Bound

We represent the target language itself as an ideal monad supplied by the user, and provide a Scope monad transformer for introducing bound variables in user supplied terms. Users supply a Monad and Traversable instance, and we traverse to find free variables, and use the Monad to perform substitution that avoids bound variables.

An untyped lambda calculus:

 {-# LANGUAGE DeriveFunctor, DeriveFoldable, DeriveTraversable #-}
 import Bound
 import Control.Applicative
 import Control.Monad (ap)
 import Prelude.Extras
 import Data.Foldable
 import Data.Traversable

 infixl 9 :@
 data Exp a = V a | Exp a :@ Exp a | Lam (Scope () Exp a)
   deriving (Eq,Ord,Show,Read,Functor,Foldable,Traversable)

 instance Eq1 Exp
 instance Ord1 Exp
 instance Show1 Exp
 instance Read1 Exp
 instance Applicative Exp where pure = V; (<*>) = ap

 instance Monad Exp where
   return = V
   V a      >>= f = f a
   (x :@ y) >>= f = (x >>= f) :@ (y >>= f)
   Lam e    >>= f = Lam (e >>>= f)

 lam :: Eq a => a -> Exp a -> Exp a
 lam v b = Lam (abstract1 v b)

 whnf :: Exp a -> Exp a
 whnf (f :@ a) = case whnf f of
   Lam b -> whnf (instantiate1 a b)
   f'    -> f' :@ a
 whnf e = e

More exotic combinators for manipulating a Scope can be imported from Bound.Scope.

You can also retain names in your bound variables by using Name and the related combinators from Bound.Name. They are not re-exported from this module by default.

The approach used in this package was first elaborated upon by Richard Bird and Ross Patterson in "de Bruijn notation as a nested data type", available from http://www.cs.uwyo.edu/~jlc/courses/5000_fall_08/debruijn_as_nested_datatype.pdf

However, the combinators they used required higher rank types. Here we demonstrate that the higher rank gfold combinator they used isn't necessary to build the monad and use a monad transformer to encapsulate the novel recursion pattern in their generalized de Bruijn representation. It is named Scope to match up with the terminology and usage pattern from Conor McBride and James McKinna's "I am not a number: I am a free variable", available from http://www.cs.st-andrews.ac.uk/~james/RESEARCH/notanum.pdf, but since the set of variables is visible in the type, we can provide stronger type safety guarantees.

There are longer examples in the examples/ folder:

https://github.com/ekmett/bound/tree/master/examples

1. Simple.hs provides an untyped lambda calculus with recursive let bindings and includes an evaluator for the untyped lambda calculus and a longer example taken from Lennart Augustsson's λ-calculus cooked four ways available from http://www.augustsson.net/Darcs/Lambda/top.pdf
2. Derived.hs shows how much of the API can be automated with DeriveTraversable and adds combinators for building binders that support pattern matching.
3. Overkill.hs provides very strongly typed pattern matching many modern language extensions, including polymorphic kinds to ensure type safety. In general, the approach taken by Derived seems to deliver a better power to weight ratio.

Manipulating user terms

substitute :: (Monad f, Eq a) => a -> f a -> f a -> f a

substitute a p w replaces the free variable a with p in w.

>>> substitute "hello" ["goodnight","Gracie"] ["hello","!!!"]
["goodnight","Gracie","!!!"]

isClosed :: Foldable f => f a -> Bool

A closed term has no free variables.

>>> isClosed []
True

>>> isClosed [1,2,3]
False

closed :: Traversable f => f a -> Maybe (f b)

If a term has no free variables, you can freely change the type of free variables it is parameterized on.

>>> closed [12]
Nothing

>>> closed ""
Just []

>>> :t closed ""
closed "" :: Maybe [b]

Scopes introduce bound variables

newtype Scope b f a

Scope b f a is an f expression with bound variables in b, and free variables in a

We store bound variables as their generalized de Bruijn representation in that we're allowed to lift (using F) an entire tree rather than only succ individual variables, but we're still only allowed to do so once per Scope. Weakening trees permits O(1) weakening and permits more sharing opportunities. Here the deBruijn 0 is represented by the B constructor of Var, while the de Bruijn succ (which may be applied to an entire tree!) is handled by F.

NB: equality and comparison quotient out the distinct F placements allowed by the generalized de Bruijn representation and return the same result as a traditional de Bruijn representation would.

Logically you can think of this as if the shape were the traditional f (Var b a), but the extra f a inside permits us a cheaper lift.

Constructors

Scope

Fields

unscope :: f (Var b (f a))

Instances

MonadTrans (Scope b)

Bound (Scope b)

Monad f => Monad (Scope b f)

The monad permits substitution on free variables, while preserving bound variables

Functor f => Functor (Scope b f)

(Typeable b, Typeable1 f) => Typeable1 (Scope b f)

(Functor f, Monad f) => Applicative (Scope b f)

Foldable f => Foldable (Scope b f)

toList is provides a list (with duplicates) of the free variables

Traversable f => Traversable (Scope b f)

(Serial b, Serial1 f) => Serial1 (Scope b f)

(Hashable b, Monad f, Hashable1 f) => Hashable1 (Scope b f)

(Monad f, Eq b, Eq1 f) => Eq1 (Scope b f)

(Monad f, Ord b, Ord1 f) => Ord1 (Scope b f)

