## Introduction

It is well known that the list comprehension notation from functional programming bares more than a passing resemblance to SQL queries:

```
SELECT c.name, o.price
FROM customer c, orders o
WHERE c.pk_cust_id = o.fk_cust_id
```

```
[ (name c, price o)
| c <- customer
, o <- orders
, pk_cust_id c == fk_cust_id o ]
```

List comprehensions offer a declarative query syntax that is very SQL-like and readable, but also much more compositional than SQL, for example we can nest comprehensions arbitrarily deep, create re-usable abstractions and use recursion. However, the choice of sets or lists to collect the rows does not model the mapping of key attributes to non-key attributes, which is essential for an efficient database implementation. In this post, we’ll explore using a map representation `Map k v`

to model the primary key. A map does force us to work with all data in memory, unlike streams, but then complex queries with grouping and ordering usually force this on us anyway. Map semantics also show up often in distributed systems, for example NoSQL stores and distributed caches.

Haskell’s Data.Map already includes implementations for many of the standard relational algebra operations, such as union, intersection and difference. But if we want to explore what a ‘map comprehension’ might look like, we need to take a look at *indexed monads*.

## Indexed Monads

Indexed monads are a generalisation of monads where a monadic type constructor is indexed by another type representing additional information that further categorises the computation or structure.

```
join : m s (m t a) -> m (s ⨂ t) a
return : m ε a
```

The monad indices are a monoid (K, ⨂, ε). Dominic Orchard has published some great slides on indexed monads here.

In order to see what a map comprehension might look like, we are going to need a structure that is both a value-level and a type-level monoid to model the key attributes. Luckily, GHC’s new DataKinds extension makes it very easy to implement a *heterogeneous list* (HList). Ideally, we probably want set semantics for our key-attributes, but an HList satisfies the minimum monoid requirement.

Firstly, since this is a literal Haskell post, we’ll need to enable some extensions and import some modules:

```
{-# LANGUAGE GADTs #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE MonadComprehensions #-}
{-# LANGUAGE TransformListComp #-}
{-# LANGUAGE RebindableSyntax #-}
```

```
import Prelude hiding (Monad(..), fmap, sum, head, tail)
import Data.Map (Map)
import qualified Data.Map as Map
import Data.Monoid
```

## Heterogeneous Lists using Data Kinds

Let’s roll our own simple HList implementation using the DataKinds extension. We will need a type-level append and a value-level append.

```
-- | The heterogeneous list
data HList :: [*] -> * where
Z :: HList '[]
(:.) :: t -> HList ts -> HList (t ': ts)
infixr 5 :.
-- | HList type-level append
type family (m :: [*]) ++ (n :: [*]) :: [*]
type instance '[] ++ ys = ys
type instance (x ': xs) ++ ys = x ': (xs ++ ys)
-- | HList value-level append
(++.) :: HList as -> HList bs -> HList (as ++ bs)
Z ++. ys = ys
(x :. xs) ++. ys = x :. (xs ++. ys)
infixr 5 ++.
head :: HList (a ': as) -> a
head (x :. _) = x
tail :: HList (a ': as) -> HList as
tail (_ :. xs) = xs
```

Now we can compute the union of two key-attributes. For example:

```
λ> (42 :. Z) ++. ("foo" :. Z)
42:."foo":.Z
```

## The Map Comprehension

To write a map comprehension, we will use the the RebindableSyntax and MonadComprehensions extensions. In theory we could bind this syntax to the methods in an indexed monad class. It should also be possible to make Map instances of an indexed functor and indexed monad class hierarchy, but that is beyond the scope of this post.

First, some useful type and constraint synonyms:

```
type K = HList
type Key k = Ord (K k)
type Union k k' = (Key k, Key k', Key (k ++ k'))
```

A Map is an indexed functor, whereby the fmap operation does not change the index type. This is consistent with Data.Map.map which leaves the keys alone:

```
-- | projection
fmap :: Key k => (v -> v') -> Map (K k) v -> Map (K k) v'
fmap = Map.map
```

The monadic join operation for Map is analogous to list concatenation. Not surprisingly, there is nothing in Data.Map for this, as we need to union the map keys. We can implement this if we make the assumption that the keys will have a monoid append operation. Here we use HList append (++.):

```
-- | monadic join or "concat"
join :: Union k k' => Map (K k) (Map (K k') v) -> Map (K (k ++ k')) v
join = Map.foldrWithKey (\k m r ->
Map.foldrWithKey (\k' v r' ->
Map.insert (k ++. k') v r') r m) Map.empty
```

