[PDF] - map rev L X l X PDF Document, free download

SLIDE 1 ✂✁ ✄ ☎ ✆ ✝ ✞ ✟✡✠ ☛ ☞ ☛ ✌ ✍✏✎ ✑✓✒✔✒✖✕ ✑ ∗ ✗ ☎ ✍✘✎ ✍✚✙✜✛✣✢✥✤ ✗✧✦ ✍ ☞ ★✪✩✫✩✜✬ ∗

(*)DISI - Univ. of Genova, Italy,

✭✯✮✰✮✲✱✴✳✶✵✷✵✹✸✰✸✷✸✻✺✽✼✿✾❁❀❂✾❃✺❅❄✷❆❇✾❉❈❋❊✫✺●✾❍✮

(+)SCS - Univ. of Tech. Sydney, Australia,

✭✯✮✰✮✰✱✻✳✶✵✷✵✹✸✷✸✰✸✻✺■❀✲❏▲❑✲❀▼✺◆❄✷✮❋❀✪✺❖❊❁✼❁❄✴✺❖P✹❄

◗

★❙❘❚✬❱❯ ✛ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ◗ ❩ ✒✶✑▼❬ map rev ❬ L ❭ X → l ❭ X ❪❴❫❛❵❝❜❡❞❣❢✷❜✐❤✹❥❋❜❧❦♥♠✷♦♣♦rq s ❢❂t✉❤✈❦♣❥❋❜●❤✡t✇❤②①✇❜❧❦♣❥❝③ ❵④①⑤❤②①❉⑥

⑦

✑❛✒■✛ ❘ ✑❙⑧ ⑨ ★ ⑦✰⑩ ❬✴❶❸❷❡❹✻❺④❺❼❻❋❽❛❾✹❿➁➀❨❻❋❽❚➂✔❾❋❶➄➃ ❹✴➀❨❻❛➅❂❾➆➃ ➇➈❻❋➉✰➂✔❾❂❶➋➊➌❹➍❻❋➎✷➂✔❾ ❶➄➏ ❢❂❥ ❻❋➉➐❾✓➏ ❤➆❵ ❻❋➉✰➂✔❾❙❶➄➏ ♠❚q ❻❋➉ ♠ ➂

⑨

➑✻✛ ❘ ✬●➒ ➓☞ ☛ ❬❛➃ ➔ →

. . .

➣↕↔✇➙ ❲➜➛✿❘✲★ ⑦ ➒ ❾ ❜➝q➐❞➞❤②① ❾ ①✖➟➡➠❝❤✹s ♠✰① ❾ ❩ ★ ✒❱✢ ◗ ★ ⑦▲❩ ➑ ✬●➒ ◗ ➢ ❜●❤✡t✇s ① ❾ ❜➝q➐❞➞❤➓♠✰①➡①⑤❦➤③❋❥④s➥❤✹❥❋❜ ❾ t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥

⑨

➑✻✛ ❘ ✛ ⑦ ✑ ➧☞ ☛ ↔✇➙ ❲➨➛➩❘✲★ ⑦ ➒ ❬ ➢ ➫ ❵➐❥✓➟⑤❜✉❢❂t✇① ➢ ➒ ➑✻✛ ❩ ✑✓✒✽✢ ↔✇➙ ❲➨➛➩❘✲★ ⑦ ➒ ❶➄➏ ♠❚q ❻❋➉ ❫ ➂ ➢ t✇❤✹❞❛t✉❤②①⑤❤✹❥❋❜➭♠✹❜❧❦♣❢❂❥④①❡❢ ➫ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①

❩➜⑦

★ ❩ ✑ ⑦ ❘✲✬ ✑ ➒ ★ ↔ ➧☞ ☛ ❪✶♠✰①✚❤❍➯✓❞➞❤➆➟⑤❜✐❤✲➦➲⑥ ❶❸❷➳➏❋➃ ❻❂➵✯➂✔❬ ➢ ❜➸q➐❞➲❤✈❦♣❥ ➫ ❤✡t✇❤✹❥✓➟➺❤➻♠✷♦➤③❋❢❂t➡❦r❜➸➠❛s ➢ ①➭❵➼❫❼➽●❤➆➟❉❜➾t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥ ❾➐➊ ➠➼❵❛t✉➟❍➠❝➚ ❿ ❢➪①➡①❉❤➆t➶❞❛t✉❢✲❞➞❤➆t✐❜➝q ➢ ①❧❜➡t✉❢❂❥❝③ ❥❛❢❂t✇s ♠✷♦♣❦➤➹❚♠✹❜❧❦♣❢❂❥ ❪❴❥❛❢➴➘➼➯❃⑥

