圏論とSwiftへの応用 / iOSDC Japan 2018

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ݍ࿦ͱɹɹɹ Swift΁ͷԠ༻ 2018/09/02 iOSDC 2018 Yasuhiro Inami / @inamiy

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ݍ࿦ Category Theory

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ݍ࿦ͱ͸ʁ • ʮ఺ʯͱʮ໼ҹʯΛ࢖ͬͨ਺ֶͷ Foundation.framework • ఺ΑΓ΋໼ҹͷํ͕ॏཁʢू߹࿦ͷߟ͑ํͱ͸ٯʣ • Ԡ༻ൣғ͕޿͍ɿ ਺ֶɺ࿦ཧֶɺܭࢉػՊֶɺཧ࿦෺ཧֶɺ ೝ஌Պֶɺ఩ֶɺੜ෺ֶɺetc • ਺ֶͷ஌͕ࣝθϩͰ΋ֶ΂Δʢඍੵɺઢܗ୅਺΋͍Βͳ͍ʣ • ؔ਺ܕϓϩάϥϛϯάͰେ׆༂

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ݍ࿦ΛֶͿ → ܕγεςϜ νϣοτσΩϧ • ࠶ؼܕʢෆಈ఺ɺfixɺreduce (fold) / sequence (unfold)ʣ • શশྔԽ ʢύϥϝʔλଟ૬ɺδΣωϦΫεʣ • ଘࡏྔԽ ʢϓϩτίϧ (existential container)ʣ • ߴϥϯΫଟ૬ʢଟ૬ؔ਺ΛҾ਺ͱͯ͠࢖͑Δʣ • ߴΧΠϯυଟ૬ʢܕίϯετϥΫλΛܕҾ਺ͱͯ͠࢖͑Δʣ • ґଘܕʢ஋ʹґଘ͢Δܕ͕࡞ΕΔʣ

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ݍ࿦ΛֶͿ ↓ ʮܕʯΛ ΑΓਂ͘ཧղͰ͖Δ ↓ Swiftͷݱঢ়ͱ ະདྷ͕ݟ͑ͯ͘Δ

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※ ͓அΓ ͜ͷεϥΠυͰ͸ɺ ʮٙࣅSwiftʯΛ࢖ͬͯղઆ͍ͯ͠·͢ɻ ( Swift 4.2ݱࡏɺݴޠػೳ͕଍Γͳ͍ͨΊ)

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ݍͷఆٛ Deﬁnition

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ݍͷఆٛ • ର৅ (object)ɿ • ࣹ (morphism)ɿ • ߃౳ࣹ (identity)ɿ • ߹੒ (composition)ɿ • ݁߹཯ (associativity)ɿ • ࢝Ҭ (domain) / ऴҬ (codomain):

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Swift ͷྫ // ߃౳ࣹʢྫɿ `f ⚬ id == f`ɺ `id ⚬ f == f`ʣ func id(_ a: A) -> A { return a } // ؔ਺ͷ߹੒ʢྫɿ `(g ⚬ f)(x) == g(f(x))`ʣ func ⚬ (_ g: B -> C, _ f: A -> B) -> A -> C { return { g(f($0)) } }

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ؔख Functor

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ؔख (Functor) • ݍΛʮର৅ʯʹɺ ͋Δݍ͔Β͋Δݍ ΁ͷߏ଄ʢ໼ҹશମͷܗʣΛอͭ ʮࣹʯʢϚοϐϯάʣ • ֤ର৅ ʹ͍ͭͯ ʹରԠ • ֤ࣹ ʹ͍ͭͯ ʹରԠ • ߏ଄ʢ߹੒ͱ߃౳ࣹʣΛอଘ͢Δ • •

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// ٙࣅSwift protocol Functor[F] { // Note: `self.map { ... }` ʢΠϯελϯεϝιουʣ // ͱͯ͠΋࢖͑Δͱ͢Δ static func map(_ self: F , _ f: A -> B) -> F } extension Functor[Array] { ... } // ࢖༻ྫɿ Array ==> Array [1, 2, 3].map(String.init) == ["1", "2", "3"]

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ؔखͱ͸ʮmap ͕Ͱ͖Δʯ͜ͱ map = ίϯςφΛม͑ͣʹ த਎Λม͑Δʁ ʢର৅͕த৺ (pointwise) ͷߟ͑ํʣ

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// લܝ protocol Functor[F] { static func map(_ self: F , _ f: A -> B) -> F } // Ҿ਺ॱংΛೖΕସ͑ͯɺΧϦʔԽ͢Δͱɾɾɾ protocol Functor[F] { // ؔ਺্࣋ͪ͛ (lifting) static func map(_ f: A -> B) -> F -> F } map ͕ f: A -> B ͔Β F(f): F -> F ΁ͷม׵ʹͳΔ

