Monads and randomness

We will first recall the Option monad and its properties.

• If α is a type Option α is a type.
• Given a: α we get a term of type Option α by writing some a.

• Given a: Option α and f : α → β we get a term of type Option β by writing a.map f.
• Given a: Option α and f: α → Option β we get a term of type Option β by writing a.bind f.

For both these it is more pleasant to use the do notation.

Equivalent to the second is that there is a function Option Option α → Option α called join.

IO Monad

Roughly wraps with the state of the real world.

For example, a random number is wrapped in this.

Question:: Why not just ℕ?

• Otherwise we will violate the principle that the value of a function is determined by its arguments.

We still do want a natural number. To get this we can run the IO computation.

def rnd (lo : ℕ) (hi : ℕ) : ℕ

A random natural number

Equations

• rnd lo hi = Option.get! (EStateM.run' (IO.rand lo hi) ())

This does not lead to a contradiction, though in a way that may be somewhat surprising.

def a : ℕ

A random number between 0 and 100

Equations

• a = rnd 0 100

def b : ℕ

A random number between 0 and 100

Equations

• b = rnd 0 100

We can run these (every run gives a different result).

#eval a -- 23
#eval b -- 96

Interestingly, we can also run the pair of them and get a result on the diagonal.

#eval (a, b) -- (87, 87)

def rndPair (lo : ℕ) (hi : ℕ) : IO (ℕ × ℕ)

A random pair

Equations

• rndPair lo hi = do let a ← IO.rand lo hi let b ← IO.rand lo hi pure (a, b)

#eval a -- 23
#eval b -- 96

#eval (a, b) -- (87, 87)