For return, we need to introduce the concept of a null-key, which we represent as an empty HList:

```
return :: v -> Map (K '[]) v
return x = Map.singleton Z x
```

Monadic bind can simply be defined in terms of join and fmap:

```
(>>=) :: Union k k' => Map (K k) v -> (v -> Map (K k') v') -> Map (K (k ++ k')) v'
m >>= f = join (fmap f m)
```

```
(>>) :: Union k k' => Map (K k) v -> Map (K k') v' -> Map (K (k ++ k')) v'
x >> y = x >>= (\_ -> y)
```

`fail = error "failed"`

Let’s try a simple cartesian product (also known as a relational cross-join). We’ll also use an HList for our non-key attributes, although the map comprehension does not require this:

```
example1 = [ x ++. y
| x <- Map.fromList [ ("foo":.Z, 1:.Z), ("bar":.Z, 2:.Z) ]
, y <- Map.fromList [ ("baz":.Z, 3:.Z), ("qux":.Z, 4:.Z) ]
]
```

```
λ> example1
fromList [("bar":."baz":.Z, 2:.3:.Z)
,("bar":."qux":.Z, 2:.4:.Z)
,("foo":."baz":.Z, 1:.3:.Z)
,("foo":."qux":.Z, 1:.4:.Z)]
```

Note that the result contains new keys which are the cartesian product of the original keys. The values are exactly what we asked for, which in this instance, is also a cartesian product. I’ve also taken the liberty of formatting the output, for ease of reading.

From the above example, we can see that a map comprehension takes care of deriving key attributes for us. It is more restrictive than combining a list comprehension with Map.fromList, as we cannot create arbitrary keys.

To filter out values, let’s rebind the monadic guard function:

```
guard :: Bool -> Map (K '[]) ()
guard True = return ()
guard False = Map.empty
```

Now we can use guards in the comprehension syntax:

```
-- | guards filter on values, they cannot access keys
example2 = [ x + y
| x <- Map.fromList [ ("foo":.Z, 1), ("bar":.Z, 2) ]
, y <- Map.fromList [ ("baz":.Z, 3), ("qux":.Z, 4) ]
, x + y < 6
]
```

```
λ> example2
fromList [("bar":."baz":.Z, 5)
,("foo":."baz":.Z, 4)
,("foo":."qux":.Z, 5)]
```

Again this is more restrictive than a list comprehension, but the restriction is useful, it forces the user out of the comprehension guard syntax, if they want to filter by key. In other words, you cannot use the comprehension guard syntax to represent an inefficient relational inner-join.

An efficient inner-join would need to be accomplished using an auxiliary function such as the one below. Again, we use an HList for value attributes, so that we can append them together.

```
innerJoin
:: (Key k, Key k') =>
(HList as -> K k')
-> Map (K k) (HList as)
-> Map (K k') (HList bs)
-> Map (K k) (HList (as ++ bs))
innerJoin f m1 m2 = Map.foldrWithKey doRow Map.empty m1
where
doRow k v m = maybe m (\v' -> Map.insert k (v ++. v') m) $ Map.lookup (f v) m2
```

## Aggregation and Grouping

In SQL, aggregation is the reduction of a row set by an associative and commutative binary operation. It is the most significant extension of the relational algebra in terms of expressiveness. Since all SQL queries must return sets of primitive atomic values, aggregation functions need to be coupled with grouping clauses. If we don’t have this restriction, we are free to completely separate aggregation from grouping.

Note that ‘groupWith’ when applied to a key constructor, yields a grouping function that conceptually is dual to concatenation (monadic join), though that’s not quite the case here as we do not split up the input key.

```
-- | grouping for potential aggregation
groupWith :: (Key k, Key k') =>
(v -> K k') -> Map (K k) v -> Map (K k') (Map (K k) v)
groupWith f = Map.foldrWithKey group Map.empty
where
group k v = Map.insertWith Map.union (f v) (Map.singleton k v)
```

```
-- | aggregate or reduce
aggregate :: Monoid m => Map (K k) m -> m
aggregate = Map.foldr mappend mempty
```

```
sum :: (Num a, Key k) => Map (K k) a -> a
sum = getSum . aggregate . fmap Sum
```

```
count :: (Num a, Key k) => Map (K k) v -> a
count = sum . fmap (const 1)
```