1

SLIDE 2 ✠ ◗ ★❙❘❚✬❱❯ ✛ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ◗ ❩ ✒✶✑

map

❬ ∀F ❬②➅ .∀X, Y. ❪ X → Y ⑥ → FX → FY ➢ ➦④♠✹❜➭♠➴❞❣❢✯♦rq➐s ❢❂t✇❞❛➠❛❦➷①✖s ❬ F ➘➼➯✓❤✲➦ ❾ X, Y ➬ ♠✷t●q ➣➻①➭➠➐♠✷❞➲❤✈❞❣❢✯♦rq➐s ❢❂t✉❞❛➠❛❦➷①✖s ❬ F ➬ ♠✷t❍❦♣❤②① ❾ X, Y ➘➼➯✓❤✲➦

L

❬❁➅ ❜➸➠❝❤✈♦♣❦➮①❧❜ ➫ ❵❼❥✓➟⑤❜✉❢❂t

rev

❬ ∀X.LX → LX t✉❤ ➬ ❤✡t❧①⑤❦♣❥❝③ ♠↕♦♣❦➮①✇❜ ➱ ➑✻✛ ❘ ➛ ✛ ❲ ✃ ✑ ✬✔❲ ↔ ✑ ⑦✯⑦ ✑❙⑧✏❐

➫

❵④♦♣♦rq ❤➡➯✓❞➐♦♣❦♣➟➺❦r❜

map L

❒ LX LX ❪ rev X ⑥ ❬ L ❭ X → L ❭ X

map L L

❒ X L ❒ X ❪ rev LX ⑥ ❬ L ❭ X → L ❭ X

❜➝q➐❞➞❤❮❦♣s

❞➐♦♣❦❰➟✹❦r❜

map L

❒ rev ❬ L ❭ X → L ❭ X

map L rev

❬ L ❭ X → L ❭ X ➣ ➫ ❵④♦♣♦rq ❦♣s ❞❼♦♣❦♣➟➺❦r❜

map rev

❬ L ❭ X → L ❭ X ✠ ❩✻❩➨⑦ ★ ✛ ➛ ➑➍✑ ➒

➊

➠➐♠✷t➡❦r❜➝q ❬ F s↕❵④①✇❜➳❫➞❤✂❤❍➯✓❞❼♦♣❦♣➟➺❦r❜

➇

♠✰①✖Ï➪❤✹♦♣♦➞❜➸q➐❞➲❤❮➟✹♦♥♠✰①➡①⑤❤②① ❬ ➟✹❢✿➦❇❤ ➫ ❢❂t map ①✖❵❛❞❝❞❼♦♣❦♣❤✲➦ ❫➼q ❵➐①❉❤➆t

❹✥➃

➔✥❬ F ❪ GX ⑥ Ð ❪ F ◦ G ⑥➺❪ X ⑥ ❾ ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ

2

SLIDE 3 Ò ✑❇✢ ✬ ⑧✻✑➲✛

F

❪ GX ⑥ Ð ❪ F ◦ G ⑥➺❪ X ⑥ ❾ ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ

F

❪ GX ⑥ ❬ F ➫ ❵❼❥✓➟⑤❜✉❢❂t ❾ GX ➦④♠✹❜➭♠

FG

❪ X ⑥ ❬ ❪ F ◦ G ⑥ ➫ ❵➐❥✓➟⑤❜✉❢❂t ❾ X ➦④♠✹❜➭♠

Ó②Ô➸Õ✹Ö❝×✐Ø●Ù✓Ú❋Û❁Ü

❶❸❷ ❤✹❥ ➵➼Ý❚➂ ❢ ➫④➫ ❵➐❥❛➟❉❜✉❢❂t❧① ❬ ➟➺❢❂s ❞❃❢➪①❍❦r❜❧❦♣❢❂❥➥❵❛❞Þ❜➭❢ß❦➮①✖❢à❪✶♠✰① ①✖❢✷♦♣❵❝❜✐❦♣❢❂❥④①✏❜✉❢à➦④❢❂s ♠✷❦♣❥ ❤➆á✿❵➲♠✹❜❧❦♣❢❂❥④①⑤⑥ãâ

