-<!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
+<!-- λ Λ ∀ ≡ α β γ ρ ω Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 -->
<!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
Introducing Monads
to instead just look at the formal definition. We'll give that to you below, but it's
sometimes sloganized as
[A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
-Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
+Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
+
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
+monadic types place values inside of *boxes*, and that monads wrap
+and unwrap boxes to expose or enclose the values inside of them. In
any case, our emphasis will be on starting with the abstract structure
-of monads, followed by instances of monads from the philosophical and
+of monads, followed in coming weeks by instances of monads from the philosophical and
linguistics literature.
-## Box types: type expressions with one free type variable
+> <small>After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. 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.</small>
+
+
+## Box types: type expressions with one free type variable ##
Recall that we've been using lower-case Greek letters
<code>α, β, γ, ...</code> as type variables. We'll
P_3 ≡ ∀α. α -> α
P_4 ≡ ∀α. α -> β
-etc.
+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
(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,
-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 type
-`int`, then in this context, `int list` is the type of a boxed integer.
+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 naturally thought of as "containers" (lists, trees, and so on, with `α`s as their "elments"), with later examples we discuss it will less intuitive to describe the box types that way. For example, where `R` is some fixed type, `R -> α` is a box type.
+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.
-The *box type* is the type `α list` (or as we might just say, `list`); the *boxed type* is some specific instantiantion of the free type variable `α`. We'll often write boxed types as a box containing the instance of the free
-type variable. So if our box type is `α list`, and `α` is instantiated with the specific type `int`, we would write:
+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:
-<u>int</u>
+<code><u>int</u></code>
-for the type of a boxed `int`. (We'll fool with the markup to make this a genuine box later; for now it will just display as underlined.)
+for the type of a boxed `int`.
-## Kleisli arrows
+## 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 -> <u>Q</u>
+<code>P -> <u>Q</u></code>
+
+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:
+
+<code>int -> <u>bool</u></code>
+
+<code>int list -> <u>int list</u></code>
+
+In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
+
+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:
+
+<code><u>int</u> -> <u>int list</u></code>
-That is, they are functions from objects of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`.
-For instance, the following are Kleisli arrows:
+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 -> <u>bool</u>
+<code><u>int</u> -> <span class="box2">int</span></code>
-int list -> <u>int list</u>
+We have to be careful though not to to unthinkingly equivocate between different kinds of boxes.
-In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <u>Q</u> is <u>bool</u>).
+Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
-Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
-if `α list` is our box type, we can write the second arrow as
+<pre>\x. [x]
+\x. [odd? x, odd? x]
+\x. prime_factors_of x
+\x. [0, 0, 0]</pre>
-<u>int</u> -> <u>Q</u>
+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.
-We'll need a number of classes of 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. (This will
-become clear shortly.)
+
+## 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.)
<code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
+> In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
+
<code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
-<code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
+> In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
+
+<code>mid (/εmaidεnt@tI/): P -> <u>P</u></code>
+
+> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `𝟭`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`.
<code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
+> 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.<*>`.
+
<code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
-<code>>=> or mpmoc (m-flipcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
+> In Haskell, this is `Control.Monad.<=<`.
+
+<code>>=> or flip mcomp : (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
+
+> In Haskell, this is `Control.Monad.>=>`. 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.
<code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
-<code>=<<mdnib (or m-flipbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
+<code>=<< or flip mbind : (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
+
+<code>join: <span class="box2">P</span> -> <u>P</u></code>
-<code>join: <u>2<u>P</u></u> -> <u>P</u></code>
+> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
-The managerie 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: <code>k <=< j ≡
-\a. (j a >>= k)</code>.
+Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, but it's too deeply entrenched to change. We've at least preprended an "m" to the front of "bind".
-In most cases of interest, instances of these systems of functions will provide
-certain useful guarantees.
+Haskell's names "return" and "pure" for `mid` are even less well chosen, and we think it will be clearer in our discussion to use a different name. (Also, in other theoretical contexts this notion goes by other names, anyway, like `unit` or `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.)
