Vinilla Lang

Vanilla is a pure functional programming language based on System F, a classic but powerful type system.

Merits

Simple as it is, Vanilla contains many features that most main-stream languages don't have:

Higher-rank polymorphism
- It allows using polymorphic functions as arguments of higher-order functions
Strong type inference
- Only polymorphic and recursive bindings need annotations
Algebraic data types
Simplicity
- Only foralls∀ and arrows→ are built-in types
- All regular types such as Unit and Bool can be declared as ADT and eliminated by case expression

Defects

No module system
For simplicity, this programming language only supports type checking and evaluation on closed terms.

Grammar

Types           A, B, C   ::= T A1...An | a | ∀a.A | A → B
Monotypes       τ, σ      ::= T τ1...τn | a | τ → σ

Data Type       T         ::= T
Constructor     K         ::= K
Declaration     D         ::= data T a1...at = K1 τ1...τm | ... | Kn σ1...σn .
Pattern         p         ::= K x1...xn

Expressions     e, f      ::=   x                                     -- variable
                              | K                                     -- constructor
                              | case e of {pi → ei}                   -- case
                              | λx.e                                  -- implicit λ
                              | λx : A.e                              -- annotated λ
                              | e1 e2                                 -- application
                              | e : A                                 -- annotation
                              | let x = e1 in e2                      -- let binding
                              | let x : A = e1 in e2                  -- annotated let binding
                              | let rec f : A = e1 in e2              -- recursive binding (sugar)
                              | fix e                                 -- fixpoint
                              | e @ A                                 -- type application

Program         P         ::= D P | e

A valid Vanilla program consists of several ADT declarations followed by a main expression.

ADT declarations are similar to ones in Haskell, except that they end with a dot for easy parsing.

Haskell-style comments (-- and {- -}) are also supported.

Usage

First of all, ghc and cabal-install should be installed in your $PATH

Examples

Higher-rank Polymorphism

Given example/cont.vn:

data Unit = Unit.

-- Can you belive that
-- type `a` and `∀r. ((a → r) → r)` are isomorphic?

let cont : ∀a. a → ∀r. ((a → r) → r) =
  λ x . λ callback . callback x
in

let runCont : ∀a. (∀r. (a → r) → r) → a =
  λ f . (let callback = λ x . x in f callback)
in

-- should output Unit
runCont (cont Unit)

Run

cabal run example/cont.vn

It should output:

Type:
Unit

Result:
Unit

Map for Lists

Given example/map.vn:

data Bool = False | True.
data List a = Nil | Cons a (List a).


let rec map : ∀a. ∀b. (a → b) → (List a) → (List b) =
  λ f. λ xs. case xs of {
      Nil → Nil,
      Cons y ys → Cons (f y) (map f ys)
    }
in

let not : Bool → Bool =
  λb. case b of {False → True, True → False}
in

let xs = Cons True (Cons False Nil) in

map not xs

Run

$ cabal run example/map.vn

It should output:

Type:
List Bool

Result:
Cons False (Cons True Nil)

Add operator for Natural Numbers

Given example/add.vn:

data Nat = Zero | Succ Nat.
data Prod a b = Prod a b.

-- add can be defined by `fixpoint`
let add : Nat → Nat → Nat =
  fix (λf. λx : Nat . λy : Nat. case x of {Zero → y, Succ a → Succ (f a y)})
in

-- or `let rec`
let rec add2 : Nat → Nat → Nat =
  λ x . λ y . case x of {Zero → y, Succ a → Succ (add2 a y)}
in

let two = Succ (Succ Zero) in
let three = Succ two in

-- 3 + 2 = 5
Prod (add three two) (add2 three two)

Run

$ cabal run example/add.vn

It should output the inferred type and evaluated value of this program:

Type:
Prod Nat Nat

Result:
Prod (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero)))))

Mutual Recursion

Mutual recursive functions can be easily defined with the fixpoint and projections.

Given example/evenodd.vn:

data Nat = Zero | Succ Nat.
data Bool = False | True.
data Prod a b = Prod a b.

let evenodd : Prod (Nat → Bool) (Nat → Bool) =
  fix (λ eo .
    let e = λ n : Nat . case n of { Zero → True, Succ x → (case eo of {Prod e o → o}) x } in
    let o = λ n : Nat . case n of { Zero → False, Succ x → (case eo of {Prod e o → e}) x } in
    (Prod e o))
in

let even = (case evenodd of {Prod e o → e}) in
let odd = (case evenodd of {Prod e o → o}) in

let five = Succ(Succ(Succ(Succ(Succ(Zero))))) in

Prod (even five) (odd five)

It reports:

Type:
Prod Bool Bool

Result:
Prod False True

Ill-typed Program

Given example/illtypedid.vn:

let id : Nat → Nat =
  (λx . x)
in
  id ()

It reports typechecking error:

Typecheck error:
cannot establish subtyping with Unit <: Nat

Unit tests

$ cabal test
...
Running 1 test suites...
Test suite vanilla-test: RUNNING...
Test suite vanilla-test: PASS
...

zehaochen19 / vanilla-lang