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theory, functional programming /mo'nad/ A technique from
category theory which has been adopted as a way of dealing
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 notaion):
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) ))