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map ʢؔखʣ͸ɾɾɾ ݍͷؔ਺ f Λɺผͷݍͷؔ਺ F(f) = map(f) ʹࣸ͢ ʢࣹ͕த৺ (pointfree) ͷߟ͑ํʣ

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ͷҙຯ 2ͭͷࣹ ʹ͍ͭͯɺઌʹ߹੒͔ͯ͠Β Λద༻ͯ͠΋ɺ ΛͦΕͧΕʹద༻͔ͯ͠Β߹੒ͯ͠΋ಉ͡

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ͷྫ let count: String -> Int = { $0.count } let isEven: Int -> Bool = { $0 % 2 == 0 } ["Hello"].map(isEven ⚬ count) // ؔ਺߹੒ ʴ 1ճ `map` == ["Hello"].map(count).map(isEven) // 2ճ `map` // ==> F.map(isEven ⚬ count) == F.map(isEven) ⚬ F.map(count) ! 2ճ map͸ɺʮؔ਺߹੒ʴ1ճ mapʯͱಉ͡ʢܭࢉྔ࡟ݮʣ

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༨ஊɿࣗݾؔख (Endofunctor) ͜͜·Ͱɺ Array ΍ Array Λɺ Int ΍ String ͱ͸ ผͷݍʢArray ͷݍʣ ͷର৅ͱͯ͠ߟ͖͑ͯͨ ɹɹɹɹɹɹɹɹɹɹɹɹɹˣ ࣮ࡍ͸ɺ͢΂ͯಉ͡SwiftͷܕͳͷͰɺ1ͭͷʮSwiftݍ ʯͷର৅ ͱͯ͠ѻͬͯྑ͍ ɹɹɹɹɹɹɹɹɹɹɹɹɹˣ ϓϩάϥϛϯάͷੈքͷ ؔख (Functor) ͱ͸ɺ࣮͸ ࣗݾؔख ʢݍ͔Βಉ͡ݍ΁ͷؔख ʹ ଟ૬ؔ਺ mapʣ ͷ͜ͱ

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ࣗવม׵ Natural Transformation

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ࣗવม׵ • 2ͭͷݍͷؒͷ"ࣹ"Ͱ͋ΔؔखΛʮର ৅ʯͱͯ͠ѻͬͨͱ͖ͷɺؔख͔Β ؔख΁ͷʮࣹʯ • ݍ ͱؔख ͕༩͑ ΒΕͨͱ͖ɺࣗવม׵ ͸ ͷࣹ ͷू߹ Ͱɺ೚ҙͷ ʹରͯ͠ɺҎ Լͷ౳ࣜʢՄ׵ਤࣜʣ Λຬͨ͢ɿ

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ͷҙຯ // ࣗવม׵ʢத਎Λม͑ͣʹίϯςφΛม͑Δʮଟ૬ؔ਺ʯʣ func alpha(_ fa: F ) -> G { ... } // Մ׵ࣜ alpha(fa.map(f)) == alpha(fa).map(f) // ==> alpha ⚬ F.map(f) == G.map(f) ⚬ alpha ! F.map(f) Λઌʹܭࢉ͔ͯ͠Β alpha Λܭࢉͯ͠΋ɺalpha Λઌʹܭࢉ͔ͯ͠Β G.map(f) Ͱ΋ɺͲͪΒͰ΋੒Γཱͭ

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ͷྫ /// Array -> Optional ΁ͷࣗવม׵ func arrFirst(_ arr: Array ) -> Optional { return arr.first } arrFirst( [1, 2, 3].map(String.init) ) // `map` ͔ͯ͠Β `first` == arrFirst([1, 2, 3]).map(String.init) // `first` ͔ͯ͠Β `map` ! ӈลͷํ͕ܭࢉྔ͕গͳ͍ → ίϯύΠϥ͕࠷దԽͰ͖Δ

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ถాͷิ୊ Yoneda Lemma

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ถాͷิ୊

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ถాͷิ୊ ʹʮࣗવม׵ʯͱʮ஋ʯͷ1:1ରԠ let fa: F = ... // ஋ func closure(_ f: A -> B) -> F { // ࣗવม׵ return fa.map(f) // `fa` Λ಺෦Ͱ࣋ͭ } faʢ஋ʣͱ closureʢ஋Λดแͨ͠ଟ૬ؔ਺ʣ͸"ಉ͡΋ͷ"ɺ ͦΕΒͷܕ͸ ಉܕʢ૬ޓʹม׵Մೳʣ ͱΈͳͤΔ

