I'm learning about the Map Reduce computation frequently used with big data.  For the fun of it, I decided to write a very high-level spec of Map Reduce.  Here is what I came up with.  Enjoy.

> module MapReduce where
> import Data.List (nub)

A high-level specification of Map Reduce as a Haskell program.  The
program uses lists to represent multisets.  As multisets have no
implied ordering, the ordering implied by lists in this specification
should be ignored.

The database is a multiset of key-value pairs.

> type Key = String
> type Value = String
> type Datum = (Key, Value)
> type Data = [Datum]

A mapper maps a datum to a finite multiset of key-value pairs.

> type Mapper = Datum -> Data

A reducer takes a key and a multiset of values and produces a finite
multiset of values.

> type Reducer = (Key, [Value]) -> [Value]

A step is a mapper followed by a reducer

> type Step = (Mapper, Reducer)

A program is a finite sequence of steps

> type Program = [Step]

The semantics of a program is provided by the run function.

> run :: Program -> Data -> Data
> run [] d = d
> run (s : p) d =
>   run p (step s d)

The three parts of a step are mapping, shuffling, and reducing.

> step :: Step -> Data -> Data
> step (m, r) d =
>   let mapped = transform m d
>       shuffled = shuffle mapped in
>   reduce r shuffled

The first part of a step is to transform the data by applying the
mapper to each datum and collecting the results.

> transform :: Mapper -> Data -> Data
> transform m d =
>   [p | u <- d, p <- m u]

Next, values with common keys are collected.  Keys are unique after
shuffling.

> shuffle :: Data -> [(Key, [Value])]
> shuffle d =
>   [(k, vs) | k <- nub (map fst d), -- nub eliminates duplicate keys
>              let vs = [v | (k', v) <- d, k' == k]]

A reducer is applied to the data associated with one key, and always
produces data with that key.

> reduce :: Reducer -> [(Key, [Value])] -> Data
> reduce r rs =
>   [(k, v) | (k, vs) <- rs, v <- r (k, vs)]