Posted on October 12, 2016

This is written in Literate Haskell.

{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE InstanceSigs #-}

module Bidirectional.Serialization where

import Control.Applicative

import Data.Bool (bool)
import Data.Binary (Binary(..), Get)
import Data.Binary.Get (runGet)
import Data.Binary.Put (runPut, PutM)
import Data.ByteString.Lazy (ByteString)
import Data.Profunctor

-- Intentional synonyms of undefined.

(...) :: omittedForBrevity
(...) = (...)

(???) :: can'tSolveThis
(???) = (???)

Recently I came across the codec_ package. It is a library to write a serializer and a deserializer as a single bidirectional artefact.

It extends the functional pearl Pickler Combinators, an earlier elementary solution by Andrew J. Kennedy (2004).

I’ve been trying to push further the ideas of these two.

The basics: Pickler Combinators

In this section I summarize the Pickler Combinators paper.

The UP (un)pickler type (“PU” in the pearl) consists of an Unpickler (deserializer) and a Pickler (serializer). The two components are parameterized over contexts r and w for unpickling (reading) and pickling (writing) respectively.

data UP r w a = UP
{ unpickle :: r a
, pickle :: a -> w ()
}

The types Get and PutM from the binary package are examples of contexts for UP. The Binary typeclass implies an (un)pickler for every instance.

type BinaryUP = UP Get PutM

binaryUP :: Binary a => BinaryUP a
binaryUP = UP get put

-- Deserialize
-- > runGet :: Get a -> ByteString -> a
binaryUnpickle :: BinaryUP a -> ByteString -> a
binaryUnpickle = runGet . unpickle

-- Serialize
-- > runPut :: PutM () -> ByteString
binaryPickle :: BinaryUP a -> a -> ByteString
binaryPickle up = runPut . pickle up

There are combinators, to (un)pickle products by concatenation:

pairUP
:: (Applicative r, Applicative w)
=> UP r w a -> UP r w b -> UP r w (a, b)
pairUP aUP bUP = UP
{ unpickle = liftA2 (,) (unpickle aUP) (unpickle bUP)
, pickle = \(a, b) -> pickle aUP a *> pickle bUP b
}

-- Infix synonym.
(>|)
:: (Applicative r, Applicative w)
=> UP r w a -> UP r w b -> UP r w (a, b)
(>|) = pairUP

To (un)pickle sums:

altUP
:: Alternative r
=> UP r w a -> UP r w b -> UP r w (Either a b)
altUP aUP bUP = UP
{ unpickle =
Left <$> unpickle aUP <|> Right <$> unpickle bUP
, pickle = either (pickle aUP) (pickle bUP)
}

The one above assumes that a and b have disjoint picklings, so that they can be distinguished by an unpickler failing. A more straightforward way to pickle sums is to precede their picklings with a tag:

eitherUP
=> UP r w Bool -> UP r w a -> UP r w b
-> UP r w (Either a b)
eitherUP boolUP aUP bUP = UP
{ unpickle = unpickle boolUP >>= bool
(Right <$> unpickle bUP) (Left <$> unpickle aUP)
, pickle = either
(\a -> pickle boolUP True *> pickle aUP a)
(\b -> pickle boolUP False *> pickle bUP b)
}

Finally, we can map over (un)picklers with isomorphisms (bijections): in other words, UP is a functor between the category of types and isomorphisms and the category of types and functions, Hask.

-- For (to, from) :: Iso a b, we assume:
--
-- > to . from = id :: b -> b
-- > from . to = id :: a -> a
type Iso a b = (a -> b, b -> a)

mapUP :: Functor r => Iso a b -> UP r w a -> UP r w b
mapUP (to, from) aUP = UP
{ unpickle = fmap to (unpickle aUP)
, pickle = pickle aUP . from
}

Using the above, we can program (un)picklers, but it is not as convenient as it might seem. Every operation involved must be invertible (obviously for mapUP, while pairUP, altUP, and eitherUP rely on pattern matching). UP definitions for large records are rather tedious as one has to write explicitly how to construct and destruct every record.

