From Tagless-Final to Typed-Final Program Transformations in the Final Style Kenichi Suzuki BizReach, Inc. [email protected]

໨࣍ • ࣗݾ঺հ • tagless-finalͱ͸ • typed-final styleͰϓϩάϥϜม׵

ࣗݾ঺հ • ླ໦ ݈Ұ (Kenichi Suzuki) • [email protected] • github.com/knih • QUEΛ (http://logic.cs.tsukuba.ac.jp/~ken/quel/) • ϓϩάϥϜݴޠ࿦ɺΫΤϦɺFunctional DDD • BizReach • yamory (https://yamory.io)

The Death of Tagless Final

Pure repository trait UserRepository[F[_]] { def findOneById(userId: UserId): F[Option[User]] } class UserService[F[_]](userRepository: UserRepository[F]) (implicit ME: MonadError[F, Error]) { def getName(userId: String) = { val userOptF: F[Option[User]] = userRepository.findOneById(userId) Monad[F].map(userOptF) { userOpt => for { user <- userOpt } yield user.name } } }

Pure repository object UserRepositoryOnJdbc extends UserRepository[IO] { override def findOneById(userId: UserId): IO[Option[User]] = ??? } object UserController extends ApplicationController { implicit val userRepository: UserRepository[IO] = UserRepositoryOnJdbc val service = new UserService[IO] ... }

What is Tagless Final?

tagless-finalͱ͸ • ܕ҆શͳຒࠐΈDSLΛߏங͢Δख๏ • ܕ෇͚͕ϝλݴޠ (ϗετݴޠ) ͷͦΕʹؼண͢Δ • ܕݕࠪΞϧΰϦζϜΛ࣮૷͠ͳͯ͘Α͍ • HOAS (ߴ֊ந৅ߏจ) ͕࢖͑Δ • ϝλݴޠʹΑͬͯείʔϓ҆શੑ͕୲อ͞ΕΔ • ݴޠίϯϙʔωϯτͷ߹੒Λαϙʔτ • ύʔαΛ࡞Βͳͯ͘Α͍ • ϗετݴޠͷΤίγεςϜ͕࢖͑Δ [J. Carette, O. Kiselyov and C.-c. Shan. (2009)] [1]

ର৅ݴޠͱϝλݴޠ • ର৅ݴޠ (object language) • දݱ͍ͨ͠ݴޠɺDSL • ϝλݴޠ (meta language) • Scala, OCaml, Haskell౳ͷຒࠐΈઌϗετݴޠ

Symantics Symantics ΠϯλϑΣʔε Symantics ΠϯλϓϦλ ߏจ(Syntax) ͱ ܕنଇ Symantics ΠϯλϓϦλ Symantics ΠϯλϓϦλ ҙຯ࿦(Semantics)

ର৅ݴޠΛઃܭ͢Δ n is an integer n : Z 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 e1 : Z e2 : Z e1 + e2 : Z AAACqHichVHLSiNBFD326IwTdczoRnDTGJRBIVTHgZGsRDcu4yPGR6Sp7lRiY7/srgScJj/gD7hw5YCI+Blu3LkamHyCuHRgNi683WlQR9RbVNWtU+fcOlVl+LYVSsY6PcqH3r6Pn/o/ZwYGh74MZ7+OrIdeMzBF2fRsL9gweChsyxVlaUlbbPiB4I5hi4qxtxjvV1oiCC3PXZMHvthxeMO16pbJJUF6drFaqwfcjISuqUW16nC5axjRVlut7u83eU0VeqH4iLZj3kyMPaG29WyO5VkS6stES5Mc0ih52TNUUYMHE004EHAhKbfBEVLbhgYGn7AdRIQFlFnJvkAbGdI2iSWIwQndo7FBq+0UdWkd1wwTtUmn2NQDUqqYZL/ZObtjV+yC3bD7V2tFSY3YywHNRlcrfH34cGz137sqh2aJ3UfVm54l6phLvFrk3U+Q+BZmV9/6eXS3WlyZjKbYL3ZL/k9Yh13SDdzWX/N0WawcI0MfoP3/3C+T9UJem80Xlr/n5hfSr+jHOCbwjd77B+axhBLKdO45rvEHHWVaKSkVZbNLVXpSzSiehWI8AA4Yo7E= e ::= n | e + e 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 ߏจ (syntax) ܕ෇͚نଇ (typing rules) දݱ͍ͨ͠ର৅ݴޠ Symantics trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] }

trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] } ର৅ݴޠΛઃܭ͢Δ n is an integer n : Z 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 e1 : Z e2 : Z e1 + e2 : Z 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 e ::= n | e + e 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 ߏจ (syntax) ܕ෇͚نଇ (typing rules) දݱ͍ͨ͠ର৅ݴޠ Symantics

trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] } ର৅ݴޠΛઃܭ͢Δ n is an integer n : Z 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 e1 : Z e2 : Z e1 + e2 : Z 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 e ::= n | e + e 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 ߏจ (syntax) ܕ෇͚نଇ (typing rules) දݱ͍ͨ͠ର৅ݴޠ Symantics Repr

Symantics • syntaxͱtyping rulesΛؔ਺ͷγάχνϟͰද͢ • දݱܕ (representation type) Ͱܕ෇͚نଇΛ ද͢ • ߴ֊ΧΠϯυΛ࢖͏ • polymorphic lifting ͸࢖Θͳ͍ • semantics Λؔ਺ͷ࣮૷Ͱද͢

ϢʔβϓϩάϥϜ trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] } class Program[S <: AddSym](val sym: S) { import sym._ val ex1 = add(int(1), int(0)) } ΠϯλϓϦλʢ࣮૷ʣʹ ґଘ͠ͳ͍Ϣʔβϓϩά ϥϜ͕هड़Ͱ͖Δ

ΠϯλϓϦλ final case class R[A](unR: A) // type Id[A] = A object interp extends AddSym { type Repr[T] = R[T] def int(n: Int): R[Int] = R(n) def add(t1: R[Int], t2: R[Int]): R[Int] = R(t1.unR + t2.unR) } def eval[A](e: R[A]): A = e.unR trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] } ܕ͕อଘ͞ΕΔ ධՁؔ਺

ಈ͔͢ object test { val evaluator = Run.interp val p1 = new Program(evaluator) def main(args: Array[String]) = { println("---------------------------------") println(evaluator.eval(p1.ex1)) // => 1 }

HOASͰλந৅Λѻ͏ • HOAS (Higher-Order Abstract Syntax; ߴ֊ந৅ߏจ) • ม਺ଋറΛϗετݴޠͷλͰදݱ͢ΔςΫχοΫ • ม਺Λਖ਼֬ʹѻ͏ͷ͸ׂͱ໘౗ • ม਺໊͕ඃΒͳ͍Α͏ʹ͢Δ • de Bruijn ΠϯσοΫεΛ࢖͏৔߹Ͱ΋ɺ୅ೖૢ࡞࣌ͷΠ ϯσοΫεͷγϑτܭࢉ͸ඞཁ • capture-avoiding substitution [Miller and Nadathur (1987); Pfenning and Elliott. (1988)][2,3]

• ม਺ଋറΛϗετݴޠʹ·͔ͤΔ HOASͰλந৅Λѻ͏ trait LamSym {
 type Repr[T] def lam[A, B](f: Repr[A] => Repr[B]): Repr[A => B] def app[A, B](e1: Repr[A => B], e2: Repr[A]): Repr[B] } [x : ⌧1] . . . e : ⌧2 x.e : ⌧1 ! ⌧2 AAAC23ichVE7b9RAEB6bV3ACOaBBorE4B1Gd1gcS6KoIGso8uCTS7em09k2OVfa8lr22Eqyr6CLaKAUVSAghfgYNf4AiEn8AUQaJhoLxQyASBcbyzrffzDczuxvESqaGsSPLPnf+wsVLc5ed+YUrVxdb165vpDpLQuyHWulkKxApKhlh30ijcCtOUEwDhZvBzuMyvpljkkodPTV7MQ6nYhLJbRkKQ9SolfMAJzIqjAgyJZJZEc4cb7Db40ZkI3/ouZw7PB9rk5bIwzrQrXiPaypddi64opZj4e66HRd7bq0mp2vYnXkOx2j8u82o1WYdVpl7GvgNaENjK7r1DjiMQUMIGUwBIQJDWIGAlL4B+MAgJm4IBXEJIVnFEWbgkDajLKQMQewOrRPaDRo2on1ZM63UIXVR9CekdGGJfWbv2TH7xD6wr+znmbWKqkY5yx75oNZiPFrcv7n+47+qKXkDz/6o/jmzgW14WM0qafa4YspThLU+f354vN5bWyrusDfsG83/mh2xj3SCKP8evl3FtVfg0AP4J6/7NNjodvx7ne7q/fbyo+Yp5uAW3Ia7dN8PYBmewAr0qe8Xy7bmrQV7aL+w9+2XdaptNZob8JfZB78AP0qzGg== e1 : ⌧1 ! ⌧2 e2 : ⌧1 e1 e2 : ⌧2 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

