Monads
======
The [[tradition in the functional programming
literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
introduce monads using a metaphor: monads are spacesuits, monads are
monsters, monads are burritos. We're part of the backlash that
prefers to say that monads are monads.
The closest we will come to metaphorical talk is to suggest that
monadic types place objects inside of *boxes*, and that monads wrap
and unwrap boxes to expose or enclose the objects inside of them. In
any case, the emphasis will be on starting with the abstract structure
of monads, followed by instances of monads from the philosophical and
linguistics literature.
### Boxes: type expressions with one free type variable
Recall that we've been using lower-case Greek letters
α, β, γ, ...
to represent types. We'll
use `P`, `Q`, `R`, and `S` as metavariables over type schemas, where a
type schema is a type expression that may or may not contain unbound
type variables. For instance, we might have
P ≡ Int
P ≡ α -> α
P ≡ ∀α. α -> α
P ≡ ∀α. α -> β
etc.
A box type will be a type expression that contains exactly one free
type variable. Some examples (using OCaml's type conventions):
α Maybe
α List
(α, P) Tree (assuming P contains no free type variables)
(α, α) Tree
The idea is that whatever type the free type variable α might be,
the boxed type will be a box that "contains" an object of type α.
For instance, if `α List` is our box type, and α is the basic type
Int, then in this context, `Int List` is the type of a boxed integer.
We'll often write box types as a box containing the value of the free
type variable. So if our box type is `α List`, and `α == Int`, we
would write
Int
for the type of a boxed Int.
At the most general level, we'll talk about *Kleisli arrows*:
P -> Q
A Kleisli arrow is the type of a function from objects of type P to
objects of type box Q, for some choice of type expressions P and Q.
For instance, the following are arrows:
Int -> Bool
Int List -> Int List
Note that the left-hand schema can itself be a boxed type. That is,
if `α List` is our box type, we can write the second arrow as
Int -> Int
We'll need a number of schematic functions to help us maneuver in the presence
of box types. We will want to define a different instance of each of
these for whichever box type we're dealing with:
mid (/εmaidεnt@tI/ aka unit, return, pure): P -> P
map (/maep/): (P -> Q) -> P -> Q
map2 (/maep/): (P -> Q -> R) -> P -> Q -> R
mapply (/εm@plai/): P -> Q -> P -> Q
mcompose (aka <=<): (Q -> R) -> (P -> Q) -> (P -> R)
mbind (aka >>=): ( Q) -> (Q -> R) -> ( R)
mflipcompose (aka >=>): (P -> Q) -> (Q -> R) -> (P -> R)
mflipbind (aka =<<) ( Q) -> (Q -> R) -> ( R)
mjoin: P -> P
Note that `mcompose` and `mbind` are interdefinable: u >=> k ≡ \a. (ja >>= k)
.
In most cases of interest, the specific instances of these types will
provide certain useful guarantees.
* ***Mappable*** ("functors") At the most general level, some box types are *Mappable*
if there is a `map` function defined for that boxt type with the type given above.
* ***MapNable*** ("applicatives") A Mappable box type is *MapNable*
if there are in addition `map2`, `mid`, and `mapply`.
* ***Monad*** ("composable") A MapNable box type is a *Monad* if
there is in addition a `mcompose` and `join`. In addition, in
order to qualify as a monad, `mid` must be a left and right
identity for mcompose, and mcompose must be associative. That
is, the following "laws" must hold:
mcompose mid k = k
mcompose k mid = k
mcompose (mcompose j k) l = mcompose j (mcompose k l)
To take a trivial example (but still useful, as we will see), consider
the identity box type Id: `α -> α`. In terms of the box analogy, the
Identity box type is an invisible box. With the following definitions
mid ≡ \p.p
mcompose ≡ \f\g\x.f(gx)
Id is a monad. Here is a demonstration that the laws hold:
mcompose mid k == (\f\g\x.f(gx)) (\p.p) k
~~> \x.(\p.p)(kx)
~~> \x.kx
~~> k
mcompose k mid == (\f\g\x.f(gx)) k (\p.p)
~~> \x.k((\p.p)x)
~~> \x.kx
~~> k
mcompose (mcompose j k) l == mcompose ((\f\g\x.f(gx)) j k) l
~~> mcompose (\x.j(kx)) l
== (\f\g\x.f(gx)) (\x.j(kx)) l
~~> \x.(\x.j(kx))(lx)
~~> \x.j(k(lx))
mcompose j (mcompose k l) == mcompose j ((\f\g\x.f(gx)) k l)
~~> mcompose j (\x.k(lx))
== (\f\g\x.f(gx)) j (\x.k(lx))
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
Id is the favorite monad of mimes everywhere.
To take a slightly less trivial example, consider the box type `α
List`, with the following operations:
mcompose f g p = [r | q <- g p, r <- f q]
In words, if g maps a P to a list of Qs, and f maps a Q to a list of
Rs, then mcompose f g maps a P to a list of Rs by first feeding the P
to g, then feeding each of the Qs delivered by g to f. For example,
let f q = [q, q+1] in
let g p = [p*p, p+p] in
mcompose f g 7 = [49, 50, 14, 15]
It is easy to see that these definitions obey the monad laws (see exercises).