Uniform Atomic Ordered Linear Logic A Meta-Circular Interpreter for - - PowerPoint PPT Presentation

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Uniform Atomic Ordered Linear Logic A Meta-Circular Interpreter for - - PowerPoint PPT Presentation

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Uniform Atomic Ordered Linear Logic A Meta-Circular Interpreter for Olli Jeff Polakow Awake Security September 8, 2017 Outline Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic

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Uniform Atomic Ordered Linear Logic

A Meta-Circular Interpreter for Olli Jeff Polakow Awake Security September 8, 2017

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Outline

Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic Meta-Circular Interpreter for Olli

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Purely Ordered Logic (Lambek Calculus)

Ω ` A

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Purely Ordered Logic (Lambek Calculus)

Ω ` A A ` Ainit

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Purely Ordered Logic (Lambek Calculus)

Ω ` A A ` Ainit Ω, A ` B Ω ` A ⇣ B ⇣R ΩL, B, ΩR ` C ΩA ` A ΩL, A ⇣ B, ΩA , ΩR ` C ⇣L

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Purely Ordered Logic (Lambek Calculus)

Ω ` A A ` Ainit Ω, A ` B Ω ` A ⇣ B ⇣R ΩL, B, ΩR ` C ΩA ` A ΩL, A ⇣ B, ΩA, ΩR ` C ⇣L A, Ω ` B Ω ` A ⇢ B⇢R ΩL, B, ΩR ` C ΩA ` A ΩL, ΩA , A ⇢ B, ΩR ` C ⇢L

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Adding Linear Hypotheses

∆ ; Ω ` A

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Adding Linear Hypotheses

∆ ; Ω ` A · ; A ` A init ∆ ; ΩL, A , ΩR ` C ∆ . / A ; ΩL, ΩR ` C place . / is non-deterministic merge

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Adding Linear Hypotheses

∆ ; Ω ` A · ; A ` A init ∆ ; ΩL, A, ΩR ` C ∆ . / A ; ΩL, ΩR ` C place ∆, A ; Ω ` B ∆ ; Ω ` A ( B(R ∆ ; ΩL, B, ΩR ` C ∆A ; · ` A ∆ . / ∆A ; ΩL, A ( B, ΩR ` C (L

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Adding Linear Hypotheses

∆ ; Ω ` A · ; A ` A init ∆ ; ΩL, A, ΩR ` C ∆ . / A ; ΩL, ΩR ` C place ∆, A ; Ω ` B ∆ ; Ω ` A ( B(R ∆ ; ΩL, B, ΩR ` C ∆A ; · ` A ∆ . / ∆A ; ΩL, A ( B, ΩR ` C (L ∆ ; Ω, A ` B ∆ ; Ω ` A ⇣ B ⇣R ∆ ; ΩL, B, ΩR ` C ∆A ; ΩA ` A ∆ . / ∆A ; ΩL, A ⇣ B, ΩA, ΩR ` C ⇣L ∆ ; A, Ω ` B ∆ ; Ω ` A ⇢ B⇢R ∆ ; ΩL, B, ΩR ` C ∆A ; ΩA ` A ∆ . / ∆A ; ΩL, ΩA, A ⇢ B, ΩR ` C ⇢L

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Ordered Uniform Linear Logic Formulas

D ::= P | 8x.D | > | D & D | G ⇣ D | G ⇢ D | G ( D | G ! D G ::= P | 8x.G | 9x.G | > | G & G | | G G | 1 | G • G | G G | D ⇣ G | D ⇢ G | ¡G | D ( G | !G | D ! G

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Ordered Uniform Linear Logic Derivations

Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P focussed judgment represents Γ; ∆; ΩL, D, ΩR ` P

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Ordered Uniform Linear Logic Derivations

Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆0; Ω0 ` G0 Γ; ∆1; Ω1 ` G1 Γ; ∆0 . / ∆1; Ω0, Ω1 ` G0 • G1