• ม਺ଋറΛϗετݴޠʹ·͔ͤΔ HOASͰλந৅Λѻ͏ trait LamSym {
 type Repr[T] def lam[A, B](f: Repr[A] => Repr[B]): Repr[A => B] def app[A, B](e1: Repr[A => B], e2: Repr[A]): Repr[B] } [x : ⌧1] . . . e : ⌧2 x.e : ⌧1 ! ⌧2 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 e1 : ⌧1 ! ⌧2 e2 : ⌧1 e1 e2 : ⌧2 AAACqXichVFNSxtBGH5cW7WxamwvhV6WBosghNm0oKSXUHvwaLTRUFeW2ckkLm52193ZQAz5A/4BD55aKNL2Z3jx2IuH+A9Kjwq99NB3P6C0or7LzjzzvM/zzjszduA6kWJsNKaNP3g4MTn1qDD9eGZ2rjj/ZCvy41DIhvBdP2zaPJKu48mGcpQrm0Eoedd25ba9v5rkt3syjBzfe6/6gdzt8o7ntB3BFVFW8Z3ZaodcDKRl6FXdVDwmYCo/gxXdPDiIeUuXVqWaJYeJ1HyTELlmWLCKJVZmaeg3gZGDEvJY94unMNGCD4EYXUh4UIRdcET07cAAQ0DcLgbEhYScNC8xRIG8MakkKTix+zR2aLWTsx6tk5pR6ha0i0t/SE4dC+yCfWFX7Jx9Yz/Y71trDdIaSS99mu3MKwNr7ujZ5q97XV2aFfb+uu7sWaGNlbRXh3oPUiY5hcj8vcPjq83qxsLgJfvEflL/H9mIndEJvN61+FyXGydIHsD4/7pvgq1K2XhVrtRfl2pv86eYwnO8wCLd9zJqWMM6GrTvV3zHCJfaklbXmtqHTKqN5Z6n+Cc08QcZZKI2

HOASͰλந৅Λѻ͏ • ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ def lam[A, B](f: R[A] => R[B]): R[A => B] = R((x: A) => f(R(x)).unR) def app[A, B](e1: R[A => B], e2: R[A]): R[B] = R(e1.unR(e2.unR))

HOASͰλந৅Λѻ͏ • ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ def lam[A, B](f: R[A] => R[B]): R[A => B] = R((x: A) => f(R(x)).unR) def app[A, B](e1: R[A => B], e2: R[A]): R[B] = R(e1.unR(e2.unR))

HOASͰλந৅Λѻ͏ • ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ def lam[A, B](f: R[A] => R[B]): R[A => B] = R((x: A) => f(R(x)).unR) def app[A, B](e1: R[A => B], e2: R[A]): R[B] = R(e1.unR(e2.unR))

HOASͰλந৅Λѻ͏ • ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ def lam[A, B](f: R[A] => R[B]): R[A => B] = R((x: A) => f(R(x)).unR) def app[A, B](e1: R[A => B], e2: R[A]): R[B] = R(e1.unR(e2.unR))

HOASͰλந৅Λѻ͏ • ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ def lam[A, B](f: R[A] => R[B]): R[A => B] = R((x: A) => f(R(x)).unR) def app[A, B](e1: R[A => B], e2: R[A]): R[B] = R(e1.unR(e2.unR))

