```{- |
Copyright   :  (c) Eduard Sergeev 2011

Maintainer  :  eduard.sergeev@gmail.com
Stability   :  experimental
Portability :  non-portable (multi-param classes, functional dependencies)

[Computation type:] Monadic computations with support for memoization.

Defines monadic interface 'MonadMemo' for memoization and simple implementation 'MemoT' (based on 'Data.Map')
-}

Memo,
runMemo,
evalMemo,
startRunMemo,
startEvalMemo,
-- * The MemoT monad transformer
MemoT(..),
runMemoT,
evalMemoT,
startRunMemoT,
startEvalMemoT,
-- * Memoization cache level access functions
memoln,
memol0,
memol1,
memol2,
memol3,
memol4,
-- * Example 1: Fibonacci numbers
-- \$fibExample

-- * Example 2: Mutualy recursive definition with memoization
-- \$mutualExample

-- * Example 3: Combining Memo with other transformers
-- \$transExample
) where

MemoT(..), runMemoT, startRunMemoT, evalMemoT, startEvalMemoT,
Memo, runMemo, startRunMemo, evalMemo, startEvalMemo )

{- \$fibExample
Memoization can be specified whenever monadic computation is taking place.
Including recursive definition. Classic example: Fibonacci number function:
Here is simple non-monadic definition of it

>fib :: (Num n) => n -> n
>fib 0 = 0
>fib 1 = 1
>fib n = fib (n-1) + fib (n-2)

To use 'Memo' monad we need to convert it into monadic form:

>fibm :: (Num n, Monad m) => n -> m n
>fibm 0 = return 0
>fibm 1 = return 1
>fibm n = do
>  n1 <- fibm (n-1)
>  n2 <- fibm (n-2)
>  return (n1+n2)

Then we can specify which computation we want to memoize with 'memo' (both recursive calls to (n-1) and (n-2)):

>fibm :: (Num n, Ord n) => n -> Memo n n n
>fibm 0 = return 0
>fibm 1 = return 1
>fibm n = do
>  n1 <- fibm `memo` (n-1)
>  n2 <- fibm `memo` (n-2)
>  return (n1+n2)

NB: 'Ord' is required since internaly Memo implementation uses 'Data.Map' to store and lookup memoized values

Then it can be run with 'startEvalMemo'

>startEvalMemo . fibm \$ 5

-}

{- \$mutualExample
In order to use memoization for both mutually recursive function we need to use nested MemoT monad transformers
(one for each cache). Let's extend our Fibonacci function with meaningless extra function @boo@ which in turn uses @fibm2@.

Memoization cache type for @fibm2@ (caches @Integer -> Integer@) will be:

>type MemoFib = MemoT Integer Integer

While cache for @boo@ (@Double -> String@):

>type MemoBoo = MemoT Double String

Stacking them together gives us te overall type for our combined memoization monad:

>type MemoFB = MemoFib (MemoBoo Identity)

>boo :: Double -> MemoFB String
>boo 0 = "boo: 0" `trace` return ""
>boo n = ("boo: " ++ show n) `trace` do
>  n1 <- boo `memol1` (n-1)         -- uses next in stack transformer (memol_1_): MemoBoo is nested in MemoFib
>  f <- fibm2 `memol0` floor (n-1)  -- uses current transformer (memol_0_): MemoFib
>  return (show n ++ show f)

>fibm2 :: Integer -> MemoFB Integer
>fibm2 0 = "fib: 0" `trace` return 0
>fibm2 1 = "fib: 1" `trace` return 1
>fibm2 n = ("fib: " ++ show n) `trace` do
>  l <- boo `memol1` fromInteger n  -- as in 'boo' we need to use 1st nested transformer here
>  f1 <- fibm2 `memol0` (n-1)       -- as in 'boo' we need to use 1st nested transformer here
>  f2 <- fibm2 `memol0` (n-2)       --
>  return (f1 + f2 + floor (read l))

>evalFibM2 = startEvalMemo . startEvalMemoT . fibm2

-}

{- \$transExample
Being transformer, @MemoT@ can be used with other monads and transformers:

With @Writer@:

>fibmw 0 = return 0
>fibmw 1 = return 1
>fibmw n = do
>  f1 <- fibmw `memo` (n-1)
>  f2 <- fibmw `memo` (n-2)
>  tell \$ show n
>  return (f1+f2)

>evalFibmw = startEvalMemo . runWriterT . fibmw

-}

```