# State Monad

The State Monad by Example

This is a short tutorial on the state monad. Emphasis is placed on intuition. The types have been simplified to protect the innocent.

## Contents

# Foundations

## Primitives

```
runState (return 'X') 1
```

('X',1)

Return set the result value but left the state unchanged. Comments:

return 'X' :: State Int Char runState (return 'X') :: Int -> (Char, Int) initial state = 1 :: Int final value = 'X' :: Char final state = 1 :: Int result = ('X', 1) :: (Char, Int)

```
runState get 1
```

(1,1)

Get set the result value to the state and left the state unchanged. Comments:

get :: State Int Int runState get :: Int -> (Int, Int) initial state = 1 :: Int final value = 1 :: Int final state = 1 :: Int

```
runState (put 5) 1
```

((),5)

Put set the result value to () and set the state value. Comments:

put 5 :: State Int () runState (put 5) :: Int -> ((),Int) initial state = 1 :: Int final value = () :: () final state = 5 :: Int

## Combinations

Because (State s) forms a monad, values can be combined together with (>>=) or do{}.

```
runState (do { put 5; return 'X' }) 1
```

('X',5)

Comments:

do { put 5; return 'X' } :: State Int Char runState (do { put 5; return 'X' }) :: Int -> (Char,Int) initial state = 1 :: Int final value = 'X' :: Char final state = 5 :: Int

```
postincrement = do { x <- get; put (x+1); return x }
runState postincrement 1
```

(1,2)

```
predecrement = do { x <- get; put (x-1); get }
runState predecrement 1
```

(0,0)

## Other Functions

```
runState (modify (+1)) 1
```

((),2)

```
runState (gets (+1)) 1
```

(2,1)

```
evalState (gets (+1)) 1
```

2

```
execState (gets (+1)) 1
```

1

# Implementation

At its heart, a value of type (State s a) is a function from initial state s to final value a and final state s: (a,s). These are usually wrapped, but shown here unwrapped for simplicity.

Return leaves the state unchanged and sets the result:

```
-- ie: (return 5) 1 -> (5,1)
return :: a -> State s a
return x s = (x,s)
```

Get leaves state unchanged and sets the result to the state:

```
-- ie: get 1 -> (1,1)
get :: State s s
get s = (s,s)
```

Put sets the result to () and sets the state:

```
-- ie: (put 5) 1 -> ((),5)
put :: s -> State s ()
put x s = ((),x)
```

The helpers are simple variations of these primitives:

```
modify :: (s -> s) -> State s ()
modify f = do { x <- get; put (f x) }
gets :: (s -> a) -> State s a
gets f = do { x <- get; return (f x) }
```

EvalState and execState just select one of the two values returned by runState. EvalState returns the final result while execState returns the final state:

```
evalState :: State s a -> s -> a
evalState act = fst . runState act
execState :: State s a -> s -> s
execState act = snd . runState act
```

Combining two states is the trickiest bit in the whole scheme. To combine do { x <- act1; act2 x } we need a function which takes an initial state, runs act1 to get an intermediate result and state, feeds the intermediate result to act2 and then runs that action with the intermediate state to get a final result and a final state:

```
(>>=) :: State s a -> (a -> State s b) -> State s b
(act1 >>= fact2) s = runState act2 is
where (iv,is) = runState act1 s
act2 = fact2 iv
```

# Complete and Concrete Example

```
-- a concrete and simple example of using the State monad
import Control.Monad.State
-- non monadic version of a very simple state example
-- the State is an integer.
-- the value will always be the negative of of the state
type MyState = Int
valFromState :: MyState -> Int
valFromState s = -s
nextState :: MyState->MyState
nextState x = 1+x
type MyStateMonad = State MyState
-- this is it, the State transformation. Add 1 to the state, return -1*the state as the computed value.
getNext :: MyStateMonad Int
getNext = State (\st -> let st' = nextState(st) in (valFromState(st'),st') )
-- advance the state three times.
inc3::MyStateMonad Int
inc3 = getNext >>= \x ->
getNext >>= \y ->
getNext >>= \z ->
return z
-- advance the state three times with do sugar
inc3Sugared::MyStateMonad Int
inc3Sugared = do x <- getNext
y <- getNext
z <- getNext
return z
-- advance the state three times without inspecting computed values
inc3DiscardedValues::MyStateMonad Int
inc3DiscardedValues = getNext >> getNext >> getNext
-- advance the state three times without inspecting computed values with do sugar
inc3DiscardedValuesSugared::MyStateMonad Int
inc3DiscardedValuesSugared = do
getNext
getNext
getNext
-- advance state 3 times, compute the square of the state
inc3AlternateResult::MyStateMonad Int
inc3AlternateResult = do getNext
getNext
getNext
s<-get
return (s*s)
-- advance state 3 times, ignoring computed value, and then once more
inc4::MyStateMonad Int
inc4 = do
inc3AlternateResult
getNext
main =
do
print (evalState inc3 0) -- -3
print (evalState inc3Sugared 0) -- -3
print (evalState inc3DiscardedValues 0) -- -3
print (evalState inc3DiscardedValuesSugared 0) -- -3
print (evalState inc3AlternateResult 0) -- 9
print (evalState inc4 0) -- -4
```