`α, β, γ, ...`

as type variables. We'll
use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
type variables. For instance, we might have
P_1 ≡ int
P_2 ≡ α -> α
P_3 ≡ ∀α. α -> α
P_4 ≡ ∀α. α -> β
and so on.
A *box type* will be a type expression that contains exactly one free
type variable. (You could extend this to expressions with more free variables; then you'd have
to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
α option
α list
(α, R) tree (assuming R contains no free type variables)
(α, α) tree
The idea is that whatever type the free type variable `α` might be instantiated to,
we will have a "type box" of a certain sort that "contains" values of type `α`. For instance,
if `α list` is our box type, and `α` instantiates to the type `int`, then in this context, `int list`
is the type of a boxed integer.
Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `α`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> α` will be one box type we work extensively with.
Also, for clarity: the *box type* is the type `α list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `α`. We'll often write boxed types as a box containing what the free
type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we write:
__int__

for the type of a boxed `int`.
## Kleisli arrows ##
A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Don't worry about why they're called that, or if you like go read some Category Theory. All we need to know is that these are functions whose type has the form:
`P -> `__Q__

That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of box and of type expressions `P` and `Q`.
For instance, the following are Kleisli arrow types:
`int -> `__bool__

`int list -> `__int list__

In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type __Q__

is __bool__

).
Note that either of the schemas `P` or `Q` are permitted to themselves be boxed
types. That is, if `α list` is our box type, we can write the second type as:
__int__ -> __int list__

And also what the rhs there is a boxing of is itself a boxed type (with the same kind of box):, so we can write it as:
__int__ -> int

We have to be careful though not to to unthinkingly equivocate between different kinds of boxes.
Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are `int -> `__int__

(that is, `int -> int list`) or `int -> `__bool__

:
\x. [x] \x. [odd? x, odd? x] \x. prime_factors_of x \x. [0, 0, 0]As semanticists, you are no doubt familiar with the debates between those who insist that propositions are sets of worlds and those who insist they are context change potentials. We hope to show you, in coming weeks, that propositions are (certain sorts of) Kleisli arrows. But this doesn't really compete with the other proposals; it is a generalization of them. Both of the other proposed structures can be construed as specific Kleisli arrow types. ## A family of functions for each box type ## We'll need a family of functions to help us work with box types. As will become clear, these have to be defined differently for each box type. Here are the types of our crucial functions, together with our pronunciation, and some other names the functions go by. (Usually the type doesn't fix their behavior, which will be discussed below.)

`map (/mæp/): (P -> Q) -> `__P__ -> __Q__

> In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
`map2 (/mæptu/): (P -> Q -> R) -> `__P__ -> __Q__ -> __R__

> In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
`mid (/εmaidεnt@tI/): P -> `__P__

> This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`. It will be a way of boxing values with your box type that plays a distinguished role in the various Laws and interdefinitions we present below.
> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. All of these names are somewhat unfortunate. First, it has little to do with `η`-reduction in the Lambda Calculus. Second, it has little to do with the `() : unit` value we discussed in earlier classes. Third, it has little to do with the `return` keyword in C and other languages; that's more closely related to continuations, which we'll discuss in later weeks. Finally, this doesn't perfectly align with other uses of "pure" in the literature. `mid`'d values _will_ generally be "pure" in the other senses, but other boxed values can be too.
> For all these reasons, we're thinking it will be clearer in our discussion to use a different name. In the class presentation Jim called it `𝟭`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".)
`m$ or mapply (/εm@plai/): `__P -> Q__ -> __P__ -> __Q__

> We'll use `m$` as a left-associative infix operator, reminiscent of (the right-associative) `$` which is just ordinary function application (also expressed by mere left-associative juxtaposition). In the class presentation Jim called `m$` `⚫`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`.
`<=< or mcomp : (Q -> `__R__) -> (P -> __Q__) -> (P -> __R__)

> In Haskell, this is `Control.Monad.<=<`.
`>=> or flip mcomp : (P -> `__Q__) -> (Q -> __R__) -> (P -> __R__)

