # · · · # T ) | ( T ⌢ · · · ⌢ T ) over an axiom linking ( ∆ = a , a ) over a sequent Γ Hybrid of focussing and proof nets: § a conflict node # represents an ALL + proof net (& , ⊕, ) & & § a concord node ⌢ ⌢ represents an MLL + ⊕ proof net ( ⊗, ⊕, & ) § ( ⊕/ & ) are not confined to a layer Correctness / sequentialization: by coalescence

De-sequentialization � � ( a , a ) a , a = � � Π Γ , A , B = � Π � Γ , A & B � � Π Γ , A = � Π � Γ , A ⊕ B � Π 1 � Π 2 Γ , A B , ∆ = � Π 1 � ⌢ ⌢ � Π 2 � Γ , A ⊗ B , ∆ � Π 1 � Π 2 Γ , A Γ , B = � Π 1 � # � Π 2 � Γ , A & B

Coalescence: MLL + ⊕ A & B C 1 . . . C k A ⊕ B C 1 . . . C k ⇓ ⇓ A & B C 1 . . . C k A ⊕ B C 1 . . . C k ⌢ (∆) ⇒ ∆ a b ( a ⌢ b ⌢ T 1 ⌢ · · · ⌢ T n ) C 1 . . . C k A ⊗ B D 1 . . . D k ⇓ c ( c ⌢ T 1 ⌢ · · · ⌢ T n ) C 1 . . . C k A ⊗ B D 1 . . . D k

2D ALL coalescence A & B A & B A & B A & B C & D C & D C & D C & D C & D C & D C & D C & D A & B A & B A & B A & B

3D ALL coalescence A & B , C & D , E & F C & D C & D C & D C & D A & B A & B A & B A & B F F F F & & & & E E E E

3D ALL coalescence A & B , C & D , E & F C & D C & D C & D C & D A & B A & B A & B A & B F F F F & & & & E E E E C & D C & D C & D C & D A & B A & B A & B A & B F F F F & & & & E E E E