Monad laws

All instances of the Monad typeclass should obey the three monad laws:

Left identity:	return a	>>= f	≡	f a
Right identity:	m	>>= return	≡	m
Associativity:	(m >>= f)	>>= g	≡	m >>= (\x -> f x >>= g)

The last law can be re-written for clarity as

      (m  >>=  (\x -> f x))  >>=  g    ≡
       m  >>=  (\x -> f x    >>=  g)

or, equally,

      (m  >>=  (\x -> f x))  >>=  (\y -> g y)    ≡
       m  >>=  (\x -> f x    >>=  (\y -> g y))

Here, p ≡ q simply means that you can replace p with q and vice-versa, and the behaviour of your program will not change: p and q are equivalent.

What is the practical meaning of the monad laws?

Let us re-write the laws in do-notation:

Left identity:	do { x′ <- return x; f x′ }	≡	do { f x }
Right identity:	do { x <- m; return x }	≡	do { m }
Associativity:	do { y <- do { x <- m; f x } g y }	≡	do { x <- m; do { y <- f x; g y } }	≡	do { x <- m; y <- f x; g y }

In this notation the laws appear as plain common-sense transformations of imperative programs.

But why should monads obey these laws?

When we see a program written in a form on the left-hand side, we expect it to do the same thing as the corresponding right-hand side; and vice versa. And in practice, people do write like the lengthier left-hand side once in a while. First example: beginners tend to write

skip_and_get = do
                   unused <- getLine
                   line <- getLine
                   return line

and it would really throw off both beginners and veterans if that did not act like (by right identity)

skip_and_get = do
                   unused <- getLine
                   getLine

Second example: Next, you go ahead to use skip_and_get:

main = do
           answer <- skip_and_get
           putStrLn answer

The most popular way of comprehending this program is by inlining (whether the compiler does or not is an orthogonal issue):

main = do
           answer <- do
                         unused <- getLine
                         getLine
           putStrLn answer

and applying associativity so you can pretend it is

main = do
           unused <- getLine
           answer <- getLine
           putStrLn answer

The associativity law is amazingly pervasive: you have always assumed it, and you have never noticed it.

Associativity of a binary operator allows for any number of operands to be combined by applying the binary operator with any arbitrary grouping to get the same well-defined result, just like the result of summing up a list of numbers is fully defined by the binary (+) operator no matter which parenthesization is used (yes, just like in folding a list of any type of monoidal values).

Whether compilers exploit the laws or not, you still want the laws for your own sake, just so you can avoid pulling your hair for counter-intuitive program behaviour that brittlely depends on how many redundant "return"s you insert or how you nest your do-blocks.

But it doesn't look exactly like an "associative law"...

Not in this form, no. To see precisely why they're called "identity laws" and an "associative law", you have to change your notation slightly.

The monad composition operator (also known as the Kleisli composition operator) is defined in Control.Monad:

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c
(m >=> n) x = do
                  y <- m x
                  n y

Using this operator, the three laws can be expressed like this:

Left identity:	return >=> g	≡	g
Right identity:	f >=> return	≡	f
Associativity:	(f >=> g) >=> h	≡	f >=> (g >=> h)

It's now easy to see that monad composition is an associative operator with left and right identities.

This is a very important way to express the three monad laws, because they are precisely the laws that are required for monads to form a mathematical category. So the monad laws can be summarised in convenient Haiku form:

Monad axioms:

Kleisli composition forms

a category.