> In Haskell, this is `Control.Monad.>=>`. We will move freely back and forth between using `<=<` (aka `mcomp`) and using `>=>`, which
is just `<=<` with its arguments flipped. `<=<` has the virtue that it corresponds more
closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
that its types flow more naturally from left to right.
> In the class handout, we gave the types for `>=>` twice, and once was correct but the other was a typo. The above is the correct typing.
`>>= or mbind : (`__Q__) -> (Q -> __R__) -> (__R__)

> Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, since it's only loosely connected with what we mean by "binding." But the name is too deeply entrenched to change. We've at least preprended an "m" to the front of "bind". In some presentations this operation is called `★`.
`=<< or flip mbind : (Q -> `__R__) -> (__Q__) -> (__R__)

`join: P -> `__P__

> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
The menagerie isn't quite as bewildering as you might suppose. Many of these will be interdefinable. For example, here is how `mcomp` and `mbind` are related: `k <=< j ≡ \a. (j a >>= k)`

. We'll state some other interdefinitions below.
These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
* ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
if there is a `map` function defined for that box type with the type given above. This
has to obey the following Map Laws:
`map (id : α -> α) == (id : `__α__ -> __α__)

`map (g ○ f) == (map g) ○ (map f)`

Essentially these say that `map` is a homomorphism from the algebra of `(universe α -> β, operation ○, elsment id)` to that of `(`__α__ -> __β__, ○', id')

, where `○'` and `id'` are `○` and `id` restricted to arguments of type _____

. That might be hard to digest because it's so abstract. Think of the following concrete example: if you take a `α list` (that's our __α__

), and apply `id` to each of its elements, that's the same as applying `id` to the list itself. That's the first law. And if you apply the composition of functions `g ○ f` to each of the list's elements, that's the same as first applying `f` to each of the elements, and then going through the elements of the resulting list and applying `g` to each of those elements. That's the second law. These laws obviously hold for our familiar notion of `map` in relation to lists.
> As mentioned at the top of the page, in Category Theory presentations of monads they usually talk about "endofunctors", which are mappings from a Category to itself. In the uses they make of this notion, the endofunctors combine the role of a box type _____

and of the `map` that goes together with it.
* ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
if there are in addition `map2`, `mid`, and `mapply`. (Given either
of `map2` and `mapply`, you can define the other, and also `map`.
Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
have to obey the following MapN Laws:
1. `mid (id : P->P) : `__P__ -> __P__

is a left identity for `m$`, that is: `(mid id) m$ xs = xs`
2. `mid (f a) = (mid f) m$ (mid a)`
3. The `map2`ing of composition onto boxes `fs` and `gs` of functions, when `m$`'d to a box `xs` of arguments == the `m$`ing of `fs` to the `m$`ing of `gs` to xs: `(mid (○) m$ fs m$ gs) m$ xs = fs m$ (gs m$ xs)`.
4. When the arguments (the right-hand operand of `m$`) are an `mid`'d value, the order of `m$`ing doesn't matter: `fs m$ (mid x) = mid ($x) m$ fs`. (Though note that it's `mid ($x)`, or `mid (\f. f x)` that gets `m$`d onto `fs`, not the original `mid x`.) Here's an example where the order *does* matter: `[succ,pred] m$ [1,2] == [2,3,0,1]`, but `[($1),($2)] m$ [succ,pred] == [2,0,3,1]`. This Law states a class of cases where the order is guaranteed not to matter.
5. A consequence of the laws already stated is that when the _left_-hand operand of `m$` is a `mid`'d value, the order of `m$`ing doesn't matter either: `mid f m$ xs == map (flip ($)) xs m$ mid f`.
* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
is in addition an associative `mcomp` having `mid` as its left and
right identity. That is, the following Monad Laws must hold:
mcomp (mcomp j k) l (that is, (j <=< k) <=< l) == mcomp j (mcomp k l)
mcomp mid k (that is, mid <=< k) == k
mcomp k mid (that is, k <=< mid) == k
You could just as well express the Monad laws using `>=>`:
l >=> (k >=> j) == (l >=> k) >=> j
k >=> mid == k
mid >=> k == k
If you studied algebra, you'll remember that a mon*oid* is a universe with some associative operation that has an identity. For example, the natural numbers form a monoid with multiplication as the operation and `1` as the identity, or with addition as the operation and `0` as the identity. Strings form a monoid with concatenation as the operation and the empty string as the identity. (This example shows that the operation need not be commutative.) Monads are a kind of generalization of this notion, and that's why they're named as they are. The key difference is that for monads, the values being operated on need not be of the same type. They *can* be, if they're all Kleisli arrows of a single type `P -> `__P__