ݴޠίϯϙʔωϯτΛ߹੒ trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(e1: Repr[Int], e2: Repr[Int]): Repr[Int] } trait SymLam { type Repr[T] def lam[A, B](f: Repr[A] => Repr[B]): Repr[A => B] def app[A, B](e1: Repr[A => B], e2: Repr[A]): Repr[B] } trait Sym extends AddSym with SymLam

ݴޠίϯϙʔωϯτΛ߹੒ trait AddSym[Repr[_]] { def int(n: Int): Repr[Int] def add(e1: Repr[Int], e2: Repr[Int]): Repr[Int] } trait SymLam[Repr[_]] { def lam[A, B](f: Repr[A] => Repr[B]): Repr[A => B] def app[A, B](e1: Repr[A => B], e2: Repr[A]): Repr[B] } trait Sym[Repr[_]] extends AddSym[Repr] with SymLam[Repr]

tagless-final ↓ typed-final

ਁಁ͍ͯ͠ͳ͍ͷͰ΋͏Ұ౓

typed-final

ʢάάϥϏϦςΟΛߟ͍͑ͯͳ͔ͬͨʣ

ϓϩάϥϜม׵ • ୯Ґݩ • n + 0 = n ... ӈ୯Ґݩ • 0 + m = m ... ࠨ୯Ґݩ • tagless-final͸ؔ਺ʹΑͬͯݴޠΛߏ੒͢Δ • ؔ਺Ͱߏ੒͞ΕͨϓϩάϥϜΛͲͷΑ͏ʹม׵͢Δ͔ val program: R[Int] = add(int(1), int(0))

ϓϩάϥϜม׵ • tagless-final typed-final ͸1ͭͷΠϯλϑΣʔ ε(symantics)ʹରͯ͠ෳ਺ͷ࣮૷(ΠϯλϓϦ λ)Λ࣋ͭ͜ͱ͕Ͱ͖Δ

ϓϩάϥϜม׵ Symantics ΠϯλϑΣʔε Symantics ΠϯλϓϦλ Symantics ΠϯλϓϦλ Symantics ΠϯλϓϦλ

ϓϩάϥϜม׵ Symantics ΠϯλϑΣʔε Symantics ΠϯλϓϦλ Symantics ΠϯλϓϦλ

ϓϩάϥϜม׵ Symantics ΠϯλϑΣʔε ධՁΠϯλϓϦλ ม׵ΠϯλϓϦλ

ϓϩάϥϜม׵ • ධՁثΠϯλϓϦλ͸ͦͷ··࢖͍͍ͨ • ม׵ϩδοΫͱධՁϩδοΫΛ෼཭͍ͨ͠ • ͞Βʹม׵ޙʹධՁ͍ͨ͠ • ܕ͸อଘ͍ͨ͠

ϓϩάϥϜม׵ Symantics ΠϯλϑΣʔε ධՁΠϯλϓϦλ ม׵ΠϯλϓϦλ

ϓϩάϥϜม׵ ධՁΠϯλϓϦλ

typed-finalϓϩάϥϜม׵ R(2) R ղऍ mul(add(int(2), int(0)),int(1)) ߲ ධՁ݁Ռ [Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

typed-finalϓϩάϥϜม׵ R(2) R ղऍ mul(int(2),in t(1)) Mul(Int(2), Int(1)) AddIdent(R) AddIdent
 ϑΝϯΫλ mul(add(int(2), int(0)),int(1)) ߲ ධՁ݁Ռ [Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

typed-finalϓϩάϥϜม׵ R(2) R ղऍ mul(int(2),in t(1)) Mul(Int(2), Int(1)) AddIdent(R) reflect reify AddIdent
 ϑΝϯΫλ mul(add(int(2), int(0)),int(1)) Ճ๏୯Ґݩنଇ ("%5 ߲ ධՁ݁Ռ [Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) Int(2) AddIdent(R) reflect reify AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ ("%5 ߲ ධՁ݁Ռ [Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

