Copyright  (C) 2012 Edward Kmett 

License  BSDstyle (see the file LICENSE) 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Stability  experimental 
Portability  portable 
Safe Haskell  SafeInferred 
Language  Haskell98 
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
)
instanceEq1
Exp instanceOrd1
Exp instanceShow1
Exp instanceRead1
Exp instanceApplicative
Exp wherepure
= V; (<*>
) =ap
instanceMonad
Exp wherereturn
= 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 reexported
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.standrews.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
 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 "λcalculus cooked four ways" available from http://www.augustsson.net/Darcs/Lambda/top.pdf
 Derived.hs shows how much of the API can be automated with DeriveTraversable and adds combinators for building binders that support pattern matching.
 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.
 substitute :: (Monad f, Eq a) => a > f a > f a > f a
 isClosed :: Foldable f => f a > Bool
 closed :: Traversable f => f a > Maybe (f b)
 newtype Scope b f a = Scope {}
 abstract :: Monad f => (a > Maybe b) > f a > Scope b f a
 abstract1 :: (Monad f, Eq a) => a > f a > Scope () f a
 instantiate :: Monad f => (b > f a) > Scope b f a > f a
 instantiate1 :: Monad f => f a > Scope n f a > f a
 class Bound t where
 (=<<<) :: (Bound t, Monad f) => (a > f c) > t f a > t f c
 data Var b a
 fromScope :: Monad f => Scope b f a > f (Var b a)
 toScope :: Monad f => f (Var b a) > Scope b f a
Manipulating user terms
substitute :: (Monad f, Eq a) => a > f a > f a > f a Source
replaces the free variable substitute
a p wa
with p
in w
.
>>>
substitute "hello" ["goodnight","Gracie"] ["hello","!!!"]
["goodnight","Gracie","!!!"]
isClosed :: Foldable f => f a > Bool Source
A closed term has no free variables.
>>>
isClosed []
True
>>>
isClosed [1,2,3]
False
closed :: Traversable f => f a > Maybe (f b) Source
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
is an Scope
b f af
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
.
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)  
(Functor f, Monad f) => Applicative (Scope b f)  
Foldable f => Foldable (Scope b f) 

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)  
Typeable (* > (* > *) > * > *) Scope  
(Monad f, Eq b, Eq1 f, Eq a) => Eq (Scope b f a)  
(Typeable * b, Typeable (* > *) 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 Source
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 Source
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 Source
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 Source
Enter a Scope
that binds one variable, instantiating it
>>>
instantiate1 "x" $ Scope [B (),F "y",F "z"]
"xyz"
Structures permitting substitution
Instances of Bound
generate left modules over monads.
This means they should satisfy the following laws:
m>>>=
return
≡ m m>>>=
(λ x → k x>>=
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
implies the above laws, and is in fact the default definition.>>>=
f ≡ m >>=
lift
.
f
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.
Nothing
Conversion to Traditional de Bruijn
"I am not a number, I am a free monad!"
A
is a variable that may either be "bound" (Var
b aB
) or "free" (F
).
(It is also technically a free monad in the same neartrivial sense as
Either
.)
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)  
Typeable (* > * > *) Var  
type Rep (Var b a) 