. But they needn't be. Their types only need to "cohere" in certain ways.
In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators `>=>` or `<=<`. This looks similar, but doesn't have the same symmetry:
u >>= (\a -> k a >>= j) == (u >>= k) >>= j
u >>= mid == u
mid a >>= k == k a
(Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.) Some authors try to make the first of those Laws look more symmetrical by writing it as:
(A >>= \a -> B) >>= \b -> C == A >>= (\a -> B >>= \b -> C)
If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others. Also, with these functions you can define `m$` and `map2` from *MapNables*. So with Monads, all you really need to get the whole system of functions are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
> In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `μ`) and `mid` (which they call `η`). These are assumed to be "natural transformations" for their box type, which means that they satisfy these equations with that box type's `map`:
> map f ○ mid == mid ○ f> The Monad Laws then take the form: >

map f ○ join == join ○ map (map f)

join ○ (map join) == join ○ join> The first of these says that if you have a triply-boxed type, and you first merge the inner two boxes (with `map join`), and then merge the resulting box with the outermost box, that's the same as if you had first merged the outer two boxes, and then merged the resulting box with the innermost box. The second law says that if you take a box type and wrap a second box around it (with `mid`) and then merge them, that's the same as if you had done nothing, or if you had instead wrapped a second box around each element of the original (with `map mid`, leaving the original box on the outside), and then merged them.

join ○ mid == id == join ○ map mid

> The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type

:
> __α__

μ ○ M(μ) == μ ○ μ> A word of advice: if you're doing any work in this conceptual neighborhood and need a Greek letter, don't use μ. In addition to the preceding usage, there's also a use in recursion theory (for the minimization operator), in type theory (as a fixed point operator for types), and in the λμ-calculus, which is a formal system that deals with _continuations_, which we will focus on later in the course. So μ already exhibits more ambiguity than it can handle. > We link to further reading about the Category Theory origins of Monads below. There isn't any single `mid` function, or single `mbind` function, and so on. For each new box type, this has to be worked out in a useful way. And as we hinted, in many cases the choice of box *type* still leaves some latitude about how they should be defined. We commonly talk about "the List Monad" to mean a combination of the choice of `α list` for the box type and particular definitions for the various functions listed above. There's also "the ZipList MapNable/Applicative" which combines that same box type with other choices for (some of) the functions. Many of these packages also define special-purpose operations that only make sense for that system, but not for other Monads or Mappables. As hinted in last week's homework and explained in class, the operations available in a Mappable system exactly preserve the "structure" of the boxed type they're operating on, and moreover are only sensitive to what content is in the corresponding original position. If you say `map f [1,2,3]`, then what ends up in the first position of the result depends only on how `f` and `1` combine. For MapNable operations, on the other hand, the structure of the result may instead be a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say `map2 f [10,20] [1,2,3]`, what ends up in the first position of the result depends only on how `f` and `10` and `1` combine. With `map`, you can supply an `f` such that `map f [3,2,0,1] == [[3,3,3],[2,2],[],[1]]`. But you can't transform `[3,2,0,1]` to `[3,3,3,2,2,1]`, and you can't do that with MapNable operations, either. That would involve the structure of the result (here, the length of the list) being sensitive to the content, and not merely the structure, of the original. For Monads (Composables), on the other hand, you can perform more radical transformations of that sort. For example, `join (map (\x. dup x x) [3,2,0,1])` would give us `[3,3,3,2,2,1]` (for a suitable definition of `dup`). ## Interdefinitions and Subsidiary notions## We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that:

μ ○ η == id == μ ○ M(η)

f : α -> β; g and h have types of the same form also sometimes these will have types of the form α -> β -> γ note that α and β are permitted to be, but needn't be, boxed types j : α ->But we may sometimes slip. Here are some ways the different notions are related:β; k and l have types of the same form u :α; v and xs and ys have types of the same form w : α

j >=> k ≡= \a. (j a >>= k) u >>= k == (id >=> k) u; or ((\(). u) >=> k) () u >>= k == join (map k u) join w == w >>= id map2 f xs ys == xs >>= (\x. ys >>= (\y. mid (f x y))) map2 f xs ys == (map f xs) m$ ys, using m$ as an infix operator fs m$ xs == fs >>= (\f. map f xs) m$ == map2 id map f xs == mid f m$ xs map f u == u >>= mid ○ fHere are some other monadic notion that you may sometimes encounter: *

`mzero`

is a value of type __α__

that is exemplified by `Nothing` for the box type `Maybe α` and by `[]` for the box type `List α`. It has the behavior that `anything m$ mzero == mzero == mzero m$ anything == mzero >>= anything`. In Haskell, this notion is called `Control.Applicative.empty` or `Control.Monad.mzero`.
* Haskell has a notion `>>` definable as `\u v. map (const id) u m$ v`, or as `\u v. u >>= const v`. This is often useful, and `u >> v` won't in general be identical to just `v`. For example, using the box type `List α`, `[1,2,3] >> [4,5] == [4,5,4,5,4,5]`. But in the special case of `mzero`, it is a consequence of what we said above that `anything >> mzero == mzero`. Haskell also calls `>>` `Control.Applicative.*>`.
* Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. map const u m$ v`. For example, `[1,2,3] <* [4,5] == [1,1,2,2,3,3]`.
* `mapconst`

is definable as `map ○ const`. For example `mapconst 4 [1,2,3] == [4,4,4]`. Haskell calls `mapconst` `<$` in `Data.Functor` and `Control.Applicative`. They also use `$>` for `flip mapconst`, and `Control.Monad.void` for `mapconst ()`.
## Examples ##
To take a trivial (but, as we will see, still useful) example,
consider the Identity box type: `α`. So if `α` is type `bool`,
then a boxed `α` is ... a `bool`. That is, __α__ == α