➟✡♠✷❥❛❢❂❥❛❦❰➟❁♠✷♦▼❦➷①✖❢➪①➌♠✰①❡➦❛♠✹❜➭♠➴➦④❦➮①❧❜➡t➡❦♣❫❛❵✿❜❧❦♣❢❂❥

❶➄➏ ♠❚q ❻❋➉ ➟ ➂

F

❪ GX ⑥

FG

❪ X ⑥

dex

❄

L

❪ L ❪ LX ⑥✉⑥ map rev

✲ L

❪ L ❪ LX ⑥✉⑥

L

❒ ❪ LX ⑥

dex

❄

map rev

✲ L

❒ ❪ LX ⑥

dex

❄

H

❪ X ⑥

map f

✲ H

❪ Y ⑥

3

SLIDE 4

➙

❲➨➛➩❘✲★ ⑦ ➒ ✗ ❘ ✢ ❩ ✑ ➒ ✗ ➒✰➛ ➑➍✑ ◗ ✛ ➣ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①➳ä

⊢ F

❬ m ❢ ➫ ♠✷t➡❦r❜➝q m

F, G

❬②❬ Ð

X | C |

å

m i | FGn | µmF

❜➝q➐❞➞❤②①❡ä

⊢ τ

❬✖æ

τ

❬②❬ Ð

X | τ

ç → τ ❒ | . . . | F ❪ τ ⑥

①➭➟❍➠❝❤✹s

♠èä

⊢ σ σ

❬②❬ Ð

τ | ∀X.σ | ∀X

❬ m.σ é ➠❝❤➆t✉❤✂ä ❬❁❬ Ð

|

ä , X ❬ m | ä , X ❬➭æ ❜➝q➐❞➞❤✂➟✹❢❂❥➪❜✐❤➡➯➐❜❚â

➙

❲➜➛✿❘✲★ ⑦ ➒❼ê ✛ ⑦ ✬✽❘ ✢ ✛ ❲ ⑧ ë ◗ ✑➲✛ ❲➜✬■❲✻✩✜ì

➟➺❢❂❥④①✇❜✉♠✷❥❋❜

➫ ❵➐❥✓➟⑤❜✉❢❂t✇① ❬ ➢ ➅▲❬❍❽ ❵➐❥❛❦r❜✏❜➝q➐❞➞❤ ➢

×,

→í❬⑤➎ ❫❛❦♣❥➐♠✷t●q ❞❛t✇❢❝➦✿❵❇➟⑤❜➈♠✷❥❇➦ ①✖❵❼s ➣Þ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥ å

m i

❬ m ❾ é ➠❝❤➆t✉❤ i ∈ m å

m i

❪ X ⑥ ∼ Ð

Xi

➣↕➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ FGn ❬ n ❾ ❦ ➫ F ❬ m ♠✷❥❇➦ Gi∈m ❬ n

FGn

❪ X ⑥ ∼ Ð

F

❪ Gi ❪ X ⑥

i∈m

⑥

❦♣❥❇➦✿❵❇➟⑤❜❧❦

➬ ❤ µmF ❬ m ❾ ❦ ➫ F ❬ m → ➅

µmF

❪ X ⑥ ∼ Ð

µY.F

❪ X, Y ⑥

4

SLIDE 5 ➱ ➑✻✛ ❘ ✛ ⑦ ✑ ➓☞ ☛ ↔✇➙ ❲➜➛✿❘✲★ ⑦ ➒ ❐ ➣è♠➴t✇❤❉➘➐❥❝❤➆sà❤➆❥➪❜➾❢ ➫ ❜➝q➐❞➞❤❉➚■➟✹❢❂❥➐①✇❜➡t✇❵❇➟❉❜✉❢❂t❧① ❬ é ❦r❜➸➠ î Ú❋ï Ø ♠✯➦❛➦④❦r❜✐❦♣❢❂❥➐♠✷♦❃①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②ð✰❞❛t✇❢✲❞➲❤➆t✐❜✐❦♣❤②① ❤❂âñ③➞â ➫ ❵➐❥❛➟❉❜✉❢❂t❧① ❾ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜✉❢❂t✇① ❾ . . .