Normalization by Evaluation • reification • ҙຯΦϒδΣΫτ͔Β߲ͷਖ਼نతදݱΛநग़͢Δ ૢ࡞ • reflection • ߲ͷਖ਼نతදݱ͔ΒҙຯΦϒδΣΫτ΁ͷม׵ • reifyͷٯؔ਺ • ਖ਼نԽ • [Per Martin-Löf. (1975); Danvy and Filinski. (1990); Dybjer and Filinski. (2000)][5,6,7] etc

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify ؍ଌ PCTFSWBUJPO

GADT (Generalized Algebraic Data Types) • දݱܕΛGADTͰߏ੒ͨ͠ܕʹ͢Δ sealed trait Term[T] final case class Unknown[A](x: sym.Repr[A]) extends Term[A] final case class IntLit(e: Int) extends Term[Int] final case class Add(e1: Term[Int], e2: Term[Int]) extends Term[Int] type Repr[T] = Term[T] trait AddSym { type Repr[T] def int(n: Int): Repr[Int] def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int] }

දݱܕͷม׵ trait RR { import cats.~> type From[_] type To[_] // Term def fwd: From ~> To // reflection def bwd: To ~> From // reification def map[A, B](f: From[A] => From[B]): To[A] => To[B] = (t: To[A]) => fwd(f(bwd(t))) def map2[A, B, C](f: (From[A], From[B]) => From[C]): (To[A], To[B]) => To[C] = (t1: To[A], t2: To[B]) => fwd(f(bwd(t1), bwd(t2))) } R.Repr AddIdent(R).Repr MulIdent(AddIdent(R)) .Repr From To From To

Reification and Reflection // AdditiveIdentity class AddIdent[FS <: AddMulSym](val sym: FS) { sealed trait Term[T] final case class Unknown[A](x: sym.Repr[A]) extends Term[A] final case class IntLit(e: Int) extends Term[Int] final case class Add(e1: Term[Int], e2: Term[Int]) extends Term[Int] object AddRR extends RR { override type From[T] = sym.Repr[T] override type To[T] = Term[T] def fwd: From ~> To = new (From ~> To) { def apply[A](x: sym.Repr[A]): Term[A] = Unknown(x) } def bwd: To ~> From = new (To ~> From) { def apply[A](term: Term[A]): sym.Repr[A] = term match { case Unknown(x) => x case IntLit(n) => sym.int(n) case Add(e1, e2) => sym.add(bwd(e1), bwd(e2)) } } } ...


Reification and Reflection // AdditiveIdentity class AddIdent[FS <: AddMulSym](val sym: FS) { sealed trait Term[T] final case class Unknown[A](x: sym.Repr[A]) extends Term[A] final case class IntLit(e: Int) extends Term[Int] final case class Add(e1: Term[Int], e2: Term[Int]) extends Term[Int] object AddRR extends RR { override type From[T] = sym.Repr[T] override type To[T] = Term[T] def fwd: From ~> To = new (From ~> To) { def apply[A](x: sym.Repr[A]): Term[A] = Unknown(x) } def bwd: To ~> From = new (To ~> From) { def apply[A](term: Term[A]): sym.Repr[A] = term match { case Unknown(x) => x case IntLit(n) => sym.int(n) case Add(e1, e2) => sym.add(bwd(e1), bwd(e2)) } } } ...


Reification and Reflection // AdditiveIdentity class AddIdent[FS <: AddMulSym](val sym: FS) { sealed trait Term[T] final case class Unknown[A](x: sym.Repr[A]) extends Term[A] final case class IntLit(e: Int) extends Term[Int] final case class Add(e1: Term[Int], e2: Term[Int]) extends Term[Int] object AddRR extends RR { override type From[T] = sym.Repr[T] override type To[T] = Term[T] def fwd: From ~> To = new (From ~> To) { def apply[A](x: sym.Repr[A]): Term[A] = Unknown(x) } def bwd: To ~> From = new (To ~> From) { def apply[A](term: Term[A]): sym.Repr[A] = term match { case Unknown(x) => x case IntLit(n) => sym.int(n) case Add(e1, e2) => sym.add(bwd(e1), bwd(e2)) } } } ...
 R.Repr AddIdent(R).Repr MulIdent(AddIdent(R)) .Repr From To From To

