Write You a Haskell Stephen Diehl

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• Call-by-value • Call-by-need Higher Order Interpreters • HOAS • PHOAS • Embedding IO 78
ere is nothing more practical than a good theory. — James C. Maxwell Hindley-Milner Inference e Hindley-Milner type system ( also referred to as Damas-Hindley-Milner or HM ) is a family of type systems that admit the serendipitous property of having a tractable algorithm for determining types from untyped syntax. is is achieved by a process known as unification, whereby the types for a wellstructured program give rise to a set of constraints that when solved always have a unique principal type. e simplest Hindley Milner type system is defined by a very short set of rules. e first four rules describe the judgements by which we can map each syntactic construct (Lam, App, Var, Let) to their expected types. We’ll elaborate on these rules shortly. x : σ ∈ Γ Γ ⊢ x : σ Γ ⊢ e 1 : τ 1 → τ 2 Γ ⊢ e 2 : τ 1 Γ ⊢ e 1 e 2 : τ 2 Γ, x : τ 1 ⊢ e : τ 2 Γ ⊢ λ x . e : τ 1 → τ 2 Γ ⊢ e 1 : σ Γ, x : σ ⊢ e 2 : τ Γ ⊢ let x = e 1 in e 2 : τ Γ ⊢ e : σ α /∈ ftv(Γ) Γ ⊢ e : ∀ α . σ Γ ⊢ e : σ 1 σ 1 ⊑ σ 2 Γ ⊢ e : σ 2 (T-Var) (T-App) (T-Lam) (T-Let) (T-Gen) (T-Inst) Milner’s observation was that since the typing rules map uniquely onto syntax, we can in effect run the typing rules “backwards” and whenever we don’t have a known type for an subexpression, we “guess” by putting a fresh variable in its place, collecting constraints about its usage induced by subsequent typing judgements. is is the essence of type inference in the ML family of languages, that by the generation and solving of a class of unification problems we can reconstruct the types uniquely from the syntax. e algorithm itself is largely just the structured use of a unification solver. 79
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Write You a Haskell Building a mode
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Contents Introduction 6 Goals . . .
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Evaluation . . . . . . . . . . . .
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Syntax Errors . . . . . . . . . . .
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• containers • unordered-contai
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File ””, line 1, in TypeError:
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] TokenSym ”x”, TokenAdd, Token
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f: mov res, arg add res, 1 ret defi
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Datatypes Constructors for datatype
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length :: [a] -> Int length [] = 0
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Higher-Kinded Types e “type of ty
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eturn a >>= f = f a Law 2 m >>= ret
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data PlatonicSolid = Tetrahedron |
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foo :: Stack () foo = Stack $ do pu
Page 29 and 30: , bar :: Int } deriving (Show) comp
Page 31 and 32: class IsString a where fromString :
Page 33 and 34: Parsing Parser Combinators For pars
Page 35 and 36: instance Monad Parser where return
Page 37 and 38: natural :: Parser Integer natural =
Page 39 and 40: print $ eval $ run a Now we can try
Page 41 and 42: module Syntax where data Expr = Tr
Page 43 and 44: cannot proceed because the reductio
Page 45 and 46: later into native CPU instructions
Page 47 and 48: λx.e ere are several syntactical c
Page 49 and 50: type Name = String data Expr = Var
Page 51 and 52: Eta-expansion (λx.ex)e ′ β →
Page 53 and 54: Y = SSK(S(K(SS(S(SSK))))K) In a unt
Page 55 and 56: parensIf :: Bool -> Doc -> Doc pare
Page 57 and 58: In this notation, the expression ab
Page 59 and 60: In all the languages which we will
Page 61 and 62: eval1 expr = case expr of Succ t ->
Page 63 and 64: TNat -> return TNat _ -> throwError
Page 65 and 66: • T-Var Variables are simply pull
Page 67 and 68: t1 throwError $ Mismatch t2 a ty -
Page 69 and 70: Full Source • Typed Arithmetic
Page 71 and 72: 1. Evaluate f to λy.e 2. Evaluate
Page 73 and 74: e 1 → e ′ 1 e 1 e 2 → e ′ 1
Page 75 and 76: case we will use a GADT to embed a
Page 77 and 78: AppP (AppP (VarP f) (VarP x)) (AppP
Page 79: e prim function will simply perform
Page 83 and 84: As before let rec expressions will
Page 85 and 86: Γ, x : τ = (Γ\x) ∪ {x : τ} Op
Page 87 and 88: where s’ = foldr Map.delete s as
Page 89 and 90: s2 Type -> Infer Subst bind a t |
Page 91 and 92: infer :: TypeEnv -> Expr -> Infer (
Page 93 and 94: App e1 e2 -> do tv
Page 95 and 96: -- | Unify two types uni :: Type ->
Page 97 and 98: unifies t (TVar v) = v ‘bind‘ t
Page 99 and 100: Interpreter Our evaluator will oper
Page 101 and 102: Our language can be compiled into a
Page 103 and 104: exec True contents -- :type command
Page 105 and 106: let rec fib n = if (n == 0) then 0
Page 107 and 108: • Higher order functions • Para
Page 109 and 110: • e PHOAS, the type-erased Core i
Page 111 and 112: If the ProtoHaskell compiler is inv
Page 113 and 114: infixl 6 + infixl 7 * f = 1 + 2 * 3
Page 115 and 116: -- Desugared f x = case x of Just _
Page 117 and 118: data Kind = KStar | KArr Kind Kind
Page 119 and 120: TypeError) -- | Inference state dat
Page 121 and 122: e expression or Expr type is the co
Page 123 and 124: data qname [var] where [tydecl] dat
Page 125 and 126: Typeclass Declarations Typeclass de
Page 127 and 128: Syntax Name Kind Description (,) Pa
Page 129 and 130: So for instance if we have three AS
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-- **Begin Haskell Syntax** { {-# O
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tokens :- -- Whitespace insensitive
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Expr : let VAR ’=’ Expr in Expr
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data Type = TVar Loc TVar | TCon Lo
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-- Start of layout ( Column: 0 ) fi
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Extensible Operators Haskell famous
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fixityspec :: Parser FixitySpec fix
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Write You a Haskell Stephen Diehl

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