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// F(A) ͔Β ∀B. (A -> B) -> F(B) ΁ͷม׵ func faToClosure(_ fa: F ) -> (A -> B) -> F { return { f in return fa.map(f) } } // ∀B. (A -> B) -> F(B) ͔Β F(A) ΁ͷม׵ func closureToFa( _ closure: /* ϥϯΫ2ଟ૬ */ (A -> B) -> F ) -> F { return closure(id) }

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ࢀߟεϥΠυ2ɿ ಉܕͷ࿩ 2 https://speakerdeck.com/inamiy/algebraic-data-type-in-swift ༨ஊɿถాͷิ୊Λ࢖͏ͱ ͷࢦ਺๏ଇ΍ɺ෼഑๏ଇͳͲ͕؆୯ʹূ໌Ͱ͖·͢

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ถాͷิ୊͔Βɺܧଓ (CPS) ͱCPSม׵ • ʢ߃౳ؔखʣͱ͓͘ͱɺCPSʢܧଓʣ • ͱ͓͘ͱɺCPSม׵ʢʹ ถాຒΊࠐΈ ʣ func cpsTransform(_ f: X -> A) -> (A -> B) -> (X -> B) { return { aToB in return { x in aToB(f(x)) } } }

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func fac(_ n: Int) -> Int { // ⚠ ຤ඌ࠶ؼͷܗͰ͸ͳ͍ʢ͕ɺཪͰίϯύΠϥ͕࠷దԽͯ͘͠ΕΔʣ return n == 0 ? 1 : n * fac (n - 1) } // `cpsTransform` Λ࢖Θͣɺखॻ͖ͰܧଓʹΑΔ຤ඌ࠶ؼ func facCPS(_ n: Int, _ next: Int -> R) -> R { return n == 0 ? next(1) : facCPS(n - 1, { next(n * $0) }) } // `cpsTransform` Λ࢖ͬͨ৔߹ʢ຤ඌ࠶ؼ࠷దԽʣ let facCPSAuto: (Int -> Int) -> Int -> Int = cpsTransform(fac) facCPS(4, id) // 24 ʢखಈͰ຤ඌ࠷దʣ facCPSAuto(id)(4) // 24 ʢࣗಈͰ຤ඌ࠷దʣ

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ਵ൐ Adjunction

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ਵ൐

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ࣗ༝ؔख ๨٫ؔख

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// ٙࣅSwift protocol Adjunction[F, U] where Functor[F], Functor[U] { /// (F(C) -> D) -> (C -> U(D)) static func leftAdjunct(_ f: F -> D) -> C -> U /// (C -> U(D)) -> (F(C) -> D) static func rightAdjunct(_ f: C -> U) -> F -> D }

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ਵ൐ͷྫ /// ੵͷܕʢλϓϧʣ typealias Tuple = (B, A) // ஫ҙɿ޲͖͕ٯ /// ؔ਺ͷܕʢReaderͱ΋ݺ͹ΕΔʣ typealias Func = A -> B

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extension Functor[Tuple] { // `Tuple ` ͸ `A` ͷΈͷ෦෼ద༻ static func map(_ f: C -> D) -> Tuple -> Tuple { return { t in return (f(t.0), t.1) } } } extension Functor[Func ] { static func map(_ f: C -> D) -> Func -> Func { return { fun in return f ⚬ fun } } }

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// Tuple ⊣ Func extension Adjunction[Tuple , Func ] { static func leftAdjunct(_ f: Tuple -> D) -> C -> Func { return { c in { a in f((c, a)) } } } static func rightAdjunct(_ f: C -> Func ) -> Tuple -> D { return { t in f(t.0)(t.1) } } }

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func curry(_ f: (A, B) -> C) -> A -> B -> C { return { a in return { b in return f(a, b) } } } func uncurry (_ f: A -> B -> C) -> (A, B) -> C { return { a, b in return f(a)(b) } }

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leftAdjunct == curry rightAdjunct == uncurry (Tuple ⊣ Func ͷͱ͖)

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ਵ൐ͷผͷఆٛ

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extension Adjunction { static func unit(_ c: C) -> U> { // `leftAdjunct` ͷୈ1Ҿ਺͸ `F -> D` ͳͷͰɺ // `id` Λద༻͢Δͱɺܕͱͯ͠ `D = F` ͕֬ఆ͢Δ return leftAdjunct(id)(c) } static func counit(_ fud: F>) -> D { return rightAdjunct(id)(fud) } }