❲❨★▼❘

➃ ➔ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇① ❬ ❫❛❵✿❜➳➟➺❢❂❵❛♦➷➦ ❫➞❤ Ø②òóÕ✹Ú✯ô➐Ø✹ô ♠✰① ➃ ➔ ①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②①

F ∗

❬✡õ❃ö❂÷ ø é ➠❝❤✹❥ F ❬ m ❾ é ➠❝❤➆t✉❤ õ❃ö❂÷✣ù✜ú û õ❃ö❂÷ ü✓ý❛þ▼ÿ ✂✁ ✄ ☎ ✆✞✝ ✟✡✠ ÿ☞☛❼ü✓ý❛þ▼ÿ ✝✍✌ ✎✑✏✓✒ ✔ ✕ ✆✖✂✁✑✗✙✘✚✜✛✢✌✣✗✙✘✚✂✁ ✄✍✗✙✘✚✜✛ ✄ ☎ ✤✙✤✙✤ ÿ✦✥✓✧★☎ ❫❛❵✿❜ ❲ ★ ❦♣❥ ➬ ❤✡t❧①❉❤ ô④Ø✹Õ✹Ú✯ô➼Ô❱ò❛Ù â

❲❨★▼❘

➇ ♠✰①✖Ï➪❤✹♦♣♦✘ð✪✩➓❢ ➫ ❤➆t✏❜➝q➐❞➞❤✈➟➺♦♥♠✰①➡①⑤❤②① ❬ ❫❛❵✿❜✈➟✹❢❂❵④♦➷➦ ❫➞❤ Ø❁òóÕ➆Ú✷ô④Ø➺ô ♠✰①❮❤✹♦♣❤✹s➥❤✹❥❋❜✇①✈❢ ➫ ♠➥❜➸q➐❞➲❤↕➟➺♦♥♠✰①➡①➥❪ ①⑤♠✷s➥❤ ❦➷➦❝❤✲♠í♠✰①➾♠❁❫❃❢ ➬ ❤②⑥ãâ ✫✭✬✯✮✱✰✳✲✵✴ ✶✵✷✹✸ ✺ ✻✽✼✿✾❁❀❃❂❅❄❇❆❉❈❊✻❋❄❇❆❍● ■✡✾ ✲❑❏▼▲❖◆✍P✓✲✵◗❘◆ ❙❁■❚❂❯■❚❂✜❱❁❲❨❳ ❄✡✻❩❂✂❱❁❲❨❳ ❀❬❄❇✾❭❈❪❂❃❆❍✼✑❀❃❂❅❄❇❆❉❈❴❫

5

SLIDE 6 ❵ ★ ◗ ✑ ➒ ✑ ◗ ✛ ❲✜❘❚✬✔➛❼➒ ↔ ★ ⑦ F ❬❚➅ ✬✔❲ ❵✻✑ ❘

F

❬❛❵ ✑ ❘ → ❵✻✑ ❘ ➫ ❵❼❥✓➟⑤❜➭❢❂t ➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ ♠✷❥❇➦ ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥④①❡❵❛❞ ❜✉❢à❤➆á✿❵➲♠✷♦♣❦r❜➸q ❥❛❢à❦♣s ❞❼♦♣❦♣➟➺❦r❜➳①✖➠➐♠✷❞➞❤✈❞❣❢✯♦rq➐s ❢❂t✇❞❛➠❛❦➷①✖s ❪❴❥❛❢❂t µmF ⑥

F

❬❛❵ ✑ ❘ → ❵✻✑ ❘ ➫ ❵❼❥✓➟⑤❜➭❢❂t é ❦r❜➸➠ t✉❤✹♦♥♠✹❜❧❦♣❢❂❥➐♠✷♦✫♠✯➟⑤❜❧❦♣❢❂❥

X

ç

f

✲ Y

ç

X

❒

R g

✲ Y

❒

S

❦♣s ❞➐♦♣❦♣❤②①

FX

ç

Ff

✲ FY

ç

FX

❒

FR Fg

✲ FY

❒

FS

F

❬❛❵ ✑ ❘ → ❵✻✑ ❘ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜➭❢❂t ❾ ❦❱â ❤❂â