  • R

Γ; ∆0; Ω0 ` G0 Γ; ∆1; Ω1 ` G1 Γ; ∆0 . / ∆1; Ω1, Ω0 ` G0 G1 R Γ; ∆; Ω, D ` G Γ; ∆; Ω ` D ⇣ G⇣R Γ; ∆; D, Ω ` G Γ; ∆; Ω ` D ⇢ G⇢R Γ; ∆, D; Ω ` G Γ; ∆; Ω ` D ( G(R Γ, D; ∆; Ω ` G Γ; ∆; Ω ` D ! G!R

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Ordered Uniform Linear Logic Derivations

Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆; ΩL, D, ΩR ` P choiceΩ Γ; ∆L, ∆R; (ΩL; ΩR) ` D P Γ; ∆L . / D, ∆R; ΩL, ΩR ` P choice∆ Γ . / D; ∆; (ΩL; ΩR) ` D P Γ . / D; ∆; ΩL, ΩR ` P choiceΓ

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Ordered Uniform Linear Logic Derivations

Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆G; ΩG ` G Γ; ∆G . / ∆; (ΩL; ΩG, ΩR) ` G ⇣ D P ⇣L Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆G; ΩG ` G Γ; ∆G . / ∆; (ΩL, ΩG; ΩR) ` G ⇢ D P ⇢L Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆G; · ` G Γ; ∆G . / ∆; (ΩL; ΩR) ` G ( D P (L Γ; ∆; (ΩL; ΩR) ` D P Γ; ·; · ` G Γ; ∆; (ΩL; ΩR) ` G ! D P !L

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Outline

Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic Meta-Circular Interpreter for Olli

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Meta-Circular Interpreter: Pure Linear Logic

Pure Linear Logic: ∆ ` G ∆ ` D P

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Meta-Circular Interpreter: Pure Linear Logic

Pure Linear Logic: ∆ ` G ∆ ` D P ∆, D ` G ∆ ` D ( G ∆ ` D P ∆ . / D ` P

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Meta-Circular Interpreter: Pure Linear Logic

Pure Linear Logic: ∆ ` G ∆ ` D P ∆, D ` G ∆ ` D ( G ∆ ` D P ∆ . / D ` P · ` P P ∆ ` D P ∆G ` G ∆ . / ∆G ` G ( D P

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . ∆ ` G ∆ ` D P goal G. focus D P.

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D =o G) o ( hyp D o goal G) . ∆, D ` G ∆ ` D ( G

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D =o G) o ( hyp D o goal G) . goal ( atm P) o hyp D, focus D P. ∆ ` D P ∆ . / D ` P

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D =o G) o ( hyp D o goal G) . goal ( atm P) o hyp D, focus D P. focus ( atm P) P. · ` P P

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D =o G) o ( hyp D o goal G) . goal ( atm P) o hyp D, focus D P. focus ( atm P) P. focus (G =o D) P o focus D P, goal G. ∆ ` D P ∆G ` G ∆ . / ∆G ` G ( D P

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . => : frm > frm > frm . bang : frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . Γ; ∆ ` G Γ; ∆ ` D P goal G. focus D P.

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . => : frm > frm > frm . bang : frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D => G) o ( hyp D > goal G) . Γ, D; ∆ ` G Γ; ∆ ` D ! G

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . => : frm > frm > frm . bang : frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D => G) o ( hyp D > goal G) . focus (G => D) P o focus D P, goal ( bang G) . Γ; ∆ ` D P Γ; · ` G Γ; ∆ ` G ! D P

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Meta-Circular Interpreter: Encoding

frm : type . atom : type . atm : atom > frm . =o : frm > frm > frm . => : frm > frm > frm . bang : frm > frm . hyp : frm > o . goal : frm > o . focus : frm > atom > o . goal (D => G) o ( hyp D > goal G) . focus (G => D) P o focus D P, goal ( bang G) . goal ( bang G) o !G. Γ; · ` G Γ; · ` !G

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Meta-Circular Interpreter: Ordered Linear Logic

Ordered Linear Logic: Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P

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Meta-Circular Interpreter: Ordered Linear Logic

Ordered Linear Logic: Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P Problem: No way to represent split ordered context.

