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

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

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On MALL proof nets Willem Heijltjes University of Bath LL2016, Lyon Intuitionistic logic natural deduction / lambda-calculus Linear logic proof nets ? Intuitionistic logic natural deduction / lambda-calculus Linear logic proof nets ? B ,

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On MALL proof nets

Willem Heijltjes University of Bath

LL2016, Lyon

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Intuitionistic logic natural deduction / lambda-calculus Linear logic proof nets ?

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Intuitionistic logic natural deduction / lambda-calculus Linear logic proof nets ? ⊢ 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

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Intuitionistic logic natural deduction / lambda-calculus Linear logic proof nets ? ⊢ 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 Motivation:

  • Computation
  • Canonicity
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MLL proof nets

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆

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MLL proof nets

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ A A A, A Γ, A, B Γ, A & B Γ A B & Γ A ⊗ B Γ A B ∆ Γ B A ∆ Γ, A, B, ∆ Γ, B, A, ∆

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MLL proof nets

A, A B, B C, C B, B ⊗ C, C A, A ⊗ B, B ⊗ C, C ∼ A, A B, B A, A ⊗ B, B C, C A, A ⊗ B, B ⊗ C, C

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MLL proof nets

A, A B, B C, C B, B ⊗ C, C A, A ⊗ B, B ⊗ C, C ∼ A, A B, B A, A ⊗ B, B C, C A, A ⊗ B, B ⊗ C, C

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MLL proof nets

A, A B, B C, C B, B ⊗ C, C A, A ⊗ B, B ⊗ C, C ∼ A, A B, B A, A ⊗ B, B C, C A, A ⊗ B, B ⊗ C, C A A ⊗ B B ⊗ C C

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Correctness conditions

✓ A A & A & A ✖ A A ⊗ A ⊗ A

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Correctness conditions

✓ A A & A & A ✖ A A ⊗ A ⊗ A

§ long-trip

[Girard 1987]

§ switching

[Danos & Regnier 1989]

§ contractibility

[Danos 1990]

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SLIDE 12

MLL contractibility

  • 1. Start from an unlabelled graph with paired

&

  • edges
  • 2. Contract by:
  • ↓

↓

  • 3. Correct ⇔ contracts to a single point

Implemented in linear time via union–find [Danos 1990, Guerrini 1999]

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A ⊗ B & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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A, A B, B (A ⊗ B), A, B (A ⊗ B), A & B (A ⊗ B) & A & B

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Properties of MLL proof nets

De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N1 ◦ N2 ✓ P-time Proof equivalence / canonicity P1

?

∼ P2 = ⇒ = ⇒ N1 = N2 ✓

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Properties of MLL proof nets

De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N1 ◦ N2 ✓ P-time Proof equivalence / canonicity P1

?

∼ P2 = ⇒ = ⇒ N1 = N2 ✓ What about other fragments of linear logic?

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking

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ax ⊗ ax ⊗ ax

A B B C A A⊗B B⊗C C

A A ⊗ B B ⊗ C C

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition

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⊗ cut &

A⊗B A & B A B A B

  • cut cut

A B A B

ax cut

A A A

  • A

A

A ⊗ A ((A & A) ⊗ A) & A ((A ⊗ A) & A) ⊗ A A & A

  • A ⊗ A

A & A

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition like lambda-calculus like categorical coherence

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app λ

(λx.M)N: B λx.M: A→B N: A M: B x: A

  • M[N/x]: B

M: B N: A x: A

A ⊗ I A A ⊗ B B ⊗ A A ⊗(B⊗C) (A⊗B)⊗ C

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition like lambda-calculus like categorical coherence

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition like lambda-calculus like categorical coherence good with exponentials good with additives

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Two traditions

Γ, A B, ∆ Γ, A ⊗ B, ∆ Γ A B ∆ ⊗ Γ A ⊗ B ∆ Computation Canonicity graph (nodes are rules) sequent + axiom linking normalization: graph rewriting normalization: path composition like lambda-calculus like categorical coherence good with exponentials good with additives interaction nets1, sharing graphs2 ??

1[Lafont 1990] 2[Lamping 1990, Asperti & Guerrini 1998]

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Additives

A , A A , A ⊕ B B , B B , A ⊕ B A & B , A ⊕ B

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Additives

A , A A , A ⊕ B B , B B , A ⊕ B A & B , A ⊕ B & ax ax ⊕1 ⊕2

A B A&B A B A⊕B

A & B A ⊕ B

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A , A A , A ⊕ A

⊕1

A , A A , A ⊕ A

⊕1

A & A , A ⊕ A ∼ A , A A , A A & A , A

⊕1

A & A , A ⊕ A & ax ax ⊕1 ⊕1

A A A&A A A A⊕A

∼ & ax ax ⊕1

A A A&A A A⊕A

A & A A ⊕ A

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Additive proof nets

A & (B ⊕ C) (A ⊕ B) & (A ⊕ C) A & B ⊕ C A ⊕ B & A ⊕ C Correctness

§ A resolution or slice deletes one child of each & § Every resolution must contain exactly one link