.
In terms of the box analogy, the Identity box type is a completely invisible box. With the following
definitions:
mid ≡ \p. p, that is, our familiar combinator I
mcomp ≡ \f g x. f (g x), that is, ordinary function composition (○) (aka the B combinator)
Identity is a monad. Here is a demonstration that the laws hold:
mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
~~> \x.(\p.p)(kx)
~~> \x.kx
~~> k
mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
~~> \x.k((\p.p)x)
~~> \x.kx
~~> k
mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
~~> mcomp (\x.j(kx)) l
≡ (\fgx.f(gx)) (\x.j(kx)) l
~~> \x.(\x.j(kx))(lx)
~~> \x.j(k(lx))
mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
~~> mcomp j (\x.k(lx))
≡ (\fgx.f(gx)) j (\x.k(lx))
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
The Identity monad is favored by mimes.
To take a slightly less trivial (and even more useful) example,
consider the box type `α list`, with the following operations:
mid : α -> [α]
mid a = [a]
mcomp : (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
mcomp k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
= foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
= [c | b <- j a, c <- k b]
In the first two definitions of `mcomp`, we give the definition first in Haskell and then in the equivalent OCaml. The three different definitions of `mcomp` (one for each line) are all equivalent, and it is easy to show that they obey the Monad Laws. (You will do this in the homework.)
In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
each `β` in that list is fed to `k`, which returns a list of `γ`s. The
final result is the concatenation of those lists of `γ`s.
For example:
let j a = [a*a, a+a] in
let k b = [b, b+1] in
mcomp k j 7 ==> [49, 50, 14, 15]
`j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
Contrast that to `m$` (`mapply`), which operates not on two *box-producing functions*, but instead on two *boxed type values*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
let js = [(\a->a*a),(\a->a+a)] in
let xs = [7, 5] in
mapply js xs ==> [49, 25, 14, 10]
These implementations of `<=<` and `m$` for lists use the "crossing" strategy for pairing up multiple lists, as opposed to the "zipping" strategy. Nothing forces that choice; you could also define `m$` using the "zipping" strategy instead. (But then you wouldn't be able to build a corresponding Monad; see below.) Haskell talks of the List Monad in the first case, and the ZipList Applicative in the second case.
Sticking with the "crossing" strategy, here's how to motivate our implementation of `<=<`. Recall that we have on the one hand, an operation `filter` for lists, that applies a predicate to each element of the list, and returns a list containing only those elements which satisfied the predicate. But the elements it does retain, it retains unaltered. On the other hand, we have the operation `map` for lists, that is capable of _changing_ the list elements in the result. But it doesn't have the power to throw list elements away; elements in the source map one-to-one to elements in the result. In many cases, we want something in between `filter` and `map`. We want to be able to ignore or discard some list elements, and possibly also to change the list elements we keep. One way of doing this is to have a function `optmap`, defined like this:
let rec optmap (f : α -> β option) (xs : α list) : β list =
match xs with
| [] -> []
| x' :: xs' ->
(match f x' with
| None -> optmap f xs'
| Some b -> b :: optmap f xs')
Then we retain only those `α`s for which `f` returns `Some b`; when `f` returns `None`, we just leave out any corresponding element in the result.
That can be helpful, but it only enables us to have _zero or one_ elements in the result corresponding to a given element in the source list. What if we sometimes want more? Well, here is a more general approach:
let rec catmap (k : α -> β list) (xs : α list) : β list =
match xs with
| [] -> []
| x' :: xs' -> List.append (k x') (catmap f xs')
Now we can have as many elements in the result for a given `α` as `k` cares to return. Another way to write `catmap k xs` is as `List.concat (map k xs)`. And this is just the definition of `mbind` or `>>=` for the List Monad. The definition of `mcomp` or `<=<`, that we gave above, differs only in that it's the way to compose two functions `j` and `k`, that you'd want to `catmap`, rather than the way to `catmap` one of those functions over a value that's already a list.
This example is a good intuitive basis for thinking about the notions of `mbind` and `mcomp` more generally. Thus `mbind` for the option/Maybe type takes an option value, applies `k` to its element (if there is one), and returns the resulting option value. `mbind` for a tree with `α`-labeled leaves would apply `k` to each of the leaves, and return a tree containing arbitrarily large subtrees in place of all its former leaves, depending on what `k` returned.
[3, 2, 0, 1] >>=Though as we warned before, only some of the Monads we'll be working with are naturally thought of "containers"; so in other cases the similarity of their `mbind` operations to what we have here will be more abstract. The question came up in class of **when box types might fail to be Mappable, or Mappables might fail to be MapNables, or MapNables might fail to be Monads**. For the first failure, we noted that it's easy to define a `map` operation for the box type `R -> α`, for a fixed type `R`. You `map` a function of type `P -> Q` over a value of the boxed type_{α list}(\a -> dup a a) ==> [3, 3, 3, 2, 2, 1] Some a >>=_{α option}(\a -> Some 0) ==> Some 0 None >>=_{α option}(\a -> Some 0) ==> None . _____ / \ . / \ . 3 >>=_{(α,unit) tree}(\a -> / \ ) ==> _/_ . / \ a a / \ / \ 1 2 . . 3 3 / \ / \ 1 1 2 2

__P__

, that is a value of type `R -> P`, by just returning a function that takes some `R` as input, first supplies it to your `R -> P` value, and then supplies the result to your `map`ped function of type `P -> Q`. (We will be working with this Mappable extensively; in fact it's not just a Mappable but more specifically a Monad.)
But if on the other hand, your box type is `α -> R`, you'll find that there is no way to define a `map` operation that takes arbitrary functions of type `P -> Q` and values of the boxed type __P__

