<theory, functional programming> /mo'nad/ A technique from category
theory which has been adopted as a way of dealing with state in functional
programming languages in such a way that the details of the state are hidden or
abstracted out of code that merely passes it on unchanged.
A monad has three components: a means of augmenting an existing type, a means of
creating a default value of this new type from a value of the original type, and
a replacement for the basic application operator for the old type that works
with the new type.
The alternative to passing state via a monad is to add an extra argument and
return value to many functions which have no interest in that state. Monads can
encapsulate state, side effects, exception handling, global data, etc. in a
purely lazily functional way.
A monad can be expressed as the triple, (M, unitM, bindM) where M is a function
on types and (using Haskell notation):
unitM :: a -> M a
bindM :: M a -> (a -> M b) -> M b
I.e. unitM converts an ordinary value of type a in to monadic form and
bindM applies a function to a monadic value after
de-monadising it. E.g. a state transformer monad:
type S a = State -> (a, State)
unitS a = \ s0 -> (a, s0)
m `bindS` k = \ s0 -> let (a,s1) = m s0
in k a s1
Here unitS adds some initial state to an ordinary value and bindS applies
function k to a value m. (`fun` is Haskell notation
for using a function as an infix operator). Both m
and k take a state as input and return a new state
as part of their output. The construction
m `bindS` k
composes these two state transformers into one while also passing the
value of m to k.
Monads are a powerful tool in functional programming. If a program is written
using a monad to pass around a variable (like the state in the example above)
then it is easy to change what is passed around simply by changing the monad.
Only the parts of the program which deal directly with the quantity concerned
need be altered, parts which merely pass it on unchanged will stay the same.
In functional programming, unitM is often called initM or returnM and bindM is
called thenM. A third function, mapM is frequently defined in terms of then and
return. This applies a given function to a list of monadic values, threading
some variable (e.g. state) through the applications:
mapM :: (a -> M b) -> [a] -> M [b]
mapM f  = returnM 
mapM f (x:xs) = f x `thenM` ( \ x2 ->
mapM f xs `thenM` ( \ xs2 ->
returnM (x2 : xs2) ))
modulo arithmetic « modulo operator « molly-guard «
» monadic » Mongolian Hordes technique » moniter