This document discusses state monads in Haskell. It defines the State monad and its operators (>>=) and return. It shows how state is threaded through computations using these operators. It provides examples of using the state monad to increment a counter and to define functions like put and get that set and get the current state. In summary: 1) It introduces the State monad and its operators for threading state through computations. 2) It gives examples like incrementing a counter to demonstrate how the state is transformed and passed between function applications in the monad. 3) It shows how functions like put and get can be defined to set and access the internal state using the monad operators.

1. State
2009 6 1 id:hiratara

2. • State
• (>>=) State

3. (1) (>>=)

4. Int State Int Int
f
Int State Int Int

5. f
Int State Int Int
∋
a
f
Int State Int Int
∋
1 (a+1, 2)
2 (a+2, 3)
...
s0 (a+s0, s0+1)

6. : f 10 2 12
1 (11, 2)
f 10 = 2 (12, 3)
...
s0 (10+s0, s0+1)

7. f
1 (2, 1)
2 (4, 2)
...
s0 (s0 × 2, s0)

8. f
1 (2, 1)
2 (4, 2)
...
s0 (s0 × 2, s0)
(=<<) f
1
1 (3, 2)
2
2 (6, 3)
... f 2+1 = 3
s0 (s0 × 2 + s0, s0 + 1)

9. (>>=)
(State x) >>= f = State $ s -> let (v,s') = x s in runState (f v) s'
State(s0 ) (v(s0), s(s0))
...
(v, s') = (s0 × 2, s0) s = (s × 2, s)
f v = f (s × 2) = (s × 2 + s0, s0 + 1)
(f v) s' = (s × 2 + s0, s0 + 1) s = (s × 2 + s, s + 1)
(s0 × 2, s0) >>= f = (s0 × 2 + s0, s0 + 1)

10. Int >>= State Int Int ∋ (s0 × 2, s0)
(=<<) f
f
Int State Int Int ∋(s0 × 2 + s0, s0 + 1)

11. (1)
•
f
• (>>=) f

12. (2) get put

13. put get
• ( return (>>=) )
State
• put get State

14. get
• State
•
( )
•

15. get
• State
•
( )
•
→ (s0, s0)

16. get
get
get = State $ s -> (s,s)
→ (s0, s0)

17. put s
•
State
•
()
• s

18. put s
•
State
•
()
• s
→ put s = ( (), s)

19. put s
put
put s = State $ _ -> ((),s)
Int State Int Int
put
() State Int ()
→ put s = ( (), s)

20. put s
>>= put
put s = State $ _ -> ((),s)
Int State Int Int
put
(=<<) put
() State Int ()
→ put s = ( (), s)

21. (3)

22. State
0 1 1
1 2 2
2 3 3
...

23. State
0 1 1
1 2 2
2 3 3
...
State Int Int
0 (1, 1)
count = 1 (2, 2)
...
s0 (s0 + 1, s0+1)

24. (s0 + 1, s0+1)
=
State Int Int
0 (1, 1)
count = 1 (2, 2)
...
s0 (s0 + 1, s0+1)

25. get put

26. Int get ∈ State Int Int
put
() State () Int
() -> n+1
return
Int State Int Int

27. put return >>=
Int get ∈ State Int Int
>>=
put
() State () Int
return . ( ¥() -> n+1 ) >>=
Int State Int Int

28. Int get = (s0, s0) ∈ State Int Int
>>=
put
() ( (), s0 + 1) ∈ State () Int
return . ( ¥() -> n+1 ) >>=
Int ( s0 + 1, s0 + 1) ∈ State Int Int

29. • >>=
• put get
• do put get
State

30. • runState evalState
(v(s0), s(s0))
runState
State Int Bool Int -> (Bool, Int)
evalState
Int
Int -> Bool
Int
(Bool, Int)
Bool