# Programming With Effects II

We're continuing to study monads by looking at examples of *specific* monads to try to understand how they work. At this point, don't panic if you don't understand the big picture, each of the specific instances are useful in their own right. For right now, think of monads as burritos, especially since they really are.

`{-# LANGUAGE NoImplicitPrelude #-}`

```
module Monads2 where
import Prelude hiding (getLine,sequence,(>>))
import Data.Map (Map)
import qualified Data.Map as Map
```

This module depends on an auxiliary module that we will define in class.

`import State`

Note: to load this module into ghci, you must first change into the current subdirectory so that ghci can find it. For me that is:

:cd /Users/sweirich/vc/cis552/13fa/lectures

Alternatively, the State module is a subset of the functionality in the standard library. We can also replace the import above with:

`-- import Control.Monad.State`

# The State Monad

Now let us consider the problem of writing functions that manipulate some kind of state. We're going to start with some examples of state manipulation, written in an awkward style, and then show how monads can cleanly abstract the sequencing necessary for such programs.

By way of an example of state manipulation, let's go back to binary trees whose leaves contains values of some type a:

```
data Tree a = Leaf a | Branch (Tree a) (Tree a)
deriving (Eq, Show)
```

Here is a simple example:

```
tree :: Tree Char
tree = Branch (Branch (Leaf 'a') (Leaf 'b')) (Leaf 'c')
```

A functional programmer would count the number of leaves in a tree with this code:

```
countF :: Tree a -> Int
countF (Leaf _) = 1
countF (Branch t1 t2) = countF t1 + countF t2
```

Now, consider how a C programmer would count the number of leaves in a tree. He might create a local reference cell, and then then walk the tree, incrementing the reference cell at each leaf. In pure code, we cannot modify the values of any variables. However, we can define a *state transformer*, a function that takes an initial state as an input and returns the new state.

In this example, the state, also called the "store", is an Int (the current count) and a state transformer is a function of type `Int -> Int`

.

`type Store = Int`

```
countI :: Tree a -> Int
countI t = aux t 0 where
aux :: Tree a -> Store -> Store
aux (Leaf _) s = s + 1
aux (Branch t1 t2) s =
aux t1 (aux t2 s)
```

In general, a state transformer takes a current state as its argument, and produces a modified state as its result, in which the modified state reflects any side effects performed by the function.

Now consider the problem of defining a function that labels each leaf in such a tree with a unique or "fresh" integer. This can be achieved by taking the next fresh integer as an additional argument to a helper function, and returning the next fresh integer as an additional result.

```
label1 :: Tree a -> Tree (a, Int)
label1 t = fst (aux t 0) where
aux :: Tree a -> Int -> (Tree(a,Int), Int)
aux (Leaf x) s = (Leaf (x, s), s+1)
aux (Branch t1 t2) s =
let (t2', s') = aux t2 s in
let (t1', s'') = aux t1 s' in
(Branch t1' t2', s'')
```

This example demonstrates that in general, we may wish to return a result value in addition to updating the store. For this reason, we generalize our type of state transformers to also return a result value, with the type of such values being a parameter of the `ST`

type:

`type ST a = Store -> (a, Store)`

The state transformer may also wish to take argument values. However, there is no need to further generalize the `ST`

type to take account of this, because this behavior can already be achieved by currying. For example, the state transformer for the tree above takes a tree and returns a labeled tree and has type `Tree a -> ST (Tree (a,Int))`

, which abbreviates the curried function type

`Tree a -> Store -> (Tree (a, Int), Store)`

The reason we are talking about state transformers is that parameterized type `ST`

is a monad. What are its definitions of return and bind? Can we abstract them from the definition of `label1`

?

```
aux :: Tree a -> Store -> (Tree (a,Int), Store)
aux (Leaf x) = \s -> ( Leaf (x,s+1) ,s)
aux (Branch t1 t2) = \s ->
let (t1', s1) = aux t1 s in
let (t2', s2) = aux t2 s1 in
(Branch t1' t2', s2)
```

```
type ST a = Store -> (a, Store)
instance Monad ST where
-- return :: a -> Store -> (a, Store)
return x = \ st -> (x,st)
-- (>>=) :: (Store -> (a, Store)) ->
(a -> (Store -> (b, Store))) -> (Store -> (b, Store))
st >>= f = \ s -> let (x, s') = st s in f x s'
```

That is, `return`

converts a value into a state transformer that simply returns that value without modifying the state.