+
+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: <code>k <=< j ≡ \a. (j a >>= k)</code>. We'll state some other interdefinitions below.
+
+We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), 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.
+
+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:
- LAWS
+ <code>map (id : α -> α) == (id : <u>α</u> -> <u>α</u>)</code>
+ <code>map (g ○ f) == (map g) ○ (map f)</code>
+
+ Essentially these say that `map` is a homomorphism from the algebra of `(universe α -> β, operation ○, elsment id)` to that of <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>. 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 <code><u>α</u></code>), 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.
+
+ > <small>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 <code><u>_</u></code> and of the `map` that goes together with it.</small>
+
* ***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:
-
-
-* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
+ have to obey the following MapN Laws:
+
+ 1. <code>mid (id : P->P) : <u>P</u> -> <u>P</u></code> 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`.
+
+<!-- Probably there's a shorter proof, but:
+ mid T m$ xs m$ mid f
+== mid T m$ ((mid id) m$ xs) m$ mid f, by 1
+== mid (○) m$ mid T m$ mid id m$ xs m$ mid f, by 3
+== mid ($id) m$ (mid (○) m$ mid T) m$ xs m$ mid f, by 4
+== mid (○) m$ mid ($id) m$ mid (○) m$ mid T m$ xs m$ mid f, by 3
+== mid ((○) ($id)) m$ mid (○) m$ mid T m$ xs m$ mid f, by 2
+== mid ((○) ($id) (○)) m$ mid T m$ xs m$ mid f, by 2
+== mid id m$ mid T m$ xs m$ mid f, by definitions of ○ and $
+== mid T m$ xs m$ mid f, by 1
+== mid ($f) m$ (mid T m$ xs), by 4
+== mid (○) m$ mid ($f) m$ mid T m$ xs, by 3
+== mid ((○) ($f)) m$ mid T m$ xs, by 2
+== mid ((○) ($f) T) m$ xs, by 2
+== mid f m$ xs, by definitions of ○ and $ and T == flip ($)
+-->
+
+* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
is in addition an associative `mcomp` having `mid` as its left and
right identity. That is, the following Monad Laws must hold:
- mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
- mcomp mid k (that is, mid <=< k) = k
- mcomp k mid (that is, k <=< mid) = k
+ mcomp (mcomp j k) l (that is, (j <=< k) <=< l) == mcomp j (mcomp k l)
+ mcomp mid k (that is, mid <=< k) == k
+ mcomp k mid (that is, k <=< mid) == k
-If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
-Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
-are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
+ You could just as well express the Monad laws using `>=>`:
-Here are some interdefinitions: TODO. Names in Haskell TODO.
+ l >=> (k >=> j) == (l >=> k) >=> j
+ k >=> mid == k
+ mid >=> k == k
-## Examples
+ 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.
-To take a trivial (but, as we will see, still useful) example,
-consider the identity box type Id: `α`. So if `α` is type `bool`,
-then a boxed `α` is ... a `bool`. In terms of the box analogy, the
-Identity box type is a completely invisible box. With the following
-definitions
+ In practice, you will often work with `>>=`. In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators. This looks similar, but doesn't have the same symmetry:
- mid ≡ \p. p
- mcomp ≡ \f g x.f (g x)
+ u >>= (\a -> k a >>= j) == (u >>= k) >>= j
+ u >>= mid == u
+ mid a >>= k == k a
-Identity is a monad. Here is a demonstration that the laws hold:
+ Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.
- 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))
-
-Id is the favorite monad of mimes.
+ > <small>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`:
+ > <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
+ > The Monad Laws then take the form:
+ > <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>
+ > 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.<p>
+ > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
+ > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre></small>
+ > 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.
-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 f g a = concat (map f (g a))
- = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
- = [c | b <- g a, c <- f b]
+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.
-These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises).
+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.
-In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
-each `β` in that list is fed to `f`, which returns a list of `γ`s. The
-final result is the concatenation of those lists of `γ`s.
+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 example:
+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`).