, that is `P -> R`, and returns values of the boxed type __Q__

.
For the second failure, that is cases of Mappables that are not MapNables, we cited box types like `(R, α)`, for arbitrary fixed types `R`. The `map` operation for these is defined by `map f (r,a) = (r, f a)`. For certain choices of `R` these can be MapNables too. The easiest case is when `R` is the type of `()`. But when we look at the MapNable Laws, we'll see that they impose constraints we cannot satisfy for *every* choice of the fixed type `R`. Here's why. We'll need to define `mid a = (r0, a)` for some specific `r0` of type `R`. The choice can't depend on the value of `a`, because `mid` needs to work for `a`s of _any_ type. Then the MapNable Laws will entail:
1. (r0,id) m$ (r,x) == (r,x)
2. (r0,f x) == (r0,f) m$ (r0,x)
3. (r0,(○)) m$ (r'',f) m$ (r',g) m$ (r,x) == (r'',f) m$ ((r',g) m$ (r,x))
4. (r'',f) m$ (r0,x) == (r0,($x)) m$ (r'',f)
5. (r0,f) m$ (r,x) == (r,($x)) m$ (r0,f)
Now we are not going to be able to write a `m$` function that inspects the second element of its left-hand operand to check if it's the `id` function; the identity of functions is not decidable. So the only way to satisfy Law 1 will be to have the first element of the result (`r`) be taken from the first element of the right-hand operand in _all_ the cases when the first element of the left-hand operand is `r0`. But then that means that the result of the lhs of Law 5 will also have a first element of `r`; so, turning now to the rhs of Law 5, we see that `m$` must use the first element of its _left_-hand operand (here again `r`) at least in those cases when the first element of its right-hand operand is `r0`. If our `R` type has a natural *monoid* structure, we could just let `r0` be the monoid's identity, and have `m$` combine other `R`s using the monoid's operation. Alternatively, if the `R` type is one that we can safely apply the predicate `(r0==)` to, then we could define `m$` something like this:
let (m$) (r1,f) (r2,x) = ((if r0==r1 then r2 else if r0==r2 then r1 else ...), ...)
But for some types neither of these will be the case. For function types, as we already mentioned, `==` is not decidable. If the functions have suitable types, they do form a monoid with `○` as the operation and `id` as the identity; but many function types won't be such that arbitrary functions of that type are composable. So when `R` is the type of functions from `int`s to `bool`s, for example, we won't have any way to write a `m$` that satisfies the constraints stated above.
For the third failure, that is examples of MapNables that aren't Monads, we'll just state that lists where the `map2` operation is taken to be zipping rather than taking the Cartesian product (what in Haskell are called `ZipList`s), these are claimed to exemplify that failure. But we aren't now in a position to demonstrate that to you.
## Further Reading ##
As we mentioned above, the notions of Monads have their origin in Category Theory, where they are mostly specified in terms of (what we call) `mid` and `join`. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in Category Theory:
[1](http://en.wikipedia.org/wiki/Outline_of_category_theory)
[2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/)
[3](http://en.wikibooks.org/wiki/Haskell/Category_theory)
[4](https://wiki.haskell.org/Category_theory), where you should follow the further links discussing Functors, Natural Transformations, and Monads.
Here are some papers that introduced Monads into functional programming:
* Eugenio Moggi, Notions of Computation and Monads: Information and Computation 93 (1) 1991. This paper is available online, but would be very difficult reading for members of this seminar, so we won't link to it. However, the next two papers should be accessible.
* [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
* [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
in M. Broy, editor, *Marktoberdorf Summer School on Program Design
Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
editors, *Advanced Functional Programming*, Springer Verlag,
LNCS 925, 1995. Some errata fixed August 2001.
Here is some other reading:
* [Yet Another Haskell Tutorial on Monad Laws](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Definition)
* [Haskell wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads)
* [Haskell wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads)
* [Haskell wiki on Monad Laws](http://www.haskell.org/haskellwiki/Monad_laws)
* [Haskell wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation)
* [Yet Another Haskell Tutorial on do-notation](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Do_Notation)
There's a long list of monad tutorials linked at the [[Haskell wiki|https://wiki.haskell.org/Monad_tutorials_timeline]] (we linked to this at the top of the page), and on our own [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you.