-- Assume for the sake of example that this type exists...
data Date

-- ... with an UP.
dateUP :: BinaryUP Date
dateUP = (...)

data User = User
{ userId :: Int
, userCreated :: Date
, userEmail :: String
}

userUP :: BinaryUP User
userUP =
mapUP
( (\(((userId, userName), userCreated), userEmail) ->
User{..})
, (\User{..} ->
) (binaryUP >| binaryUP >| dateUP >| binaryUP)

Half the definition of userUP is boilerplate for restructuring a tuple into/out of a User.

A possible improvement is to derive the isomorphism generically, or with meta-programming.

However, we can design a much nicer interface by spending some effort to fit common abstractions in Haskell: applicative functors and monads.

I found something that works but I can really see that it looks good a posteriori, whereas I have trouble giving an a priori motivation to work in that direction. One is that that functional programmers are already familiar with these abstractions, and that we can reasonably expect the r and w context to be instances of Applicative or even Monad, so it might make sense that a “product” of those inherits of such structure.

Applicative Codec

UP r w is not a Haskell Functor (endofunctor of Hask), because pickling is contravariant (of type a -> w ()).

The Trick

The codec package dissociates the types being parsed (i.e., unpickled, deserialized) and produced (i.e., pickled, serialized).

data Codec r w x a = Codec
{ parse :: r a
, produce :: x -> w ()
}

We easily get Functor, Applicative and even Alternative.

instance Functor r => Functor (Codec r w x) where
fmap f codec = codec { parse = fmap f (parse codec) }

instance (Applicative r, Applicative w)
=> Applicative (Codec r w x) where

pure a = Codec (pure a) (\_ -> pure ())

f <*> a = Codec
{ parse = parse f <*> parse a
, produce = \x -> produce f x *> produce a x
}

instance (Alternative r, Alternative w)
=> Alternative (Codec r w x) where

empty = Codec empty (\_ -> empty)

a <|> a' = Codec
{ parse = parse a <|> parse a'
, produce = \x ->
produce a x <|> produce a' x
}

UP r w a is isomorphic to Codec r w a a; we’re indeed working with a generalization of (un)picklers.

upToCodec :: UP r w a -> Codec r w a a
upToCodec (UP parse produce) = Codec parse produce

codecToUP :: Codec r w a a -> UP r w a
codecToUP (Codec unpickle pickle) = UP unpickle pickle

However if we work only with Codec r w a a, we cannot use Applicative, because the context Codec r w a :: * -> * is related to the content a :: *.

To modify the context, we note that Codec r w x a is contravariant with respect to x. In fact, we have a Profunctor.

instance Functor r => Profunctor (Codec r w) where

lmap :: (y -> x) -> Codec r w x a -> Codec r w y a
lmap from = liftA2 Codec parse ((. from) . produce)

rmap :: (a -> b) -> Codec r w x a -> Codec r w x b
rmap = fmap

In the produce direction, lmap makes the Codec accept a larger structure, a y containing an x that can be extracted with from.

As an aside, notice that

-- dimap
--   :: (y -> x) -> (a -> b)
--   -> Codec r w x a -> Codec r w y b
-- dimap from to = lmap from . rmap to

generalizes, with (y -> x) ~ (b -> a),

-- mapUP
--   :: (a -> b, b -> a)     -- Iso a b
--   -> UP r w a -> UP r w b

An example of from :: y -> x function is a field getter; we can now easily define a Codec for a record.

Assume we have a Codec for each field of User:

type BinaryCodec a = Codec Get PutM a a

-- For Int, String, etc.
binaryCodec :: Binary a => BinaryCodec a
binaryCodec = Codec get put

dateCodec :: BinaryCodec Date
dateCodec = (...)

