Characterisation of Parallel Independence in AGREE-Rewriting Michael - PowerPoint PPT Presentation

AGREE: Local Copies (t, l) ● T L ● x x 𝜃 t l r L K n 17 ICGT 2018

AGREE: Local Copies (t, l) ● T L ● x x 𝜃 t l r L K m ● n m G x x 17 ICGT 2018

AGREE: Local Copies (t, l) ● T L ● x x 𝜃 t l r L K m ● n m n’ G D g x x x x 17 ICGT 2018

AGREE: Global Copies (t, l) ● T L ● x x x 𝜃 t l r L K n 18 ICGT 2018

AGREE: Global Copies (t, l) ● T L ● x x x 𝜃 t l r L K m ● n m n’ G D g x x x x x x 18 ICGT 2018

AGREE: Local Deletion (t, l) ● T L ● x x 𝜃 t l r L K n 19 ICGT 2018

AGREE: Local Deletion (t, l) ● T L ● x x 𝜃 t l r L K m ● n m n’ G D g x x x x 19 ICGT 2018

AGREE: Global Deletion (t, l) ● T L ● x 𝜃 t l r L K n 20 ICGT 2018

AGREE: Global Deletion (t, l) ● T L ● x 𝜃 t l r L K m ● n m n’ G D g x x 20 ICGT 2018

AGREE: Local Addition (t, l) ● T L ● x x 𝜃 t l r L K R 21 ICGT 2018

AGREE: Local Addition (t, l) ● T L ● x x 𝜃 t l r L K R m ● n m n’ p’ G D D g x x x x x x 21 ICGT 2018

Gluing Construction l r L K R m X n G 22 ICGT 2018

Gluing Construction (r, l) ◦ (q, p) = (n, m) ◦ (h, g) l r L K R (PB) (FPC) m u p x y X Z Y n v q (FPC) (PO) G D H g h 22 ICGT 2018

Gluing for DPO-Rewriting l r L K R (PB) (FPC) m u p x y X Z Y n v q (FPC) (PO) G D H g h 23 ICGT 2018

Gluing for DPO-Rewriting Monomorphism l r L K R (PB) (FPC) m u p Isomorphism x y X Z Y n v q Pushout (FPC) (PO) Complement G D H g h 23 ICGT 2018

Gluing for SPO-Rewriting Monomorphism l r L K R (PB) (FPC) m u p Isomorphism x y X Z Y n v q (FPC) (PO) G D H g h 24 ICGT 2018

Gluing for SqPO-Rewriting l r L K R (PB) (FPC) m u p Isomorphism x y X Z Y n v q (FPC) (PO) G D H g h 25 ICGT 2018

Gluing for AGREE-Rewriting l r L K R (PB) (FPC) m u p x y X Z Y n v q (FPC) (PO) G D H g h 26 ICGT 2018

Gluing for AGREE-Rewriting Induced by base rule Monomorphism l r L K R Induced by monic (PB) (FPC) m u p base match x y X Z Y n v q Special (FPC) (PO) monomorphism G D H g h 26 ICGT 2018

Gluing Construction l r L K R (PB) (FPC) m u p x y X Z Y n v q (FPC) (PO) G D H g h 27 ICGT 2018

Gluing Construction l r L K R (PB) (FPC) m u p x y X Z Y n v q (FPC) (PO) G D H g h 27 ICGT 2018

Gluing Construction L K R X Z Y G D H 28 ICGT 2018

Gluing Construction K’’ L K R K’ R’ Z’’ X Z Y Y D’’ G D H H’ 28 ICGT 2018

Gluing Construction K’’ L K R K’ R’ Z’’ X Z Y Z’ Y D’’ G D H D’ H’ 28 ICGT 2018

Gluing Construction K’’ L K R K’ R’ Z’’ X Z Y Z’ Y D’’ G D H D’ H’ Gluing diagrams compose and decompose like pushouts 28 ICGT 2018

Parallel Independence (l, r) (m, n) (p, q) (m’, n’) (g, h) (l’, r’) (g’, h’) (p’, q’) 29 ICGT 2018

Parallel Independence (l, r) Residual? (g’, h’) ◦ (m, n) (m, n) Residual? (p, q) (m’, n’) (g, h) (g, h) ◦ (m’, n’) (l’, r’) (g’, h’) (p’, q’) 29 ICGT 2018