(Functor f, Show b, Show1 f) => Show1 (Scope b f)

(Functor f, Read b, Read1 f) => Read1 (Scope b f)

(Monad f, Eq b, Eq1 f, Eq a) => Eq (Scope b f a)

(Typeable b, Typeable1 f, Data a, Data (f (Var b (f a)))) => Data (Scope b f a)

(Monad f, Ord b, Ord1 f, Ord a) => Ord (Scope b f a)

(Functor f, Read b, Read1 f, Read a) => Read (Scope b f a)

(Functor f, Show b, Show1 f, Show a) => Show (Scope b f a)

(Binary b, Serial1 f, Binary a) => Binary (Scope b f a)

(Serial b, Serial1 f, Serial a) => Serial (Scope b f a)

(Serialize b, Serial1 f, Serialize a) => Serialize (Scope b f a)

(Hashable b, Monad f, Hashable1 f, Hashable a) => Hashable (Scope b f a)

Abstraction over bound variables

abstract :: Monad f => (a -> Maybe b) -> f a -> Scope b f a

Capture some free variables in an expression to yield a Scope with bound variables in b

>>> :m + Data.List
>>> abstract (`elemIndex` "bar") "barry"
Scope [B 0,B 1,B 2,B 2,F "y"]

abstract1 :: (Monad f, Eq a) => a -> f a -> Scope () f a

Abstract over a single variable

>>> abstract1 'x' "xyz"
Scope [B (),F "y",F "z"]

Instantiation of bound variables

instantiate :: Monad f => (b -> f a) -> Scope b f a -> f a

Enter a scope, instantiating all bound variables

>>> :m + Data.List
>>> instantiate (\x -> [toEnum (97 + x)]) $ abstract (`elemIndex` "bar") "barry"
"abccy"

instantiate1 :: Monad f => f a -> Scope n f a -> f a

Enter a Scope that binds one variable, instantiating it

>>> instantiate1 "x" $ Scope [B (),F "y",F "z"]
"xyz"

Structures permitting substitution

class Bound t where

Instances of Bound generate left modules over monads.

This means they should satisfy the following laws:

 m >>>= return ≡ m
 m >>>= (λ x → k >>= h) ≡ (m >>>= k) >>>= h

This guarantees that a typical Monad instance for an expression type where Bound instances appear will satisfy the Monad laws (see doc/BoundLaws.hs).

If instances of Bound are monad transformers, then m >>>= f ≡ m >>= lift . f implies the above laws, and is in fact the default definition.

This is useful for types like expression lists, case alternatives, schemas, etc. that may not be expressions in their own right, but often contain expressions.

Methods

(>>>=) :: Monad f => t f a -> (a -> f c) -> t f c

Perform substitution

If t is an instance of MonadTrans and you are compiling on GHC >= 7.4, then this gets the default definition:

m >>>= f = m >>= lift . f

Instances

Bound IdentityT

Bound ListT

Bound MaybeT

Bound (ContT c)

Error e => Bound (ErrorT e)

Bound (ReaderT r)

Bound (StateT s)

Monoid w => Bound (WriterT w)

Bound (Scope b)

Bound (Scope b)

Monoid w => Bound (RWST r w s)

(=<<<) :: (Bound t, Monad f) => (a -> f c) -> t f a -> t f c

A flipped version of (>>>=).

(=<<<) = flip (>>>=)

Conversion to Traditional de Bruijn

data Var b a

"I am not a number, I am a free monad!"

A Var b a is a variable that may either be "bound" (B) or "free" (F).

(It is also technically a free monad in the same near-trivial sense as Either.)

Constructors

B b

this is a bound variable

F a

this is a free variable

Instances

Typeable2 Var

Bitraversable Var

Bifunctor Var

Bifoldable Var

Serial2 Var

Hashable2 Var

Eq2 Var

Ord2 Var

Show2 Var

Read2 Var

Monad (Var b)

Functor (Var b)

Applicative (Var b)

Foldable (Var b)

Traversable (Var b)

Serial b => Serial1 (Var b)

Hashable b => Hashable1 (Var b)

Eq b => Eq1 (Var b)

Ord b => Ord1 (Var b)

Show b => Show1 (Var b)

Read b => Read1 (Var b)

(Eq b, Eq a) => Eq (Var b a)

(Data b, Data a) => Data (Var b a)

(Ord b, Ord a) => Ord (Var b a)

(Read b, Read a) => Read (Var b a)

(Show b, Show a) => Show (Var b a)

Generic (Var b a)

(Binary b, Binary a) => Binary (Var b a)

(Serial b, Serial a) => Serial (Var b a)

(Serialize b, Serialize a) => Serialize (Var b a)

(Hashable b, Hashable a) => Hashable (Var b a)

fromScope :: Monad f => Scope b f a -> f (Var b a)

fromScope quotients out the possible placements of F in Scope by distributing them all to the leaves. This yields a more traditional de Bruijn indexing scheme for bound variables.

Since,

fromScope . toScope ≡ id

we know that

fromScope . toScope . fromScope ≡ fromScope

and therefore (toScope . fromScope) is idempotent.

toScope :: Monad f => f (Var b a) -> Scope b f a

Convert from traditional de Bruijn to generalized de Bruijn indices.

This requires a full tree traversal