F

❪ X ⑥

dataX

✲ L

❪ X ⑥

S ∼

Ð

F

❪ ➅ ⑥

F

❪❧Ñ➄⑥

❄

❜ Ð

data

ç

✲ L

❪ ➅ ⑥ ∼ Ð

N

❄

L

❪✇Ñ➄⑥

F

❪ X ⑥ ∼ Ð ❝ s ❬ S.X ❞ s ❥➐♠✹❜❧❵❛t➡♠✷♦♣♦rq ❦♣❥ X ➣ F t✉❤✹❞❛t➆â S ∈ ❵✻✑ ❘ , ❜ ❬ S → N ❢ ➫ ①➭➠➐♠✷❞➲❤✹♦rq ➫ ❵➐❥❛➟❉❜✉❢❂t

F

❪ X ⑥ Ð➻Ð ❝ s ❬ S.X ❞ s å ç ❡ Ð➻Ð

➅ , λ

❬②➅ . ➅ ➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ ♠✷❥❇➦ ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥④①❡❢❂❥❛♦rq ❵❛❞ ❜➭❢è❦➮①➭❢ ❦♣s ❞➐♦♣❦♣➟➺❦r❜❡①✖➠➐♠✷❞➞❤✈❞❃❢✷♦rq➐s ❢❂t✉❞④➠❛❦➮①➭s ❪✶♠✷❥❇➦ µmF ⑥

6

SLIDE 7 ✆ ✑ ⑦➪◗ ➒ ✛ ✒■✛ ☎ ➑ ➙ ⑦ ➛ ➑

❜●❤✡t✇s

①❡ä ❢❍❣ ⊢ t ❬ σ

t

❬❁❬ Ð

x | c | λx.t | t

ç t ❒ | ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒ é ➠❝❤➆t✉❤❤❣ ❬②❬ Ð

|

❣ , x ❬ σ ❪✽❦ ➫ ä

⊢ σ

⑥ ❜●❤✡t✇s ➟✹❢❂❥➪❜✐❤➡➯➐❜❚â ✞ ➙ ✒❖✑ ➒ ↔ ★ ⑦ ❘ ✢ ❩ ✑ ✛ ➒❂➒➪✬❖✩ ❲ ◗ ✑ ❲✜❘ ❪ c ⑥ ä ❢❍❣ ⊢ ä ❢❍❣ ⊢ c ❬ σc ❪ x ⑥ ä ❢❍❣ ⊢ ä ❢❍❣ ⊢ x ❬ σ σ Ð ❣❨❪ x ⑥ ❪■➣❥✐ ❞❝❞

n

⑥ ä ❢❪❣ ⊢ t ❬ ∀X ❬ n.σ ä

⊢ F

❬ n ä ❢❍❣ ⊢ t ❬ σ{F/X} ❪■➣❥❦

n

⑥ ä , Y ❬ n ❢❪❣ ⊢ t ❬ σ{Y/X} ä ❢❉❣ ⊢ t ❬ ∀X ❬ n.σ

Y ∈

❧♥♠ ❪❱ä ⑥ ❪♦✐ ❞❝❞❣⑥ ä ❢❉❣ ⊢ t ❬ ∀X.σ ❣ ⊢ τ ä ❢❍❣ ⊢ t ❬ σ{τ/X} ❪♦❦➾⑥ ä , Y ❬➭æ ❢❉❣ ⊢ t ❬ σ{Y/X} ä ❢❉❣ ⊢ t ❬ ∀X.σ

Y ∈

❧♥♠ ❪❱ä ⑥ ❪✶♠✷❞❝❞➲⑥ ä ❢❪❣ ⊢ t ❬ τ ç → τ ❒ ä ❢❍❣ ⊢ t ç ❬ τ ç ä ❢❍❣ ⊢ ❪ t t ç ⑥ ❬ τ ❒ ❪ λ ⑥ ä ❢❍❣ , y ❬ τ ç ⊢ t{y/x} ❬ τ ❒ ä ❢❍❣ ⊢ ❪ λx.t ⑥ ❬ τ ç → τ ❒

y ∈

❧♥♠ ❪✽❣ ⑥ ❪✽♦♣❤❉❜❉⑥ ä ❢❪❣ ⊢ t ç ❬ σ ç ä ❢❪❣ , y ❬ σ ç ⊢ t ❒ {y/x} ❬ σ ❒ ä ❢❉❣ ⊢ ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒ ❬ σ ❒