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Composition is path composition A ⊕ B B & A B ⊕ A A & B

  • A ⊕ B

A & B

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ALL: Coalescence

A B ⊕ C → A B ⊕ C ← A B ⊕ C A B & C → A B & C Correct ⇔ contracts to a single link between roots [H & Hughes 2015]

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ALL: Coalescence

A , A B ⊕ A , A B , B B ⊕ A , B B ⊕ A , A & B B ⊕ A A & B

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ALL: Coalescence

A , A B ⊕ A , A B , B B ⊕ A , B B ⊕ A , A & B B ⊕ A A & B

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ALL: Coalescence

A , A B ⊕ A , A B , B B ⊕ A , B B ⊕ A , A & B B ⊕ A A & B

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ALL: Coalescence

A , A B ⊕ A , A B , B B ⊕ A , B B ⊕ A , A & B B ⊕ A A & B

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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

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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 P ∪ Q Q P ∪ Q P P ∩ P

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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 P ∪ Q Q P ∪ Q P P ∩ P But distributivity destroys coalescence: (A ∪ B) ∩ (A ∪ C) A ∪ (B ∩ C)

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Properties of ALL proof nets

De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N1 ◦ N2 ✓ P-time Proof equivalence / canonicity P1

?

∼ P2 = ⇒ = ⇒ N1 = N2 ✓

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ALLU

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

  • ⊤ ⊕ A

⊤ ⊕ B

  • ⊤ ⊕ A

⊤ ⊕ B

  • ⊤ ⊕ A

⊤ ⊕ B

  • ⊤ ⊕ A

⊤ ⊕ B

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Saturation

⊤ A ⊕ B ∼ ⊤ A ⊕ B ∼ ⊤ A ⊕ B

  • ⊤

A ⊕ B

  • ⊤

A ⊕ B

  • ⊤

A ⊕ B

  • ⊤

A & B ∼ ⊤ A & B

  • ⊤

A & B

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Properties of ALLU proof nets

De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N1 ◦ N2 ✓ P-time Proof equivalence / canonicity P1

?

∼ P2 = ⇒ = ⇒ N1 = N2 ✓

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MLLU

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

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Equivalence

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

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SLIDE 63

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

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SLIDE 64

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 65

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 67

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 68

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 69

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 70

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 71

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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SLIDE 75

⊥ ⊗ ⊥ 1 1 1 ⊥ ⊗ ⊥

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Properties of MLLU proof nets

De-sequentialization P = ⇒ N ✓ linear-time Sequentialization / correctness N = ⇒ P ✓ linear-time Composition / cut-elimination N1 ◦ N2 ✓ P-time Proof equivalence / canonicity P1

?

∼ P2 = ⇒ = ⇒ N1 = N2 ✖ PSPACE

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Proof nets and complexity

MLL MLLU ALL ALLU De-sequentialization ✓ ✓ ✓ ✓ Sequentialization / correctness ✓2 ✓ ✓6 ✓6 Composition / cut-elimination ✓ ✓ ✓ ✓ Proof equivalence / canonicity ✓1 ✖5 ✓3 ✓4

1[Girard 1987]; 2[Guerrini 1999]; 3[Hu 1999]; 4[H 2011]; 5[H & Houston 2014]; 6[H & Hughes 2015]

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SLIDE 78

MALL

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Monomial nets

Γ, A Γ, B Γ, A & B Γ, Ai Γ, A1 ⊕ A2 Γp Γp Ap Bp &p Γ A ⊗ B Γ Ai ⊕i Γ A ⊗ B Links are indexed by monomial weights: elements p1 · p2 · p3 . . . pn · q1 · q2 · q3 . . . qm from a boolean algebra (P, 0, 1, +, ·, ) whose atomic elements p, p ∈ P indicate the two branches of a subformula A &p B

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SLIDE 80

&p ax ax ⊕1 ⊕1

w w·p w·p w·p w·p A&A A⊕A

∼ &p ax ax ⊕1

w w·p w·p w A&A A⊕A

Distributivity: (w · p) + (w · p) = w · (p + p) = w · 1 = w

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SLIDE 81

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 ✖

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Sequent + linking

A , A B , B A , A ⊗ B , B ⊕ B A , A 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 ⊕ B A , A B , B A , A ⊗ B , B ⊕ B A & A , A ⊗ B , B ⊕ B