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Meta-Circular Interpreter: Ordered Linear Logic

Ordered Linear Logic: Γ; ∆; Ω ` G Γ; ∆; (ΩL; ΩR) ` D P Problem: No way to represent split ordered context. Solution: Remove need for splitting ordered context.

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Outline

Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic Meta-Circular Interpreter for Olli

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Residuation

Logically “compile” clause into new goal. Removes need to split ordered context when focussing on non-ordered clause. Γ; ∆L, ∆R; (ΩL; ΩR) ` D P Γ; ∆L, D, ∆R; ΩL, ΩR ` P choice∆ Γ . / D; ∆; (ΩL; ΩR) ` D P Γ . / D; ∆; ΩL, ΩR ` P choiceΓ

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO G ; P P \ G GI ; D P \ GO GI ; 8x.D P \ 9x.GO

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO G ; P P \ G GI ; D P \ GO GI ; 8x.D P \ 9x.GO G ; > P \ 0 GI ; D0 P \ G0 GI ; D1 P \ G1 GI ; D0 & D1 P \ G0 G1

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO G ; P P \ G GI ; D P \ GO GI ; 8x.D P \ 9x.GO G ; > P \ 0 GI ; D0 P \ G0 GI ; D1 P \ G1 GI ; D0 & D1 P \ G0 G1 G GI ; D P \ GO GI ; G ⇣ D P \ GO G • GI ; D P \ GO GI ; G ⇢ D P \ GO

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO G ; P P \ G GI ; D P \ GO GI ; 8x.D P \ 9x.GO G ; > P \ 0 GI ; D0 P \ G0 GI ; D1 P \ G1 GI ; D0 & D1 P \ G0 G1 G GI ; D P \ GO GI ; G ⇣ D P \ GO G • GI ; D P \ GO GI ; G ⇢ D P \ GO ¡G • GI ; D P \ GO GI ; G ( D P \ GO !G • GI ; D P \ GO GI ; G ! D P \ GO

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Residuation

Logically “compile” clause into new goal. GI ; D P \ GO New choice rules: 1 ; D P \ G Γ; ∆; Ω ` G Γ; ∆ . / D; Ω ` P choice∆ 1 ; D P \ G Γ . / D; ∆; Ω ` G Γ . / D; ∆; Ω ` P choiceΓ No split ordered contexts.

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Remove Ordered Choice

Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆; ΩL, D, ΩR ` P choiceΩ

I We cannot residuate away ordered choice context split.

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Remove Ordered Choice

Γ; ∆; (ΩL; ΩR) ` D P Γ; ∆; ΩL, D, ΩR ` P choiceΩ

I We cannot residuate away ordered choice context split. I So let’s just remove ordered choice entirely.

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Use Placeholders

· ; · ; P2, P1 ⇣ P2 ⇢ P , P1 ` P

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Use Placeholders

· ; · ; P2, P1 ⇣ P2 ⇢ P , P1 ` P can be transformed to · ; QP ⇣ P1 ⇣ P2 ⇢ P ; P2, QP , P1 ` P

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Use Placeholders

Ξ ·; ·; P2 , QP, P1 ` P2 • P1 QP 1 choice∆ ·; QP ⇣ P1 ⇣ P2 ⇢ P; P2, QP, P1 ` P Ξ = 1 ; QP ⇣ P1 ⇣ P2 ⇢ P P \ P2 • P1 QP 1

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Use Placeholders

Ξ ·; ·; P2 ` P2 ·; ·; QP, P1 ` P1 QP 1

  • R

·; ·; P2, QP, P1 ` P2 • P1 QP 1 choice∆ ·; QP ⇣ P1 ⇣ P2 ⇢ P; P2, QP, P1 ` P Ξ = 1 ; QP ⇣ P1 ⇣ P2 ⇢ P P \ P2 • P1 QP 1

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Use Placeholders

Ξ ·; ·; P2 ` P2 ·; ·; P1 ` P1 ·; ·; QP ` QP 1 R ·; ·; QP, P1 ` P1 QP 1 •R ·; ·; P2, QP, P1 ` P2 • P1 QP 1 choice∆ ·; QP ⇣ P1 ⇣ P2 ⇢ P; P2, QP, P1 ` P Ξ = 1 ; QP ⇣ P1 ⇣ P2 ⇢ P P \ P2 • P1 QP 1

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Outline

Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic Meta-Circular Interpreter for Olli

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Uniform Atomic Ordered Linear Logic Syntax

I Distinguished placeholder predicate: QX (x is a term).