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counit: F> -> D ʹɺ D = F Λ୅ೖͯ͠ΈΔͱ counit: F>> -> F ͜ΕΛ͞Βʹ ؔख U ͰϦϑτ͢Δͱ U.map(counit): U>>> -> U> extension Adjunction { static func join(_ x: U>>>) -> U> { return map(counit) } }

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੔ཧͯ͠ΈΔɿ extension Adjunction { static func unit(_ c: C) -> U> { return leftAdjunct(id)(c) } static func join(_ x: U>>>) -> U> { return map(counit) } } ͜͜Ͱɺ U> = M ͱஔ͍ͯΈΔͱɾɾɾ

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protocol Monad[M] where Functor[M] { static func unit(_ c: C) -> M static func join(_ mmc: M>) -> M } extension Monad { static func flatMap(_ f: C -> M) -> M -> M { return { (join ⚬ map(f))($0) } } }

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Ϟφυ Monad

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ਵ൐ ↓ Ϟφυ

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ྫɿ Tuple ⊣ Func ΑΓɺ Func> ʹʮؔ਺ͷதʹλϓϧΛೖΕΔʯʹ ঢ়ଶϞφυ struct State { let runState: S -> (A, S) init(_ fun: Func>) { // `Func>` ͱ // `S -> (A, S)` ͸࣮͸ಉ͡ʂ self.runState = fun } }

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͸Ϟφυ Ͱ͸ɺٯͷ ͸lĤĤĤ

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// W ͸ Ϟφυ M ͷٯ protocol Comonad[W] where Functor[W] { // Ϟφυͷ `unit: C -> M` ͷٯ static func counit(_ wc: W) -> D // Ϟφυͷ `join: M> -> M` ͷٯ static func duplicate(_ wc: W) -> W> }

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༨Ϟφυ Comonad

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ਵ൐ ↓ ༨Ϟφυ

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ྫɿ Tuple ⊣ Func ΑΓɺ Tuple > ʹʮλϓϧͷதʹؔ਺ΛೖΕΔʯʹ ༨ঢ়ଶ༨Ϟφυ struct Store { // CoState ͱ΋ݴΘΕΔ let setter: A -> S let getter: A init(_ t: Tuple >) { self.setter = t.0 self.getter = t.1 } }

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ͦͯ͠ɺ S -> Store ʢ༨୅਺ɺCoalgebraʣΛߟ͑Δ struct Lens { let setter: S -> A -> S let getter: S -> A init(_ f: S -> Store ) { self.setter = { s in f(s).setter } self.getter = { s in f(s).getter } } }

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Ϩϯζ͸ ༨ঢ়ଶ༨Ϟφυͷ ༨୅਺ Lenses are Costate Comonad Coalgebras

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·ͱΊ • ݍɺ߹੒ɺ߃౳ࣹ • ؔख (Functor) → ݍͷࣹΛɺผͷݍʹࣸ͢ (map) • ࣗવม׵ (Natural Transformation) → Մ׵ਤࣜɺଟ૬ؔ਺ • ถాͷิ୊ (Yoneda Lemma) → ܧଓ (CPS) ͳͲ • ਵ൐ (Adjunction) → ΧϦʔԽɺϞφυɺ༨Ϟφυ • Ϟφυ / ༨Ϟφυ ((Co-)Monad) → ঢ়ଶϞφυɺϨϯζ

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͞ΒʹֶͿͨΊʹ • ۃݶ (Limit) l ੵɺΠίϥΠβɺҾ͖໭͠ɺetc • Τϯυ / ༨Τϯυ (End / Coend) • ୅਺ / ༨୅਺ (Algebra / Coalgebra) • ࣗ༝ؔखɾࣗ༝Ϟφυ (Free Functor, Free Monad) • Χϯ֦ு (Kan Extension) • ϞϊΠυݍɾ๛য়ݍ (Monoidal / Enriched Category)

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ࢀߟจݙʢॳֶऀ޲͚ʣ • Bartosz Milewski's Programming Cafe | Category Theory, Haskell, Concurrency, C++ • ! Bartosz Milewski - YouTube • ! TheCatsters - YouTube • Category theory for beginners • ݍ࿦ษڧձ @ ϫʔΫεΞϓϦέʔγϣϯζ

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ࢀߟจݙʢதʙ্ڃऀ޲͚ʣ • ! ݍ࿦ ݪஶୈ2൛ | Steve Awodey • ! ϕʔγοΫݍ࿦ ීวੑ͔Βͷ଎शίʔε | Tom Leinster • ! ݍ࿦ͷجૅ | Saunders MacLane • ݍ࿦ | ұେ੔Ҭ • ᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه • nLab

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Thanks! Yasuhiro Inami @inamiy