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SLIDE 83

Slice nets

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

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SLIDE 84

Slice nets

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

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SLIDE 85

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 ✖ ✓

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SLIDE 86

The problem: size v canonicity

Π1 A , C Π3 D A , C ⊗ D Π2 B , C Π3 D B , C ⊗ D A & B , C ⊗ D ∼ Π1 A , C Π2 B , C A & B , C Π3 D A & B , C ⊗ D Distributivity: (a · x) + (a · b) + (y · b) (a · (x + b)) + (y · b) (a · x) + ((a + y) · b)

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SLIDE 87

Γ , A, B, C, D & Γ , A, B, C & D & Γ , A & B, C & D ↔ Γ , A, B, C, D & Γ , A & B, C, D & Γ , A & B, C & D Γ ,A,C Γ ,A,D& Γ , A, C & D Γ ,B,C Γ ,B,D& Γ, B, C & D & Γ, A & B, C & D ↔ Γ ,A,C Γ ,B,C & Γ, A & B, C Γ ,A,D Γ ,B,D & Γ, A & B, D& Γ, A & B, C & D Γ , A B, ∆, C D, Σ ⊗ B, ∆, C ⊗ D, Σ ⊗ Γ , A ⊗ B, ∆, C ⊗ D, Σ ↔ Γ , A B, ∆, C ⊗ Γ , A ⊗ B, ∆, C D, Σ ⊗ Γ , A ⊗ B, ∆, C ⊗ D, Σ Γ , Ai, Bj ⊕j Γ , A1, B1 ⊕ B2 ⊕i Γ , A1 ⊕ A2, B1 ⊕ B2 ↔ Γ , Ai, Bj ⊕i Γ , A1 ⊕ A2, B, Cj ⊕j Γ , A1 ⊕ A2, B1 ⊕ B2 Γ , A B, ∆, C, D & B, ∆, C & D ⊗ Γ , A ⊗ B, ∆, C & D ↔ Γ , A B, ∆, C, D ⊗ Γ , A ⊗ B, ∆, C, D & Γ , A ⊗ B, ∆, C & D Γ , A, Ci Γ , B, Ci & Γ , A & B, Ci ⊕i Γ , A & B, C1 ⊕ C2 ↔ Γ , A, Ci ⊕i Γ , A, C1 ⊕ C2 Γ , B, Ci ⊕i Γ , B, C1 ⊕ C2 & Γ , A & B, C1 ⊕ C2 Γ , A B, ∆, Ci ⊕i B, ∆, C1 ⊕ C2 ⊗ Γ , A ⊗ B, ∆, C1 ⊕ C2 ↔ Γ , A B, ∆, Ci ⊗ Γ , A ⊗ B, ∆, Ci ⊕i Γ , A ⊗ B, ∆, C1 ⊕ C2 Γ , Ai, B, C & Γ , A1, B & C ⊕i Γ , A1 ⊕ A2, B & C ↔ Γ , Ai, B, C ⊕i Γ , A1 ⊕ A2, B, C & Γ , A1 ⊕ A2, B & C Γ , A, B, C Γ , A, B, D & Γ , A, B, C & D & Γ , A & B, C & D ↔ Γ , A, B, C & Γ , A & B, C Γ , A, B, D & Γ , A & B, D & Γ , A & B, C & D

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SLIDE 88

Conflict nets: idea

(strong) canonicity: invariance under all commutations local canonicity: invariance under local commutations

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SLIDE 89

A, A A, (A ⊕ A)

⊕2

B, B B, B B, B & B

&

A, (A ⊕ A) ⊗ B, B & B

⊗

A, A A, (A ⊕ A)

⊕1

B, B A, (A ⊕ A) ⊗ B, B

⊗

A, A A, (A ⊕ A)

⊕2

B, B A, (A ⊕ A) ⊗ 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 & ⊗ ⊗ ⊗ & & & &

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SLIDE 90

A, A A, (A ⊕ A)

⊕2

B, B B, B B, B & B

&

A, (A ⊕ A) ⊗ B, B & B

⊗

A, A A, (A ⊕ A)

⊕1

B, B A, (A ⊕ A) ⊗ B, B

⊗

A, A A, (A ⊕ A)

⊕2

B, B A, (A ⊕ A) ⊗ 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 & ⊗ ⊗ ⊗ & &

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SLIDE 91

A, A A, (A ⊕ A)

⊕2

B, B B, B B, B & B

&

A, (A ⊕ A) ⊗ B, B & B

⊗

A, A A, (A ⊕ A)

⊕1

B, B A, (A ⊕ A) ⊗ B, B

⊗

A, A A, (A ⊕ A)

⊕2

B, B A, (A ⊕ A) ⊗ 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 & ⊗ ⊗ ⊗ &

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SLIDE 92

A, A A, (A ⊕ A)

⊕2

B, B B, B B, B & B

&

A, (A ⊕ A) ⊗ B, B & B

⊗

A, A A, (A ⊕ A)