In turn, `>>=`

provides a means of sequencing state transformers: `st >>= f`

applies the state transformer `st`

to an initial state `s`

, then applies the function `f`

to the resulting value `x`

to give a second state transformer `(f x)`

, which is then applied to the modified state `s'`

to give the final result.

Note that `return`

could also be defined by `return x s = (x,s)`

.

However, we prefer the above definition in which the second argument `s`

is shunted to the body of the definition using a lambda abstraction, because it makes explicit that `return`

is a function that takes a single argument and returns a state transformer, as expressed by the type `a -> ST a`

: A similar comment applies to the above definition for `>>=`

.

We conclude this section with a technicality. In Haskell, types defined using the `type`

mechanism cannot be made into instances of classes. Hence, in order to make ST into an instance of the class of monadic types, in reality it needs to be redefined using the "data" mechanism, which requires introducing a dummy constructor (called `S`

for brevity):

`-- type ST a = Store -> (a, Store)`

`newtype ST2 a = S { apply :: Store -> (a, Store) }`

S :: (Store -> (a,Store)) -> ST2 a apply :: ST2 a -> (Store -> (a, Store))

It is convenient to define our own application function for this type, which simply removes the dummy constructor. We do so using a record type declaration, which lets us name the arguments to a data constructor.

```
*Monads2> :type apply
apply :: ST2 a -> Store -> (a, Store)
```

In turn, ST2 is now defined as a monadic type as follows:

```
instance Monad ST2 where
-- return :: a -> ST2 a
return x = S (\ s -> (x,s))
-- (>>=) :: ST2 a -> (a -> ST2 b) -> ST2 b
st >>= f = S (\s -> let (x,s') = apply st s in apply (f x) s')
```

(*Aside*: there is no runtime overhead for manipulating the dummy constructor because we defined ST2 using the `newtype`

mechanism of Haskell, rather than the `data`

mechanism.)

Now, let's rewrite the tree labeling function with the State monad.

In order to generate a fresh integer, we define a special state transformer that simply returns the current state as its result, and the next integer as the new state:

```
fresh :: ST2 Int --- Store -> (Int, Store)
fresh = S (\ s -> (s , s + 1))
```

Using this, together with the `return`

and `>>=`

primitives that are provided by virtue of `ST`

being a monadic type, it is now straightforward to define a function that takes a tree as its argument, and returns a state transformer that produces the same tree with each leaf labelled by a fresh integer:

```
mlabel :: Tree a -> ST2 (Tree (a,Int))
mlabel (Leaf x) = -- fresh >>= \v -> return (Leaf (x, v))
do v <- fresh
return (Leaf (x,v))
mlabel (Branch l r) = do l' <- mlabel l -- ST2 (Tree (a,Int))
r' <- mlabel r -- ST2 (Tree (a,Int))
return (Branch l' r')
```

Note that the programmer does not have to worry about the tedious and error-prone task of dealing with the plumbing of fresh labels, as this is handled automatically by the state monad.

Finally, we can now define a function that labels a tree by simply applying the resulting state transformer with zero as the initial state, and then discarding the final state:

```
label :: Tree a -> Tree (a, Int)
label t = fst $ apply (mlabel t) 0
```

For example, `label tree`

gives the following result:

```
ghci> label tree
Branch (Branch (Leaf ('a', 0)) (Leaf ('b',1))) (Leaf ('c', 2))
```

## Exercise

Define a function

`app :: (Store -> Store) -> ST2 Store`

, such that fresh can be redefined by`fresh = app (+1)`

.Define a function

`run :: ST2 a -> Store -> a`

, such that label can be redefined by`label t = run (mlabel t) 0`

.

