```{- |
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,
-- * Adapter for memoization of multi-argument functions
for2,
for3,
for4,
-- * 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

-- * Example 4: Memoization of multi-argument function
-- \$multiExample
) 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 <- memo fibm (n-1)
>  n2 <- memo fibm (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 = return ""
>boo n = do
>  n1 <- memol1 boo (n-1)           -- uses next in stack transformer (memol_1_): MemoBoo is nested in MemoFib
>  fn <- memol0 fibm2 floor (n-1)   -- uses current transformer (memol_0_): MemoFib
>  return (show fn ++ n1)

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

>evalFibM2 = startEvalMemo . startEvalMemoT . fibm2

-}

{- \$transExample

>fibmw :: (Num n, MonadWriter String m, MonadMemo n n m) => n -> m n
>fibmw 0 = return 0
>fibmw 1 = return 1
>fibmw n = do
>  f1 <- memo fibmw (n-1)
>  f2 <- memo fibmw (n-2)
>  tell \$ show n
>  return (f1+f2)

>evalFibmw = startEvalMemo . runWriterT . fibmw

-}

{- \$multiExample
Functions with more than one argument (in curried form) can also be memoized with a help of @forX@ set of function:
For two-argument function we can use 'for2' function adapter:

>-- Ackerman function classic definition
>ack :: Num n => n -> n -> n
>ack 0 n = n+1
>ack m 0 = ack (m-1) 1
>ack m n = ack (m-1) (ack m (n-1))
>
>-- Ackerman function memoized definition
>ackm :: (Num n, Ord n, MonadMemo (n, n) n m) => n -> n -> m n
>ackm 0 n = return (n+1)
>ackm m 0 = for2 memo ackm (m-1) 1
>ackm m n = do
>  n1 <- for2 memo ackm m (n-1)
>  for2 memo ackm (m-1) n1
>
>evalAckm :: (Num n, Ord n) => n -> n -> n
>evalAckm n m = startEvalMemo \$ ackm n m

-}
```