Define an infix synonym for niceness:

(=.)
:: Functor r
=> (y -> x) -> Codec r w x a -> Codec r w y a
(=.) = lmap

The following definition looks much nicer than the one using mapUP.

userCodec :: BinaryCodec User
userCodec = User
<$> userId =. binaryCodec <*> userName =. binaryCodec <*> userCreated =. dateCodec <*> userEmail =. binaryCodec We can move fields around, (de)serializing them in a different order, with one less location to modify compared to an UP definition (the to component being mostly implicit here), though it still looks unwieldly. userReversedCodec :: BinaryCodec User userReversedCodec = (\email created name id -> User id name created email) <$> userEmail =. binaryCodec
<*> userCreated =. dateCodec
<*> userId =. binaryCodec

Magic record construction

The codec package actually does not work in the way I just presented. It provides an Applicative instance, but is missing the Profunctor instance, or more specifically an (=.) (lmap), to work with Applicative.

In fact, codec takes another approach. With some boilerplate generated via Template Haskell, it allows to define Codecs with a syntax very similar to the above. It has the additional feature that permuting the fields does not require rewriting the constructor as I did in userReversedCodec.

All you need to do is provide a de/serializer for every record field in any order you like, and you get a de/serializer for the whole structure. The type system ensures that you provide every field exactly once.

After getting an Applicative, one is naturally led to wonder whether there is a Monad as well.

If we try to implement it, we realize Codec is unfortunately not endowed with such a structure. parse is fine, but there is no way to obtain a produce from the second operand.

-- Failed
=> Monad (Codec r w x) where
a >>= f = Codec
{ parse = parse a >>= parse . f
, produce = \x ->
produce a x *>
(???) -- Can't apply f
}

From here on, I have gone through a succession of choices, that I haven’t considered in detail individually, but I’m seeing something promising at the end.

Carry a projection

A simple fix is to make explicit the intent that in Codec r w x a, the x should contain an a.

data Codec0 r w x a = Codec0
{ parse0 :: r a
, produce0 :: x -> w ()
, project0 :: x -> a
}

instance Functor (Codec0 r w x) where fmap = (...)
instance Applicative (Codec0 r w x) where
pure = (...) ; (<*>) = (...)
-- No Alternative?
instance Profunctor (Codec0 r w) where dimap = (...)

=> Monad (Codec0 r w x) where
a >>= f = Codec0
{ parse0 = parse0 a >>= parse0 . f
, produce0 = \x ->
produce0 a x *>
produce0 (f (project0 a x)) x
, project0 = \x ->
project0 (f (project0 a x)) x
}

The issue with that definition is that there is a duplication of code between produce0 :: x -> w () and project0 :: x -> a, made evident if we unroll a composition of (>>=) and lmap:

-- lmap g a >>= f
-- =
-- Codec0
--   { produce0 = \x ->
--       produce0 a (g x) *>
--       produce0 (f (project0 a (g x))) x
--   , ..
--   }

(g x) occurs twice, and we would like to factor it, but the compiler won’t see it.

Factor the projection

That duplication might be avoided by factoring project out of produce:

data Codec1 r w x a = Codec1
{ parse1 :: r a
, produce1 :: a -> w ()
, project1 :: x -> a
}

But that is just UP with a new field, and we face again contravariance with respect to a, and lose so much niceness (though how much of an inconvenience it causes is still unclear).

The Trick (bis)

I would try to apply again the trick that led from UP to Codec in the first place, splitting the covariant and contravariant occurences of a:

-- (Maybe come up with another name?)
-- The ordering here is chosen to be compatible
-- with the Profunctor typeclass...
data Codec3 r w k x a = Codec3
{ parse3 :: r a
, produce3 :: k -> w ()
, project3 :: x -> k
}

-- ... but I really have some diagram x -> k -> a in mind.
type Codec' r w x k = Codec3 r w k x

instance Functor (Codec3 r w k x) where fmap = (...)
instance Applicative (Codec3 r w k x) where
pure = (...) ; (<*>) = (...)
instance Profunctor (Codec3 r w k) where dimap = (...)

Monad is unfortunately still out of reach. It now seems quite foolish, I erased the link that I made just earlier between x and a.

After spending some time with this puzzle, I would generalize the Haskell Monad as follows:
bindCodec'
bindCodec' = (...)