y ∈

❧♥♠ ❪✽❣ ⑥

7

SLIDE 8 ☎ ★ ❲➜➒✲❘ ✛ ❲✜❘✯➒ ✛ ❲ ⑧ ❘ ➑ ✑ ✬ ⑦ ➒✰➛ ➑➍✑ ◗ ✛ s ♠✷❞

m

❬ ∀F ❬ m.∀X, Y . ❪ Xi → Yi ⑥

i∈m → F

❪ X ⑥ → F ❪ Y ⑥ ❞➞❤➡➯

m,i

❬ ∀X.Xi → å

m i

❪ X ⑥ ❞❛❦♣❥

m,i

❬ ∀X. å

m i

❪ X ⑥ → Xi ➦❝❤➡➯

m,n

❬ ∀F ❬ m.∀G ❬ n.∀X.F ❪ Gi ❪ X ⑥

i∈m

⑥ → FGn ❪ X ⑥ ➦④❦♣❥

m,n

❬ ∀F ❬ m.∀G ❬ n.∀X.FGn ❪ X ⑥ → F ❪ Gi ❪ X ⑥

i∈m

⑥ ❦♣❥❋❜✖t✉❢

m

❬ ∀F ❬ m → ➅ .∀X.F ❪ X, µmF ❪ X ⑥✉⑥ → µmF ❪ X ⑥ ➫ ❢✯♦➷➦

m

❬ ∀F ❬ m → ➅ .∀X, Y. ❪ F ❪ X, Y ⑥ → Y ⑥ → µmF ❪ X ⑥ → Y ❵➐❥ ❬í➅ ❞➲♠✷❦♣t ❬ ∀X ❡ , X ç .X ❡ → X ç → X ❡ × X ç ❞④❦

j

❬ ∀X ❡ , X ç .X ❡ × X ç → Xj ❦♣❥

j

❬ ∀X ❡ , X ç .Xj → X ❡ →

X

ç ➟❁♠✰①⑤❤ ❬ ∀X ❡ , X ç , Y. ❪ X ❡ → Y ⑥ → ❪ X ç → Y ⑥ → X ❡ →

X

ç → Y ➘➼➯ ❬ ∀X. ❪ X → X ⑥ → X

◗

✛ ✬■❲ ⑦ ✑ ➒ ➙ ✒ ❘ ❬ ➃ ➔ ➚■♦♣❦♣Ï➪❤➁❜➝q➐❞➞❤❮❦♣❥ ➫ ❤➆t✉❤✹❥✓➟➺❤➻♠✷♦➤③❋❢❂t➡❦r❜➝➠④s â

8

SLIDE 9 ✆ ✢ ❩ ✑q♣ ↔ ⑦ ✑④✑ ✞ ✑❙⑧ ➙ ➛➩❘✲✬❖★✜❲ ❪ λx.t ❒ ⑥ t ç > t ❒ {t ç /x} ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒

> t

❒ {t ç /x} ❞❛❦

j

❪✽❞➞♠✷❦♣t t ❡ t ç ⑥ > tj ➟✡♠✰①❉❤ f ❡ f ç ❪✽❦♣❥

j t

⑥ > fj t ❞❛❦♣❥

m,i

❪✽❞➞❤➡➯

m,i t

⑥ > t ➦④❦♣❥

m,n

❪❱➦❝❤➡➯

m,n t

⑥ > t s ♠✷❞

m fk∈m

❪✽❞➞❤➡➯

m,i t

⑥ > ❞➞❤➡➯

m,i

❪ fi t ⑥ s ♠✷❞

n fk∈n

❪❱➦❝❤➡➯

m,n t

⑥ > ➦❝❤➡➯

m,n

❪✽s ♠✷❞

m

❪✽s ♠✷❞

n f

⑥

i∈m t

⑥ s ♠✷❞

m fi∈m

❪✽❦♣❥❋❜✖t✉❢

m t

⑥ > ❦♣❥❋❜✖t✉❢

m

❪✽s ♠✷❞

m

✤ ç f ❪❴s ♠✷❞

m f

⑥ t ⑥ s ♠✷❞ ❒ f ❡ f ç ❪✽❞➞♠✷❦♣t t ❡ t ç ⑥ > ❞➲♠✷❦♣t➓❪ f ❡ t ❡ ⑥➁❪ f ç t ç ⑥ s ♠✷❞ ❒ f ❡ f ç ❪✽❦♣❥