- let f b = [b, b+1] in
- let g a = [a*a, a+a] in
- mcomp f g 7 ==> [49, 50, 14, 15]
+<!--
+Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the primitive Monad operations? For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`.
+-->
-`g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
-Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
+## Interdefinitions and Subsidiary notions##
- let gs = [(\a->a*a),(\a->a+a)] in
- let xs = [7, 5] in
- mapply gs xs ==> [49, 25, 14, 10]
+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:
+<pre>
+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 -> <u>β</u>; k and l have types of the same form
+u : <u>α</u>; v and xs and ys have types of the same form
-As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to identify the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the Monadic functions of the Option/Maybe box type.
+w : <span class="box2">α</span>
+</pre>
+But we may sometimes slip.
-Safe division
--------------
+Here are some ways the different notions are related:
-Integer division presupposes that its second argument
-(the divisor) is not zero, upon pain of presupposition failure.
-Here's what my OCaml interpreter says:
+<pre>
+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 ○ f
+</pre>
- # 12/0;;
- Exception: Division_by_zero.
-Say we want to explicitly allow for the possibility that
-division will return something other than a number.
-To do that, we'll use OCaml's `option` type, which works like this:
+Here are some other monadic notion that you may sometimes encounter:
- # type 'a option = None | Some of 'a;;
- # None;;
- - : 'a option = None
- # Some 3;;
- - : int option = Some 3
+* <code>mzero</code> is a value of type <code><u>α</u></code> 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`.
-So if a division is normal, we return some number, but if the divisor is
-zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
+* 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.*>`.
-<pre>
-let safe_div (x:int) (y:int) =
- match y with
- | 0 -> None
- | _ -> Some (x / y);;
-
-(*
-val safe_div : int -> int -> int option = fun
-# safe_div 12 2;;
-- : int option = Some 6
-# safe_div 12 0;;
-- : int option = None
-# safe_div (safe_div 12 2) 3;;
-# safe_div (safe_div 12 2) 3;;
- ~~~~~~~~~~~~~
-Error: This expression has type int option
- but an expression was expected of type int
-*)
-</pre>
+* 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]`. You might expect Haskell to call `<*` `<<`, but they don't. They used to use `<<` for `flip (>>)` instead, but now they seem not to use `<<` anymore.
-This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
-just the behavior we were hoping for. But we want to be able to use
-the output of the safe-division function as input for further division
-operations. So we have to jack up the types of the inputs:
+* <code>mapconst</code> 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 ()`.
-<pre>
-let safe_div2 (u:int option) (v:int option) =
- match u with
- None -> None
- | Some x -> (match v with
- Some 0 -> None
- | Some y -> Some (x / y));;
-
-(*
-val safe_div2 : int option -> int option -> int option = <fun>
-# safe_div2 (Some 12) (Some 2);;
-- : int option = Some 6
-# safe_div2 (Some 12) (Some 0);;
-- : int option = None
-# safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
-- : int option = None
-*)
-</pre>
-Beautiful, just what we need: now we can try to divide by anything we
-want, without fear that we're going to trigger any system errors.
-I prefer to line up the `match` alternatives by using OCaml's
-built-in tuple type:
+## Examples ##
-<pre>
-let safe_div2 (u:int option) (v:int option) =
- match (u, v) with
- | (None, _) -> None
- | (_, None) -> None
- | (_, Some 0) -> None
- | (Some x, Some y) -> Some (x / y);;
-</pre>
+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, <code><u>α</u> == α</code>.
+In terms of the box analogy, the Identity box type is a completely invisible box. With the following
+definitions:
-So far so good. But what if we want to combine division with
-other arithmetic operations? We need to make those other operations
-aware of the possibility that one of their arguments has triggered a
-presupposition failure:
+ 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)
-<pre>
-let safe_add (u:int option) (v:int option) =
- match (u, v) with
- | (None, _) -> None
- | (_, None) -> None
- | (Some x, Some y) -> Some (x + y);;
-
-(*
-val safe_add : int option -> int option -> int option = <fun>
-# safe_add (Some 12) (Some 4);;
-- : int option = Some 16
-# safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
+Identity is a monad. Here is a demonstration that the laws hold:
-This works, but is somewhat disappointing: the `safe_add` operation
-doesn't trigger any presupposition of its own, so it is a shame that
-it needs to be adjusted because someone else might make trouble.
