Category theory and Haskell

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ݍ࿦ͱHaskell Ӊࠤݟ ެี ୈ̎ճ ؔ੢೔༵਺ֶ ༑ͷձ

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ࣗݾ঺հ • Ӊࠤݟ ެีʢ͏͞Έ ͜͏͚͢ʣ • twitter: @usamik26 / connpass: usami-k • େֶ࣌୅͸਺ֶઐ߈ • Ͱ΋ݍ࿦͸΄ͱΜͲ΍ͬͯͳ͍ • ຊ৬͸ϓϩάϥϚʢϑΣϯϦϧגࣜձࣾʣ • Ͱ΋Haskell͸࢖ͬͯͳ͍

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ݍ࿦ͱHaskell • ݍ࿦͸਺ֶཧ࿦ͷͻͱͭ • ࣸ૾ʹண໨ͨ͠ந৅୅਺࿦ • Haskell͸ϓϩάϥϛϯάݴޠͷͻͱͭ • ୅දతͳΞϓϦͱͯ͠ Pandoc ͳͲ͕͋Δ • ݴޠઃܭͱͯ͠ݍ࿦Λҙ͍ࣝͯ͠Δ

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ݍʢCategoryʣ

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ݍͷఆٛ • ର৅ , , , ... • ࣹ • ࣹͷ߹੒ ʢ͜͜Ͱ , ʣ • ݁߹཯ • ߃౳ࣹ ʢ೚ҙͷର৅ ʹ͍ͭͯଘࡏ͢Δʣ • ୯Ґݩ

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ྫɿର৅͕;ͨͭͷݍ

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(1) ର৅ؒͷࣹ͕ͳ͍ʢ߃౳ࣹ͚ͩʣ • ͜Ε͸ݍʹͳ͍ͬͯΔʢ߹੒Մೳͳࣹ͸ͳ͍ʣ

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(2) ର৅ؒͷࣹ͕ͻͱͭ • ͜Ε͸ݍʹͳ͍ͬͯΔ

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(3) ର৅ؒͷࣹ͕;ͨͭɿͦͷ1 • ͜Ε͸ݍʹͳ͍ͬͯΔʢ ʣ

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(4) ର৅ؒͷࣹ͕;ͨͭɿͦͷ2 • ͜Ε͸ݍʹͳ͍ͬͯΔʢ ͷࣹ͕ෳ਺͋ͬͯ΋໰୊ͳ͍ʣ

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(5) ର৅ؒͷࣹ͕Έͬͭʁʁ • ͜Μͳݍ͸ͳ͍ʢͦΕ͸ͳͥͰ͠ΐ͏͔ʁʣ

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(5) ର৅ؒͷࣹ͕Έͬͭʁʁ • ͱ ʹ஫ҙ͢ΔɿͳͥͳΒɺͦΕʹ૬౰ ͢Δࣹ͕ଞʹଘࡏ͠ͳ͍͔Β ͦͷ͜ͱ͔Βɺ • • • ͕ͨͬͯ͠ ͱͳΔʢԾఆʹ൓͢Δʣ

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(6) ର৅ؒͷࣹ͕Έͬͭ • ͜ͷ৔߹͸ ͕ Ͱͳ͍ࣹʹͳΓ͏Δ

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Haskell ʹ͓͚Δݍ

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ର৅͸ Haskell ͷܕͱ͢Δ • جຊσʔλܕ : Int, Char, ... • Ϧετ : [Int], [Char], ... • Maybe ܕ : Maybe Int, Maybe Char, ... ͳͲͷ΋ͷ͕ର৅ • Maybe ܕʹ͍ͭͯิ଍ : Maybe Int ͷ஋͸ Just 0, Just 1, ... ·ͨ͸ Nothing

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ࣹ͸ Haskell ͷؔ਺ͱ͢Δ incr :: Int -> Int incr x = x + 1 • incr ͸ Int ܕͷ஋Λड͚औΓ Int ܕͷ஋Λฦؔ͢਺ • ͜ΕΛ Int ͔Β Int ΁ͷࣹͱΈͳ͢ • ͜ͷର৅ͱࣹʹΑͬͯݍ͕Ͱ͖Δ : Ҏޙ ͱॻ͘ • ͳ͓ɺ߃౳ࣹ͸߃౳ؔ਺ idɺ߹੒͸ؔ਺߹੒ (.)

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Ҿ਺Λෳ਺࣋ͭؔ਺͸ʁ add :: Int -> Int -> Int add x y = x + y • add ͸ೋͭͷ Int ܕͷ஋Λड͚औΓ Int ܕͷ஋Λฦؔ͢਺ • ɾɾɾͱ͍͏ղऍͷ··ͩͱ ͷࣹͰ͸ͳ͍

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ΧϦʔԽ add :: Int -> (Int -> Int) • add ͸ Int ܕͷ஋Λड͚औΓ Int -> Int ܕͷ஋Λฦ͢ add 3 :: Int -> Int (add 3) y = 3 + y -- `add 3` ͸ʮ3 Λ଍ؔ͢਺ʯͰ͋Δ • Int -> Int ͸ؔ਺ͷܕͰ͋Γɺ ͷର৅ • add ͸ର৅ Int ͔Β Int -> Int ΁ͷ ͷࣹ