The TransformListComp extension allows us to use SQL-like “group by” clauses in our monad comprehensions. In the example below, we group and count the numeric values from an input map. A rather idiosyncratic feature of the TransformListComp extension is that the variable `x`

gets rebound to be the grouping result, in this case a map from the grouping key to the group. Here the rebinding is convenient and we can simply call ‘count’ on it to perform an aggregation.

```
example3 = [ count x
| x <- Map.fromList [ ("foo":.Z, 1), ("bar":.Z, 2)
, ("baz":.Z, 2), ("qux":.Z, 3)]
, then group by (x:.Z) using groupWith
]
```

```
λ> example3
fromList [(1:.Z, 1])
,(2:.Z, 2])
,(3:.Z, 1])]
```

It is quite straightforward to create a flat pivot table using a compound grouping key:

```
example4 = [ sum (fmap orderAmt x)
| x <- orders
, then group by (orderIsin x :. orderTrader x :.Z) using groupWith
]
```

```
orders = Map.fromList
[ (1:.Z, "US68389X1054":."John":.10.0:.Z)
, (2:.Z, "US0378331005":."Joe" :. 2.0:.Z)
, (3:.Z, "US68389X1054":."John":. 2.0:.Z)
, (4:.Z, "US0378331005":."Joe" :. 5.0:.Z)
, (5:.Z, "US5949181045":."John":.10.0:.Z) ]
orderIsin = head -- ^ we need some Template Haskell here
orderTrader = head . tail
orderAmt = head . tail . tail
```

```
λ> example4
fromList [("US0378331005":."Joe" :.Z, 7.0)
,("US5949181045":."John":.Z,10.0)
,("US68389X1054":."John":.Z,12.0)]
```

However, if we want to produce an output with successive nested maps, the TransfromListComp syntax is not much help. It is easier to use simple function composition of a flat grouping function ‘GroupBy’:

```
-- | an alternative grouping function which performs aggregation
groupBy :: (Key k, Key k') =>
(v -> K k') -> (Map (K k) v -> v') -> Map (K k) v -> Map (K k') v'
groupBy f g = fmap g . groupWith f
```

We can compose the groupBy function according to how many dimensions we want to pivot on. This works because the aggregation function passed to ‘GroupBy’ can itself be another grouping operation. Such a transformation would be very useful if we were, for example, converting a flat CSV file into a hierarchical XML file.

```
-- | from a flat representation to a hierarchy
example5 = ( groupBy ((:.Z) . orderIsin)
. groupBy ((:.Z) . orderTrader)
) (fmap orderAmt) orders
```

```
λ> example5
fromList [("US0378331005":.Z
,fromList [("Joe":.Z
,fromList [(2:.Z, 2.0)
,(4:.Z, 5.0)])])
,("US5949181045":.Z
,fromList [("John":.Z
,fromList [(5:.Z,10.0)])])
,("US68389X1054":.Z
,fromList [("John":.Z
,fromList [(1:.Z,10.0)
,(3:.Z, 2.0)])])]
```

The reverse of this operation is a series of corresponding cartesian products using the map comprehension. This is the type of transformation we would need to go from a hierarchical data representation to a flat one, for example from XML to CSV.

```
-- | from a hierarchy to a flat representation
example6 = [ amt
| isin <- example5
, trader <- isin
, amt <- trader
]
```

```
λ> example5
fromList [("US0378331005":."Joe" :.2:.Z, 2.0)
,("US0378331005":."Joe" :.4:.Z, 5.0)
,("US5949181045":."John":.5:.Z,10.0)
,("US68389X1054":."John":.1:.Z,10.0)
,("US68389X1054":."John":.3:.Z,2.0)]
```

That’s about all I’ve explored of the ‘map comprehension’ idea to date.

Note: the literal haskell for this entire post can be found here.

## References

[2] Giving Haskell a Promotion

[3] Comprehensive Comprehensions

## Appendix

#### HList instances

```
instance Show (HList '[]) where
show _ = "Z"
instance (Show x, Show (HList xs)) => Show (HList (x ': xs)) where
show (x:.xs) = show x ++ ":." ++ show xs
```

```
instance Eq (HList '[]) where
_ == _ = True
instance (Eq x, Eq (HList xs)) => Eq (HList (x ': xs)) where
(x:.xs) == (y:.ys) = x == y && xs == ys
```

```
instance Ord (HList '[]) where
_ `compare` _ = EQ
instance (Ord x, Ord (HList xs)) => Ord (HList (x ': xs)) where
(x:.xs) `compare` (y:.ys) = (x, xs) `compare` (y, ys)
```