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Uniform Atomic Ordered Linear Logic Syntax

I Distinguished placeholder predicate: QX (x is a term). I Extend goal formulae with placeholders:

G ::= Qx | P | . . .

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Uniform Atomic Ordered Linear Logic Syntax

I Distinguished placeholder predicate: QX (x is a term). I Extend goal formulae with placeholders:

G ::= Qx | P | . . .

I New kind of (modified) clause formulae:

E ::= D | Qx ⇣ D

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Uniform Atomic Ordered Linear Logic Syntax

I Distinguished placeholder predicate: QX (x is a term). I Extend goal formulae with placeholders:

G ::= Qx | P | . . .

I New kind of (modified) clause formulae:

E ::= D | Qx ⇣ D

I Demoted ordered context:

! ::= · | !, Qx where x not in !

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Uniform Atomic Ordered Linear Logic Syntax

I Distinguished placeholder predicate: QX (x is a term). I Extend goal formulae with placeholders:

G ::= Qx | P | . . .

I New kind of (modified) clause formulae:

E ::= D | Qx ⇣ D

I Demoted ordered context:

! ::= · | !, Qx where x not in !

I Modified linear context:

  • ::=

· | , E

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G Γ; , Qx ⇣ D ; !, Qx ` G Γ; ; ! ` D ⇣ G ⇣R0 (x new)

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G Γ; , Qx ⇣ D; !, Qx ` G Γ; ; ! ` D ⇣ G ⇣R0 (x new) Γ; , Qx ⇣ D; Qx, ! ` G Γ; ; ! ` D ⇢ G ⇢R0 (x new)

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G Γ; , Qx ⇣ D; !, Qx ` G Γ; ; ! ` D ⇣ G ⇣R0 (x new) Γ; , Qx ⇣ D; Qx, ! ` G Γ; ; ! ` D ⇢ G ⇢R0 (x new) Γ; ·; Qx ` Qx choiceω

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G Γ; , Qx ⇣ D; !, Qx ` G Γ; ; ! ` D ⇣ G ⇣R0 (x new) Γ; , Qx ⇣ D; Qx, ! ` G Γ; ; ! ` D ⇢ G ⇢R0 (x new) Γ; ·; Qx ` Qx choiceω 1 ; E P \ G Γ; ; ! ` G Γ; . / E; ! ` P choiceδ

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Uniform Atomic Ordered Linear Logic Derivations

Γ; ; ! ` G Γ; , Qx ⇣ D; !, Qx ` G Γ; ; ! ` D ⇣ G ⇣R0 (x new) Γ; , Qx ⇣ D; Qx, ! ` G Γ; ; ! ` D ⇢ G ⇢R0 (x new) Γ; ·; Qx ` Qx choiceω 1 ; E P \ G Γ; ; ! ` G Γ; . / E; ! ` P choiceδ 1 ; E P \ G Γ . / E; ; ! ` G Γ . / E; ; ! ` P choiceΓ

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Outline

Ordered Linear Logic Meta-Circular Interpreters Unsplitting Ordered Contexts Uniform Atomic Ordered Linear Logic Meta-Circular Interpreter for Olli

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OLL syntax in Olli

trm : type. frm : type. atom : type atm : atom -> frm. place : trm -> atm.

  • ne : frm.

# : frm -> frm -> frm. zero : frm. & : frm -> frm -> frm. top : frm. forall : (trm -> frm) -> frm. exists : (trm -> frm) -> frm.

  • >> : frm -> frm -> frm.

>-> : frm -> frm -> frm.

  • -o : frm -> frm -> frm.
  • -> : frm -> frm -> frm.

* : frm -> frm -> frm. <> : frm -> frm -> frm. gnab : frm -> frm. bang : frm -> frm. # ⌘ * ⌘ • <> ⌘

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1. resid Gi (forall D) P (exists Go) ⌘ 8y . resid Gi (D y) P (Go y).

