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

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 Meta-Circular Interpreter for Olli

Purely Ordered Logic (Lambek Calculus) Ω ` A

Purely Ordered Logic (Lambek Calculus) Ω ` A A ` Ainit

Purely Ordered Logic (Lambek Calculus) Ω ` A A ` Ainit Ω L , B , Ω R ` C Ω A ` A Ω , A ` B Ω ` A ⇣ B ⇣ R ⇣ L Ω L , A ⇣ B , Ω A , Ω R ` C

Purely Ordered Logic (Lambek Calculus) Ω ` A A ` Ainit Ω , A ` B Ω L , B , Ω R ` C Ω A ` A Ω ` A ⇣ B ⇣ R ⇣ L Ω L , A ⇣ B , Ω A , Ω R ` C Ω L , B , Ω R ` C Ω A ` A A , Ω ` B Ω ` A ⇢ B ⇢ R ⇢ L Ω L , Ω A , A ⇢ B , Ω R ` C

Adding Linear Hypotheses ∆ ; Ω ` A

Adding Linear Hypotheses ∆ ; Ω ` A ∆ ; Ω L , A , Ω R ` C · ; A ` A init place ∆ . / A ; Ω L , Ω R ` C . / is non-deterministic merge

Adding Linear Hypotheses ∆ ; Ω ` A ∆ ; Ω L , A , Ω R ` C · ; A ` A init / A ; Ω L , Ω R ` C place ∆ . ∆ , A ; Ω ` B ∆ ; Ω L , B , Ω R ` C ∆ A ; · ` A ∆ ; Ω ` A ( B ( R ( L ∆ . / ∆ A ; Ω L , A ( B , Ω R ` C

Adding Linear Hypotheses ∆ ; Ω ` A ∆ ; Ω L , A , Ω R ` C · ; A ` A init / A ; Ω L , Ω R ` C place ∆ . ∆ , A ; Ω ` B ∆ ; Ω L , B , Ω R ` C ∆ A ; · ` A ∆ ; Ω ` A ( B ( R ( L / ∆ A ; Ω L , A ( B , Ω R ` C ∆ . ∆ ; Ω , A ` B ∆ ; Ω L , B , Ω R ` C ∆ A ; Ω A ` A ∆ ; Ω ` A ⇣ B ⇣ R / ∆ A ; Ω L , A ⇣ B , Ω A , Ω R ` C ⇣ L ∆ . ∆ ; A , Ω ` B ∆ ; Ω L , B , Ω R ` C ∆ A ; Ω A ` A ∆ ; Ω ` A ⇢ B ⇢ R / ∆ A ; Ω L , Ω A , A ⇢ B , Ω R ` C ⇢ L ∆ .

Ordered Uniform Linear Logic Formulas D ::= P | 8 x . D | > | D & D | G ⇣ D | G ⇢ D | G ( D | G ! D G ::= P | 8 x . G | 9 x . G | | G & G > | | G � G 0 | 1 | G • G | G � G | D ⇣ G | D ⇢ G | ¡ G | D ( G | ! G | D ! G

Ordered Uniform Linear Logic Derivations Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P focussed judgment represents Γ ; ∆ ; Ω L , D , Ω R ` P

Ordered Uniform Linear Logic Derivations Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ 0 ; Ω 0 ` G 0 Γ ; ∆ 1 ; Ω 1 ` G 1 • R Γ ; ∆ 0 . / ∆ 1 ; Ω 0 , Ω 1 ` G 0 • G 1 Γ ; ∆ 0 ; Ω 0 ` G 0 Γ ; ∆ 1 ; Ω 1 ` G 1 � R Γ ; ∆ 0 . / ∆ 1 ; Ω 1 , Ω 0 ` G 0 � G 1 Γ ; ∆ ; Ω , D ` G Γ ; ∆ ; D , Ω ` G Γ ; ∆ ; Ω ` D ⇣ G ⇣ R Γ ; ∆ ; Ω ` D ⇢ G ⇢ R Γ ; ∆ , D ; Ω ` G Γ , D ; ∆ ; Ω ` G Γ ; ∆ ; Ω ` D ! G ! R Γ ; ∆ ; Ω ` D ( G ( R

Ordered Uniform Linear Logic Derivations Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P choice Ω Γ ; ∆ ; Ω L , D , Ω R ` P Γ ; ∆ L , ∆ R ; ( Ω L ; Ω R ) ` D � P / D , ∆ R ; Ω L , Ω R ` P choice ∆ Γ ; ∆ L . / D ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ . choice Γ Γ . / D ; ∆ ; Ω L , Ω R ` P

Ordered Uniform Linear Logic Derivations Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ G ; Ω G ` G ⇣ L Γ ; ∆ G . / ∆ ; ( Ω L ; Ω G , Ω R ) ` G ⇣ D � P Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ G ; Ω G ` G / ∆ ; ( Ω L , Ω G ; Ω R ) ` G ⇢ D � P ⇢ L Γ ; ∆ G . Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; ∆ G ; · ` G ( L Γ ; ∆ G . / ∆ ; ( Ω L ; Ω R ) ` G ( D � P Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Γ ; · ; · ` G ! L Γ ; ∆ ; ( Ω L ; Ω R ) ` G ! D � P

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

Meta-Circular Interpreter: Pure Linear Logic Pure Linear Logic: ∆ ` G ∆ ` D � P

Meta-Circular Interpreter: Pure Linear Logic Pure Linear Logic: ∆ ` G ∆ ` D � P ∆ , D ` G ∆ ` D � P ∆ ` D ( G ∆ . / D ` P

Meta-Circular Interpreter: Pure Linear Logic Pure Linear Logic: ∆ ` G ∆ ` D � P ∆ , D ` G ∆ ` D � P ∆ ` D ( G ∆ . / D ` P ∆ ` D � P ∆ G ` G · ` P � P ∆ . / ∆ G ` G ( D � P

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.

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

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

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

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

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.

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

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

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

Meta-Circular Interpreter: Ordered Linear Logic Ordered Linear Logic: Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P

Meta-Circular Interpreter: Ordered Linear Logic Ordered Linear Logic: Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Problem: No way to represent split ordered context.

Meta-Circular Interpreter: Ordered Linear Logic Ordered Linear Logic: Γ ; ∆ ; Ω ` G Γ ; ∆ ; ( Ω L ; Ω R ) ` D � P Problem: No way to represent split ordered context. Solution: Remove need for splitting ordered context.

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

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

Residuation Logically “compile” clause into new goal. G I ; D � P \ G O

Residuation Logically “compile” clause into new goal. G I ; D � P \ G O G I ; D � P \ G O G ; P � P \ G G I ; 8 x . D � P \ 9 x . G O

Residuation Logically “compile” clause into new goal. G I ; D � P \ G O G I ; D � P \ G O G ; P � P \ G G I ; 8 x . D � P \ 9 x . G O G I ; D 0 � P \ G 0 G I ; D 1 � P \ G 1 G ; > � P \ 0 G I ; D 0 & D 1 � P \ G 0 � G 1