Parallel Independence (l, r) Residual? (g’, h’) ◦ (m, n) (m, n) Residual? (p, q) (m’, n’) (g, h) (g, h) ◦ (m’, n’) (l’, r’) (g’, h’) (p’, q’) 29 ICGT 2018

Parallel Independence (l, r) Residual? (g’, h’) ◦ (m, n) (m, n) Residual? (p, q) (m’, n’) (g, h) (g, h) ◦ (m’, n’) (l’, r’) (g’, h’) (p’, q’) 29 ICGT 2018

Residual g h G D H 30 ICGT 2018

Residual L ● m gh ● m ● g h G D H 𝜃 L m m gh L 30 ICGT 2018

Residual L ● m gh ● m ● g h G D H 𝜃 L m m gh m’ g ◦ m’ = m, h ◦ m’ = m gh L 30 ICGT 2018

Residual m ● ◦ g = m’ ● , m gh ● ◦ h = m’ ● L ● m gh ● m’ ● m ● g h G D H 𝜃 L m m gh m’ g ◦ m’ = m, h ◦ m’ = m gh L 30 ICGT 2018

Residual L ● m gh ● m’ ● m ● g h G D H 𝜃 L m m gh m’ (m’, id L ) pullback of (m, g) and (h, m gh ) L 30 ICGT 2018

Residual L ● x m gh ● m ● G D H g h 𝜃 L m L m gh 31 ICGT 2018

Residual L ● x m gh ● m ● G D H g h 𝜃 L m L m’ m gh g ◦ m’ = m, h ◦ m’ = m gh 31 ICGT 2018

Residual m ● ◦ g ≠ m’ ● , m gh ● ◦ h ≠ m’ ● L ● x m gh ● m’ ● m ● G D H g h x 𝜃 L m L m’ m gh g ◦ m’ = m, h ◦ m’ = m gh 31 ICGT 2018

Residual L ● m gh ● m’ ● m ● g h G D H 𝜃 L m m gh m’ L (m’, id L ) pullback of (m, g) and (h, m gh ) 32 ICGT 2018

Residual L ● m’ ● m ● g h G D H 𝜃 L m m’ L (m’, id L ) pullback of (m, g) and h ◦ m’ = m gh 32 ICGT 2018

Residual g h G D H 33 ICGT 2018

Residual L ● (h ◦ m’) ● m’ ● m ● g 𝜃 L h L G D H m (t, l) ● m’ l K t T 33 ICGT 2018

Residual L ● (h ◦ m’) ● m’ ● m ● g 𝜃 L h L G D H m (t, l) ● m’ l x v (PB) K X Y (PB) t h’ y T w 33 ICGT 2018

Residual L ● (h ◦ m’) ● m’ ● m ● g 𝜃 L h L G D H m (t, l) ● m’ (FPC) l x v (PB) K X Y (PB) t h’ y T w 33 ICGT 2018

Characterising Independence L 1 m 1 L 2 ● L 2 G m 2 𝜃 2 (t 2 , l 2 ) ● l 2 T 2 K 2 t 2 34 ICGT 2018

Characterising Independence Match m 1 for rule 1 has residual after applying rule 2 L 1 at m 2 , only if m 1 1. everything that m 1 needs (locally copies, deletes, or preserves) is neither copied nor deleted (neither L 2 ● L 2 G m 2 𝜃 2 locally nor globally) by rule 2 at match m 2 . (t 2 , l 2 ) ● l 2 T 2 K 2 t 2 34 ICGT 2018

Characterising Independence L 1 m 1 L 2 ● L 2 G m 2 𝜃 2 (t 2 , l 2 ) ● l 2 T 2 K 2 t 2 35 ICGT 2018

Characterising Independence L 1 𝜌 2 L* m 1 𝜌 1 ( 𝜌 2 , 𝜌 1 ) ● L 2 ● L 2 G m 2 𝜃 2 (t 2 , l 2 ) ● l 2 T 2 K 2 t 2 35 ICGT 2018

Characterising Independence L 1 𝜌 2 L* m 1 𝜌 1 ( 𝜌 2 , 𝜌 1 ) ● L 2 ● L 2 G m 2 𝜃 2 (t 2 , l 2 ) ● l 2 id LI (PB) T 2 K 2 t 2 L 1 35 ICGT 2018