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1. resid Gi (forall D) P (exists Go) ⌘ 8y . resid Gi (D y) P (Go y). resid Gi (G ->> D) P Go ⌘ resid (G <> Gi) D P Go.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1. resid Gi (forall D) P (exists Go) ⌘ 8y . resid Gi (D y) P (Go y). resid Gi (G ->> D) P Go ⌘ resid (G <> Gi) D P Go. resid Gi (G >-> D) P Go ⌘ resid (G * Gi) D P Go.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1. resid Gi (forall D) P (exists Go) ⌘ 8y . resid Gi (D y) P (Go y). resid Gi (G ->> D) P Go ⌘ resid (G <> Gi) D P Go. resid Gi (G >-> D) P Go ⌘ resid (G * Gi) D P Go. resid Gi (G --o D) P Go ⌘ resid (gnab G * Gi) D P Go.

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Encoding of Residuation

resid : frm -> frm -> atm -> frm -> o. resid G (atm P) P G. resid G top P zero. resid G (D0 & D1) P (G0 # G1) ⌘ resid G D0 P G0 • resid G D1 P G1. resid Gi (forall D) P (exists Go) ⌘ 8y . resid Gi (D y) P (Go y). resid Gi (G ->> D) P Go ⌘ resid (G <> Gi) D P Go. resid Gi (G >-> D) P Go ⌘ resid (G * Gi) D P Go. resid Gi (G --o D) P Go ⌘ resid (gnab G * Gi) D P Go. resid Gi (G --> D) P Go ⌘ resid (bang G * Gi) D P Go.

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Encoding of Derivations

hyp : frm -> o. goal : frm -> o.

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Encoding of Derivations

hyp : frm -> o. goal : frm -> o. goal top ⌘ >. goal (G0 & G1) ⌘ goal G0 & goal G1. goal (G0 # G1) ⌘ goal G0 goal G1.

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SLIDE 71

Encoding of Derivations

hyp : frm -> o. goal : frm -> o. goal top ⌘ >. goal (G0 & G1) ⌘ goal G0 & goal G1. goal (G0 # G1) ⌘ goal G0 goal G1. goal (forall G) ⌘ 8x . goal (G x). goal (exists G) ⌘ goal (G X).

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Encoding of Derivations

hyp : frm -> o. goal : frm -> o. goal top ⌘ >. goal (G0 & G1) ⌘ goal G0 & goal G1. goal (G0 # G1) ⌘ goal G0 goal G1. goal (forall G) ⌘ 8x . goal (G x). goal (exists G) ⌘ goal (G X). goal (gnab G) ⌘ ¡ (goal G). goal (bang G) ⌘ ! (goal G).

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Encoding of Derivations

hyp : frm -> o. goal : frm -> o. goal top ⌘ >. goal (G0 & G1) ⌘ goal G0 & goal G1. goal (G0 # G1) ⌘ goal G0 goal G1. goal (forall G) ⌘ 8x . goal (G x). goal (exists G) ⌘ goal (G X). goal (gnab G) ⌘ ¡ (goal G). goal (bang G) ⌘ ! (goal G). goal one ⌘ 1. goal (G * H) ⌘ goal G • goal H. goal (G <> H) ⌘ goal G goal H.

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Encoding of Derivations Continued

goal (D ->> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇣ goal G.

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Encoding of Derivations Continued

goal (D ->> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇣ goal G. goal (D >-> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇢ goal G.

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Encoding of Derivations Continued

goal (D ->> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇣ goal G. goal (D >-> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇢ goal G. goal (D --o G) ⌘ hyp D ( goal G. goal (D --> G) ⌘ hyp D ! goal G.

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SLIDE 77

Encoding of Derivations Continued

goal (D ->> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇣ goal G. goal (D >-> G) ⌘ 8x . hyp (atm (place x) ->> D) ( hyp (atm (place x)) ⇢ goal G. goal (D --o G) ⌘ hyp D ( goal G. goal (D --> G) ⌘ hyp D ! goal G. goal (atm P) ⌘ hyp D • resid one D P G • goal G.