-
-But we can automate the adjustment. The standard way in OCaml,
-Haskell, and other functional programming languages, is to use the monadic
-`bind` operator, `>>=`. (The name "bind" is not well chosen from our
-perspective, but this is too deeply entrenched by now.) As mentioned above,
-there needs to be a different `>>=` operator for each Monad or box type you're working with.
-Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
-symbol and it will calculate from the context of the surrounding type constraints what
-monad you must have meant. In OCaml, the `>>=` or `bind` operator is not pre-defined, but we will
-give you a library that has definitions for all the standard monads, as in Haskell.
-For now, though, we will define our `bind` operation by hand:
+ 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.
-<pre>
-let bind (u: int option) (f: int -> (int option)) =
- match u with
- | None -> None
- | Some x -> f x;;
+To take a slightly less trivial (and even more useful) example,
+consider the box type `α list`, with the following operations:
-let safe_add3 (u: int option) (v: int option) =
- bind u (fun x -> bind v (fun y -> Some (x + y)));;
+ 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]
-(* This is really just `map2 (+)`, using the `map2` operation that corresponds to
- definition of `bind`. *)
+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.)
-let safe_div3 (u: int option) (v: int option) =
- bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;
+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.
-(* This goes back to some of the simplicity of the original safe_div, without the complexity
- introduced by safe_div2. *)
-</pre>
+For example:
-The above definitions look even simpler if you focus on the fact that `safe_add3` can be written as simply `map2 (+)`, and that `safe_div3` could be written as `u >>= fun x -> v >>= fun y -> if 0 = y then None else Some (x / y)`. Haskell has an even more user-friendly notation for this, namely:
+ let j a = [a*a, a+a] in
+ let k b = [b, b+1] in
+ mcomp k j 7 ==> [49, 50, 14, 15]
- safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
- safe_div3 u v = do {x <- u;
- y <- v;
- if 0 == y then Nothing else return (x `div` y)}
+`j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
-Let's see our new functions in action:
+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:
-<pre>
-(*
-# safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
-- : int option = Some 2
-# safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
-- : int option = None
-# safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
-- : int option = None
-*)
-</pre>
+ let js = [(\a->a*a),(\a->a+a)] in
+ let xs = [7, 5] in
+ mapply js xs ==> [49, 25, 14, 10]
+
+
+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 <code><u>P</u></code>, 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 <code><u>P</u></code>, that is `P -> R`, and returns values of the boxed type <code><u>Q</u></code>.
+
+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.
-Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
-for `safe_add3` shows what it looks like to equip an ordinary operation to
-survive in dangerous presupposition-filled world. Note that the new
-definition of `safe_add3` does not need to test whether its arguments are
-None objects or real numbers---those details are hidden inside of the
-`bind` function.
-
-The definition of `safe_div3` shows exactly what extra needs to be said in
-order to trigger the no-division-by-zero presupposition. Here, too, we don't
-need to keep track of what presuppositions may have already failed
-for whatever reason on our inputs.
-
-(Linguistics note: Dividing by zero is supposed to feel like a kind of
-presupposition failure. If we wanted to adapt this approach to
-building a simple account of presupposition projection, we would have
-to do several things. First, we would have to make use of the
-polymorphism of the `option` type. In the arithmetic example, we only
-made use of `int option`s, but when we're composing natural language
-expression meanings, we'll need to use types like `N option`, `Det option`,
-`VP option`, and so on. But that works automatically, because we can use
-any type for the `'a` in `'a option`. Ultimately, we'd want to have a
-theory of accommodation, and a theory of the situations in which
-material within the sentence can satisfy presuppositions for other
-material that otherwise would trigger a presupposition violation; but,
-not surprisingly, these refinements will require some more
-sophisticated techniques than the super-simple Option monad.)
+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.