# A Generic State Transformer

Often, the *store* that we want to have will have multiple components, eg multiple variables whose values we might want to *update*. This is easily accomplished by using a different type for `Store`

above, for example, if we want two integers, we might use the definition

`type Store = (Int, Int)`

and so on.

However, it would be good to write reusable code, which would work with any store.

The file State ( lhs version ) contains a generic library for that purpose.

# Using a Generic State Transformer

Let us use our generic state monad to rewrite the tree labeling function from above. Note that the actual type definition of the generic transformer is *hidden* from us, so we must use only the publicly exported functions: `get`

, `put`

and `runState`

(in addition to the monadic functions we get for free.)

First, we write an action that returns the next fresh integer (Note, the first type argument is the store, the second is the result type of the monadic action.)

```
freshS :: State Int Int
freshS = do num <- get
put $ num + 1
return num
```

Now, the labeling function is straightforward

```
mlabelS :: Tree t -> State Int (Tree (t, Int))
mlabelS (Leaf x) = do num <- freshS
return $ Leaf (x, num)
mlabelS (Branch l r) = do l' <- mlabelS l
r' <- mlabelS r
return $ Branch l' r'
```

Easy enough!

`ghci> runState (mlabelS tree) 0`

We can *execute* the action from any initial state of our choice

`ghci> runState (mlabelS tree) 1000`

Now, what's the point of a generic state transformer if we can't have richer states? Next, let us extend our `fresh`

and `label`

functions so that

each node gets a new label (as before), and

the state also contains a map of the

*frequency*with which each leaf value appears in the tree.

Thus, our state will now have two elements, an integer denoting the *next* fresh integer, and a `Map a Int`

denoting the number of times each leaf value appears in the tree. (Documentation for the Data.Map module. )

```
data MySt a = M { index :: Int
, freq :: Map a Int }
deriving (Eq, Show)
-- or, without fancy record notation:
-- data MySt a = M Int (Map a Int)
```

We write an *action* that returns the next fresh integer:

```
freshM :: State (MySt a) Int
freshM = do state@(M { index = i }) <- get
put $ state { index = i + 1 }
return i
-- or, without fancy record notation:
freshM' :: State (MySt a) Int
freshM' = do M ind fr <- get
put $ M (ind+1) fr
return ind
```

Similarly, we want an action that updates the frequency of a given element `k`

.

```
updFreqM :: Ord a => a -> State (MySt a) ()
updFreqM k = do state@(M { freq = fr }) <- get
let fr' = Map.alter (\maybe_freq ->
case maybe_freq of
Nothing -> Just 1
Just f -> Just (f + 1)) k fr
put $ state { freq = fr' }
-- or, without fancy record notation:
updFreqM' k = do M ind fr <- get
let fr' = Map.alter (\maybe_freq ->
case maybe_freq of
Nothing -> Just 1
Just f -> Just (f + 1)) k fr
put $ M ind fr'
```

And with these two, we are done

```
mlabelM :: Ord a => Tree a -> State (MySt a) (Tree (a, Int))
mlabelM (Leaf x) = do updFreqM x
label <- freshM
return $ Leaf (x, label)
mlabelM (Branch l r) = do l' <- mlabelM l
r' <- mlabelM r
return $ Branch l' r'
```

Now, our *initial* state will be something like

```
initM :: MySt a
initM = M 0 Map.empty
```

and so we can label the tree

```
ghci> let tree2 = Branch tree tree
ghci> let (lt, s) = runState (mlabelM tree) initM
ghci> lt
Branch (Branch (Branch (Leaf ('a',0)) (Leaf ('b',1))) (Leaf ('c',2))) (Branch (Branch (Leaf ('a',3)) (Leaf ('b',4))) (Leaf ('c',5)))
ghci> s
M {index = 6, freq = fromList [('a',2),('b',2),('c',2)]}
```

## Credit

This lecture is a revised version of the lecture notes by Graham Hutton, January 2011

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