j t

⑥ > ❦♣❥

j

❪ fj t ⑥ s ♠✷❞ ❡ ❵➐❥

>

❵❼❥ ➫ ❢✷♦➷➦

m f

❪✽❦♣❥❋❜✖t✉❢

m t

⑥ > f ❪✽s ♠✷❞

m

✤ ç ❪ λx.x ⑥

i∈m

❪ ➫ ❢✯♦➷➦

m f

⑥ t ⑥ ◗ ✛ ✬✔❲ ⑦ ✑ ➒ ➙ ✒ ❘✯➒

➊

➠❋❵❛t➭➟➡➠❇➚ ❿ ❢❋①➡①⑤❤✡t✏❞❛t✇❢✲❞➲❤➆t✐❜➸q✿❢

❺

❵❋❫➐➽➝❤✡➟⑤❜➾t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥❭❢

❺

❜✖t✉❢❂❥❝③í❥❛❢❂t✇s ♠✷♦♣❦➤➹❚♠✹❜❧❦♣❢❂❥ ❪ é ➠❝❤✹❥ ❥❛❢➴➘➼➯❃⑥ãâ

9

SLIDE 10 ❵✥✛ ◗ ❩ ✒❖✑ ⑦ ✑✣⑧ ➙ ➛➩❘❚✬✶★✜❲➨➒

➫

❢✷♦➷➦ ❡ f ❪✽❦♣❥❋❜✖t✉❢ ❡ t ⑥ > f ❪❴s ♠✷❞ ç ❪ ➫ ❢✯♦➷➦ ❡ f ⑥ t ⑥

F

❪ µ ❡

F

⑥ ❦♣❥❋❜➡t✇❢ ❡

✲ µ

❡

F F

❪ Y ⑥ s ♠✷❞ ç ❪ ➫ ❢✯♦➷➦ ❡ f ⑥

❄

f

✲ Y ❄

➫ ❢✯♦➷➦ ❡ f

s

♠✷❞ ç f ❪❴❦♣❥❋❜➡t✇❢ ç t ⑥ > ❦♣❥❋❜✖t✉❢ ç ❪✽s ♠✷❞ ❒ f ❪✽s ♠✷❞ ç f ⑥ t ⑥

F

❪ X, µ ç F ❪ X ⑥✉⑥ ❦♣❥❋❜➡t✇❢ ç

✲ µ

ç F ❪ X ⑥

F

❪ Y, µ ç F ❪ Y ⑥✉⑥ s ♠✷❞ ❒ ❪ f, s ♠✷❞ ç f ⑥

❄

❦♣❥❋❜➡t✇❢ ç

✲ µ

ç F ❪ Y ⑥

❄

s ♠✷❞ ç f

10

SLIDE 11 ☎ ★✜❲➨➛ ✒ ➙ ➒➪✬✶★✜❲➨➒

❹✥➃

➔ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①➾♠✰①★✐ ❧ æ ➢ ①⑤❤ ➬ ❤✡t✖♠✷♦ Ô❱ï❤r❁s Ø②ï Ø②ò✪×➝Ö❇×❖Ô➝Ú❋ò î❡❞❣❢❋①➡①❍❦♣❫❼♦♣❤ ➢ ♠➴①➭s ❢✿❢✷❜➸➠ é ♠❚q ❜✉❢è❤❍➯➐❜●❤✹❥❇➦ ➃ ➔❘t ✉✇✈ ❺✓➔✥❾ ➟➺❢❂t✉❤⑤➚ ➃ ➔ ⊂ ❹✴➃ ➔ ➢ ❫④❵❝❜➳t➡♠✹❜➸➠❝❤✡t✏♦♣❢ é ➚■♦♣❤ ➬ ❤✹♦ ❾ ❤❂âñ③➞â ♦♣❦➷①✇❜✇① L Ð

µ

ç F é ➠❝❤✡t✇❤ F Ð → ➅ ❒ , × ❥④❦♣♦ Ð ❪❴❦♣❥❋❜➡t✇❢ ç ◦ ➦❝❤➡➯ ❒ , ❒ ◦ ❦♣❥ ❡ ◦ ➦❇❤➡➯ ❡ , ❒ ⑥➜❵➐❥ ❬ LX