Characterising Independence Match m 1 for rule 1 has residual after applying rule 2 L 1 at m 2 , only if 𝜌 2 L* m 1 1. everything that m 1 needs (locally copies, deletes, or 𝜌 1 ( 𝜌 2 , 𝜌 1 ) ● preserves) is neither copied nor deleted (neither L 2 ● L 2 G m 2 𝜃 2 locally nor globally) by rule 2 at match m 2 . (t 2 , l 2 ) ● l 2 2. everything that rule 1 adds is neither (globally) id LI (PB) T 2 K 2 copied nor deleted by rule 2 at match m 2 . t 2 L 1 35 ICGT 2018

Characterising Independence Match m 1 for rule 1 has residual after applying rule 2 add L 1 at m 2 , only if 𝜌 2 L* m 1 1. everything that m 1 needs (locally copies, deletes, or copy (globally) copy (globally) 𝜌 1 ( 𝜌 2 , 𝜌 1 ) ● preserves) is neither copied nor deleted (neither add L 2 ● ≠ L 2 G m 2 𝜃 2 locally nor globally) by rule 2 at match m 2 . (t 2 , l 2 ) ● l 2 2. everything that rule 1 adds is neither (globally) id LI (PB) T 2 K 2 copied nor deleted by rule 2 at match m 2 . t 2 L 1 35 ICGT 2018

Characterising Independence L 1 𝜌 2 L* m 1 𝜌 1 L 2 ● L 2 G m 2 𝜃 2 (t 2 , l 2 ) ● l 2 T 2 K 2 t 2 36 ICGT 2018

Characterising Independence l 1 r 1 r 1 L 1 K 1 R 1 𝜌 2 L* m’’ 2 n’ 1 m 1 𝜌 1 g 1 L 2 ● L 2 G D 1 m 2 𝜃 2 (t 2 , l 2 ) ● l 2 m’ 2 T 2 K 2 t 2 36 ICGT 2018

Characterising Independence (r 1 ◦ m’’ 2 , 𝜌 1 ) ● l 1 r 1 r 1 L 1 K 1 R 1 𝜌 2 L* m’’ 2 n’ 1 m 1 𝜌 1 g 1 L 2 ● L 2 G D 1 m 2 𝜃 2 (t 2 , l 2 ) ● l 2 m’ 2 T 2 K 2 t 2 36 ICGT 2018

Characterising Independence (r 1 ◦ m’’ 2 , 𝜌 1 ) ● l 1 r 1 r 1 L 1 K 1 R 1 𝜌 2 L* m’’ 2 n’ 1 m 1 𝜌 1 g 1 L 2 ● L 2 G D 1 m 2 𝜃 2 id RI (t 2 , l 2 ) ● l 2 m’ 2 T 2 K 2 t 2 (PB) R 1 36 ICGT 2018

Characterising Independence (r 1 ◦ m’’ 2 , 𝜌 1 ) ● Match m 1 for rule 1 has residual after applying rule 2 l 1 r 1 r 1 L 1 K 1 R 1 at m 2 , if and only if 𝜌 2 L* m’’ 2 n’ 1 m 1 1. everything that m 1 needs (locally copies, deletes, or 𝜌 1 g 1 preserves) is neither copied nor deleted (neither L 2 ● L 2 G D 1 m 2 𝜃 2 locally nor globally) by rule 2 at match m 2 . id RI (t 2 , l 2 ) ● l 2 m’ 2 2. everything that rule 1 adds is neither (globally) T 2 K 2 copied nor deleted by rule 2 at match m 2 . t 2 (PB) R 1 36 ICGT 2018

Characterising Independence (r 1 ◦ m’’ 2 , 𝜌 1 ) ● l 1 r 1 L 1 K 1 R 1 𝜌 2 L* m’’ 2 n’ 1 m 1 𝜌 1 ( 𝜌 2 , 𝜌 1 ) ● g 1 L 2 ● L 2 G D 1 m 2 𝜃 2 id RI (t 2 , l 2 ) ● l 2 m’ 2 id LI (PB) T 2 K 2 t 2 (PB) L 1 R 1 37 ICGT 2018

Conclusion AGREE-rewriting is instance of the Gluing Construction! There is a precise notion of residual! Gluing and mutual residuals provides Church-Rosser! Residuals can be characterized syntactically! ——————————————————————— Are global effects useful? 38 ICGT 2018