GADTͱNbEͰܕอଘͭͭ͠ม׵ • GADTʹΑͬͯSymanticsͷܕʹै͏ class AddIdent[FS <: AddMulSym](val sym: FS) { ... val addIdentInterp = new GenericTrans(AddRR)(sym) { def add(t1: Term[Int], t2: Term[Int]): Term[Int] = (t1, t2) match { case (IntLit(0), _) => t2 case (_, IntLit(0)) => t1 case (_, _) => Add(t1, t2) } } } ࠨ୯Ґݩ ӈ୯Ґݩ

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify ؍ଌ PCTFSWBUJPO

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify ؍ଌ PCTFSWBUJPO ͦΕͧΕͷϞδϡʔϧ͸ 4ZNBOUJDTΛ९कͯ͠࿈࠯͢Δ

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify ؍ଌ PCTFSWBUJPO

typed-finalϓϩάϥϜม׵ R(2) R ղऍ int(2) ߲ ධՁ݁Ռ Int(2) AddIdent(R) reflect AddIdent
 ϑΝϯΫλ add(int(2), int(0) Ճ๏୯Ґݩنଇ mul(add(int(2), int(0)),int(1)) Add(Int(2),Int(0)) MulIdent(AddIdent(R)) ... ... ... MulIdent
 ϑΝϯΫλ ৐๏୯Ґݩنଇ reify reify ؍ଌ PCTFSWBUJPO ίʔυੜ੒ثΛ༩͑Δ ͜ͱ΋Ͱ͖Δ

Finally, Tagless ҙ༁: ࠷ޙʹλά͕ͳ͚Ε͹͍͍ΑͶ

typed-final

[Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

·ͱΊ

·ͱΊ • typed-final͸DSLΛߏங͢ΔΞϓϩʔν • දݱܕreprͰର৅֎ͷߏ੒ࢠ͕ೖΓࠐ·ͳ͍ Α͏ʹ͢Δ • GADTͱNbEతΞϓϩʔνΛซ༻͢Δ͜ͱͰɺ ܕΛอଘ͠ͳ͕Βfinal styleͰϓϩάϥϜม׵ • tagless-final͸typed-finalʹͳͬͨ

ࢀߟจݙ • [1] Jacques Carette, Oleg Kiselyov, and Chung-chieh Shan. 2009. Finally tagless, partially evaluated: Tagless staged interpreters for simpler typed languages. J. Funct. Program. 19, 5 (September 2009), 509-543. • [2] Miller, Dale & Nadathur, Gopalan (1987) A logic programming approach to manipulating formulas and programs. In IEEE Symposium on Logic Programming, Seif Haridi (ed). Washington, DC: IEEE Comp. Society Press, pp. 379–388. • [3] Pfenning, Frank & Elliott, Conal (1988) Higher-order abstract syntax. In PLDI 88: Proceedings of the ACM Conference on Programming Language Design and Implementation. ACM SIGPLAN Notices, vol. 23(7). New York: ACM, pp. 199–208. • [4] Kenichi Suzuki, Oleg Kiselyov, and Yukiyoshi Kameyama. 2016. Finally, safely-extensible and efficient language-integrated query. PEPM '16. ACM, New York, NY, USA, 37-48. • [5] Per Martin-Löf. About models for intuitionistic type theories and the notion of definitional equality. In Proceedings of the Third Scandinavian Logic Symposium (1972), volume 82 of Studies in Logic and the Foundation of Mathematics, pages 81–109. North-Holland, 1975. • [6] Olivier Danvy and Andrzej Filinski. Abstracting control. In Mitchell Wand, editor, Proceedings of the 1990 ACM Conference on Lisp and Functional Programming, pages 151–160, Nice, France, June 1990. ACM Press • [7] Peter Dybjer and Andrzej Filinski. Normalization and partial evaluation. In Gilles Barthe, Peter Dybjer, Luís Pinto, and João Saraiva, editors, Applied Semantics – Advanced Lectures, number 2395 in Lecture Notes in Computer Science, pages 137–192, Caminha, Portugal, September 2000. Springer-Verlag.