①✇❜✐❵✓➦❇q

❹✥➃ ➔ ❜✐❤➆t✉s ①➓♠ ♦♥♠ ➊ ❵❛t✇t✐q ❬ t➡❦♣➟➡➠❝❤✡t➁s ❢✿➦❇❤✹♦❰① ❾ ❫➲❤❉❜✔❜✐❤➆t❮❢✲❞❇➚ ❜❧❦♣s ❦➤➹❚♠➆❜✐❦♣❢❂❥④① ❶❸➇✂➃ ❻❋➉✲➂

➟✡♠✷❥❛❢❂❥❛❦❰➟❁♠✷♦▼❦➷①✖❢➪①❡❦♣❥

➫ ❤✡t✇❤➆❥❛➟✹❤✂❞❛t✇❢❂❫➐♦♣❤✹s

❹✥➃

➔ ♠✰① ❦♣❥❋❜✐❤➆t✉sà❤✲➦④❦♥♠✹❜✐❤ ♦♥♠✷❥❇③❋❵➞♠✲③❋❤ ➫ ❢❂tÞ♠✷❥➐♠✷♦rq❼①❍❦➷① ❢ ➫ ♠✯➟➆➟➺❤②①➡①➓❜✉❢ ➦④♠✹❜➭♠ ❪✽➟❁♠✷❥④❢❂❥❛❦♣➟✡♠✷♦▼❦➮①✖❢❂s ❢❂t✉❞❛➠④❦➮①✖s ①❉⑥ ❶❸➀➶➏✷❻❛➅❂❾❛➔ ❤✡t ❻❋➎❚➂

11

SLIDE 12 ①✇②④③⑤②⑦⑥❉②⑦⑧✖⑨✞②❶⑩

[Ben67]

J. Benabou. Introduction to bicategories, volume 47. Springer, 1967.

[BFSS90] E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35–64, 1990. [BJM96]

G. Bell`

e,

C. B. Jay,

and E. Moggi. Functorial ML. available from ftp://ftp.disi.unige.it/person/MoggiE/functorial ml.dvi, 1996. [CF92] J.R.B. Cockett and T. Fukushima. About charity. Technical Report 92/480/18, University of Calgary, 1992. [HM95]

R. Harper and G. Morrisett. Compiling polymorphism using intensional type ana-
lysis. In Conference Record of POPL ’95: 22nd ACM SIGPLAN-SIGACT Sym-

posium on Principles of Programming Languages, pages 130–141, San Francisco, California, January 1995. [Jay95a] C.B. Jay. Polynomial polymorphism. In R. Kotagiri, editor, Proceedings of the Eighteenth Australasian Computer Science Conference: Glenelg, South Australia 1–3 February, 1995, volume 17, pages 237–243. A.C.S. Communications, 1995. [Jay95b] C.B. Jay. A semantics for shape. Science of Computer Programming, 25:251–283, 1995. [Jay95c] C.B. Jay. Shape analysis for parallel computing. In Parallel Computing Workshop ’95 at Fujitsu Parallel Computing Centre, Imperial College, 1995. [Jeu95]

J. Jeuring. Polytypic pattern matching. In Conference on Functional Programming

Languages and Computer Architecture, pages 238–248, 1995. [Jon95] M.P. Jones. A system of constructor classes: overloading and implicit higher-order

polymorphism. J. of Functional Programming, 5(1), 1995.

[Ler92]

X. Leroy. Unboxed objects and polymorphic typing. In 19th Symp. on Principle
f Programming Languages. ACM Press, 1992.

[MFP91] E. Meijer, M. Fokkinga, and R. Paterson. Functional programming with bananas, lenses, envelopes and barbed wire. In J. Hughes, editor, Procceding of the 5th ACM Conference on Functional Programming and Compter Architecture, volume 523 of LNCS, pages 124–44. Springer Verlag, 1991. [MH95]

E. Meijer and G. Hutton. Bananas in space: extending fold and unfold to exponen-

tial types. In Procedings 7th International Conference on Functional Programming and Computer Architecture, San Diego, California, June 1995. ACM Press, 1995. [PJ91]

S. Peyton Jones. Unboxed values as first-class citizens. In Functional Programming

and Computer Architecture, volume 523 of LNCS, 1991. [RP90]

J. Reynolds and G.D. Plotkin. On functors expressible in polymorphic lambda-