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༨ஊ • ଟҾ਺ؔ਺Λ1Ҿ਺ؔ਺ͱΈͳ͢ํ๏ͱͯ͠λϓϧ΋ߟ͑ΒΕΔ • (Int, Int) -> Int • ͔͠͠ɺHaskell ͷΧϦʔԽͷ΄͏͕ɺݍ࿦ʹద߹ͤ͞Δͱ ͍͏ҙຯͰ͸ΤϨΨϯτͩͱࢥΘΕΔ

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ͷ෦෼ݍ • ಛఆͷܕ͚ͩΛूΊͯ෦෼ݍΛߟ͑Δ͜ͱ͕Ͱ͖Δ • ྫɿϦετܕ͚ͩΛର৅ͱͨ͠෦෼ݍ • ྫɿMaybe ܕ͚ͩΛର৅ͱͨ͠෦෼ݍ

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ؔखʢFunctorʣ

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ؔखͷఆٛ • ݍ͔Βݍ΁ͷؔख • ͕ ͷର৅ ͸ ͷର৅ • ͕ ͷࣹ ͸ ͷࣹ • ͷ߃౳ࣹ ʹ͍ͭͯ • ͷ߹੒ ʹ͍ͭͯ

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Haskell ʹ͓͚Δؔख class Functor f where fmap :: (a -> b) -> (f a -> f b) • Functor ͱ͍͏ܕΫϥε͕͋Δ • ͜Ε͕࣮ࡍʹݍ࿦ʹ͓͚Δؔखͷ֓೦ʹ͋ͨΔ΋ͷ • ΑΓ໌ࣔతʹݴ͏ͱɺFunctor ܕΫϥεͷΠϯελϯε͕ؔख

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ؔखͷྫ Maybe • Maybe ͸ Functor ܕΫϥεͷΠϯελϯε • ͜Ε͸ɺݍ ͔Βݍ ΁ͷؔखͰ͋Δ • Maybe ͸೚ҙͷܕ T ͔Β৽͍͠ܕ Maybe T Λ࡞Δ • Int ͔Β Maybe Int, Char ͔Β Maybe Char, ... • ͭ·Γɺ ͷର৅Λ ͷର৅ʹ͏ͭ͢

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ؔखͷྫ Maybe fmap :: (a -> b) -> (Maybe a -> Maybe b) fmap f :: Maybe a -> Maybe b fmap f (Just x) = Just (f x) fmap f Nothing = Nothing • f :: a -> b Λ fmap f :: Maybe a -> Maybe b ʹ • ͭ·Γɺ ͷࣹΛ ͷࣹʹ͏ͭ͢ • ͕ͨͬͯ͠ɺMaybe ͸ ͔Β ΁ͷؔख

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Functor ͰԿ͕خ͍͠ͷ͔ • Maybe Int ܕͷ஋Λͦͷ··ѻ͓͏ͱ͢Δͱ৔߹෼͚͕ඞཁ • ஋Λ࣋ͭ Just 1 ͳͲͷ৔߹ • ஋Λ࣋ͨͳ͍ Nothing ͷ৔߹ • Functor ͷߟ͑Λ࢖͑͹ɺNothing ͷ͜ͱ͸๨Ε͍͍ͯ • Int ʹ͍͚ͭͯͩؔ਺ఆٛ͢Ε͹͍͍

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Haskell ͷ Functor ͷ஫ҙ • Functor ܕΫϥεͷΠϯελϯεΛ࡞ͬͯ΋ɺͦΕ͕ؔखͷఆ ٛΛຬ͍ͨͯ͠Δ͜ͱ͸อূ͞Εͳ͍ • ର৅ͱର৅ͷରԠɺࣹͱࣹͱରԠ͸ɺࣗવʹͰ͖Δ • ߃౳ࣹͱ߹੒ʹ͍ͭͯ͸ɺఆٛΛຬͨ͢Α͏ʹࣗ෼ͰؾΛ͚ͭ Δ

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ྫ : ؔखʹͳΒͳ͍΋ͷ CMaybe a = CNothing | CJust Int a fmap f CNothing = CNothing fmap f (CJust counter x) = CJust (counter + 1) (f x) ҎԼͷΑ͏ʹ߃౳ࣹΛอͨͳ͍ɺ·ͨ߹੒΋อͨͳ͍ fmap id (CJust 0 "abc") -- CJust 1 "abc" id (CJust 0 "abc") -- CJust 0 "abc"

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͞ΒͳΔ࿩୊

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͞ΒͳΔ࿩୊ • ݍ࿦ʹ͓͚ΔϞφυ : ࣗݾؔखͰࣗવม׵Λ࣋ͭ΋ͷ • Haskell Ͱॏࢹ͞ΕΔͷ͸ϞφυʢMonadʣ • Monad ಉ࢜ΛνΣΠϯͤ͞Δ͜ͱ͕Ͱ͖Δ • ෭࡞༻Λ Monad ʹด͡ࠐΊΔ͜ͱ͕Ͱ͖Δ • ͦΕ͸·ͨͦͷ͏ͪɾɾɾ