⊕1

B, B A, (A ⊕ A) ⊗ B, B

⊗

A, A A, (A ⊕ A)

⊕2

B, B A, (A ⊕ A) ⊗ 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 & ⊗ ⊗ ⊗ &

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SLIDE 93

A, A A, (A ⊕ A)

⊕2

B, B B, B B, B & B

&

A, (A ⊕ A) ⊗ B, B & B

⊗

A, A A, (A ⊕ A)

⊕1

B, B A, (A ⊕ A) ⊗ B, B

⊗

A, A A, (A ⊕ A)

⊕2

B, B A, (A ⊕ A) ⊗ 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 # ⌢ ⌢ ⌢ #

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SLIDE 94

((A & A) & ((A ⊕ A) ⊗ B)) & (B & B)

a b c d f e g a b c d e f g # ⌢ ⌢ ⌢ #

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SLIDE 95

Conflict nets

Data: (#/⌢ ⌢) alternating, n-ary conflict tree T T · ·· ·= ∆ ⊆ Γ | (T # · · · # T) | (T ⌢ · · · ⌢ T)

  • ver 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

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SLIDE 96

De-sequentialization

  • a, a
  • =

(a, a)

  • Π

Γ, A, B Γ, A & B

  • =

Π

  • Π

Γ, A Γ, A ⊕ B

  • =

Π Π1 Γ, A Π2 B, ∆ Γ, A ⊗ B, ∆

  • =

Π1 ⌢ ⌢ Π2 Π1 Γ, A Π2 Γ, B Γ, A & B

  • =

Π1 # Π2

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SLIDE 97

Coalescence: MLL + ⊕

A & B C1 . . . Ck A ⊕ B C1 . . . Ck ⇓ ⇓ A & B C1 . . . Ck A ⊕ B C1 . . . Ck

⌢(∆) ⇒ ∆

C1 . . . Ck A ⊗ B D1 . . . Dk

a b

(a ⌢ b ⌢ T1 ⌢ · · · ⌢ Tn) ⇓ C1 . . . Ck A ⊗ B D1 . . . Dk

c

(c ⌢ T1 ⌢ · · · ⌢ Tn)

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SLIDE 98

2D ALL coalescence

A & B C & D A & B C & D A & B C & D A & B C & D A & B C & D A & B C & D A & B C & D A & B C & D

slide-99
SLIDE 99

3D ALL coalescence

A & B , C & D , E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F

slide-100
SLIDE 100

3D ALL coalescence

A & B , C & D , E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F A & B C & D E & F

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SLIDE 101

A & B , C & D , E & F

(A, C, E) # (B, C, E) # (A, C, F) # (B, C, F) # (B, D, F) # (B, D, E) # (A, D, E) # (A, D, F)

⇓

(A & B, C, E) # (A, C, F) # (B, C & D, F) # (B, D, E) # (A, D, E & F)

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SLIDE 102

Coalescence: ALL + &

A & B C1 . . . Ck

a b

(a # b) ⇓ A & B C1 . . . Ck

c

c

# (∆) ⇒ ∆

A & B

an a1 b1 bm

(a1 # · · · # an # b1 # . . . bm) ⇓

  • (a1 # · · · # an) # (b1 # . . . bm)
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SLIDE 103

Example

A & A A ⊗ B B & B

a b1 b2 c2 d2 d1 c1

(a ⌢ (b1 # b2)) # (c1 ⌢ d1) # (c2 ⌢ d2)

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SLIDE 104

Example

A & A A ⊗ B B & B

a b c2 d2 d1 c1

(a ⌢ b) # (c1 ⌢ d1) # (c2 ⌢ d2)

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SLIDE 105

Example

A & A A ⊗ B B & B

a c2 d2 d1 c1

a # (c1 ⌢ d1) # (c2 ⌢ d2)

slide-106
SLIDE 106

Example

A & A A ⊗ B B & B

a c2 c1

a # c1 # c2

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SLIDE 107

Example

A & A A ⊗ B B & B

a c2 c1

a # (c1 # c2)

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SLIDE 108

Example

A & A A ⊗ B B & B

a c

a # c

slide-109
SLIDE 109

Example

A & A A ⊗ B B & B

a

a

slide-110
SLIDE 110

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 ✖ ✓ ∼ ∼ ∼

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SLIDE 111

Cut-elimination

Four rewrite steps:

§ axiom–cut–axiom § tensor–cut–par § with–cut–plus § conflict–cut

  • r

cut–conflict Three logical steps and one duplication step Π1 A , C Π2 B , C A & B , C Π3 C , Γ A & B , Γ ∼ Π1 A , C Π3 C , Γ A , Γ Π2 B , C Π3 C , Γ B , Γ A & B , Γ Bureaucracy or computation?

slide-112
SLIDE 112

Thank you