+++ /dev/null
-Monads in General
------------------
-
-We've just seen a way to separate thinking about error conditions
-(such as trying to divide by zero) from thinking about normal
-arithmetic computations. We did this by making use of the `option`
-type: in each place where we had something of type `int`, we put
-instead something of type `int option`, which is a sum type consisting
-either of one choice with an `int` payload, or else a `None` choice
-which we interpret as signaling that something has gone wrong.
-
-The goal was to make normal computing as convenient as possible: when
-we're adding or multiplying, we don't have to worry about generating
-any new errors, so we would rather not think about the difference
-between `int`s and `int option`s. We tried to accomplish this by
-defining a `bind` operator, which enabled us to peel away the `option`
-husk to get at the delicious integer inside. There was also a
-homework problem which made this even more convenient by defining a
-`lift` operator that mapped any binary operation on plain integers
-into a lifted operation that understands how to deal with `int
-option`s in a sensible way.
-
-So what exactly is a monad? We can consider a monad to be a system
-that provides at least the following three elements:
-
-* A complex type that's built around some more basic type. Usually
- the complex type will be polymorphic, and so can apply to different basic types.
- In our division example, the polymorphism of the `'a option` type
- provides a way of building an option out of any other type of object.
- People often use a container metaphor: if `u` has type `int option`,
- then `u` is a box that (may) contain an integer.
-
- type 'a option = None | Some of 'a;;
-
-* A way to turn an ordinary value into a monadic value. In OCaml, we
- did this for any integer `x` by mapping it to
- the option `Some x`. In the general case, this operation is
- known as `unit` or `return.` Both of those names are terrible. This
- operation is only very loosely connected to the `unit` type we were
- discussing earlier (whose value is written `()`). It's also only
- very loosely connected to the "return" keyword in many other
- programming languages like C. But these are the names that the literature
- uses. [The rationale for "unit" comes from the monad laws
- (see below), where the unit function serves as an identity,
- just like the unit number (i.e., 1) serves as the identity
- object for multiplication. The rationale for "return" comes
- from a misguided desire to resonate with C programmers and
- other imperative types.]
-
- The unit/return operation is a way of lifting an ordinary object into
- the monadic box you've defined, in the simplest way possible. You can think
- of the singleton function as an example: it takes an ordinary object
- and returns a set containing that object. In the example we've been
- considering:
-
- let unit x = Some x;;
- val unit : 'a -> 'a option = <fun>
-
- So `unit` is a way to put something inside of a monadic box. It's crucial
- to the usefulness of monads that there will be monadic boxes that
- aren't the result of that operation. In the Option/Maybe monad, for
- instance, there's also the empty box `None`. In another (whimsical)
- example, you might have, in addition to boxes merely containing integers,
- special boxes that contain integers and also sing a song when they're opened.
-
- The unit/return operation will always be the simplest, conceptually
- most straightforward way to lift an ordinary value into a monadic value
- of the monadic type in question.
-
-* Thirdly, an operation that's often called `bind`. As we said before, this is another
- unfortunate name: this operation is only very loosely connected to
- what linguists usually mean by "binding." In our Option/Maybe monad, the
- bind operation is:
-
- let bind u f = match u with None -> None | Some x -> f x;;
- val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
-
- Note the type: `bind` takes two arguments: first, a monadic box
- (in this case, an `'a option`); and second, a function from
- ordinary objects to monadic boxes. `bind` then returns a monadic
- value: in this case, a `'b option` (you can start with, e.g., `int option`s
- and end with `bool option`s).
-
- Intuitively, the interpretation of what `bind` does is this:
- the first argument is a monadic value `u`, which
- evaluates to a box that (maybe) contains some ordinary value, call it `x`.
- Then the second argument uses `x` to compute a new monadic
- value. Conceptually, then, we have
-
- let bind u f = (let x = unbox u in f x);;
-
- The guts of the definition of the `bind` operation amount to
- specifying how to unbox the monadic value `u`. In the `bind`
- operator for the Option monad, we unboxed the monadic value by
- matching it with the pattern `Some x`---whenever `u`
- happened to be a box containing an integer `x`, this allowed us to
- get our hands on that `x` and feed it to `f`.
-
- If the monadic box didn't contain any ordinary value,
- we instead pass through the empty box unaltered.
-
- In a more complicated case, like our whimsical "singing box" example
- from before, if the monadic value happened to be a singing box
- containing an integer `x`, then the `bind` operation would probably
- be defined so as to make sure that the result of `f x` was also
- a singing box. If `f` also wanted to insert a song, you'd have to decide
- whether both songs would be carried through, or only one of them.
- (Are you beginning to realize how wierd and wonderful monads
- can be?)
-
- There is no single `bind` function that dictates how this must go.
- For each new monadic type, this has to be worked out in an
- useful way.
-
-So the "Option/Maybe monad" consists of the polymorphic `option` type, the
-`unit`/return function, and the `bind` function.
-
-
-A note on notation: Haskell uses the infix operator `>>=` to stand for
-`bind`: wherever you see `u >>= f`, that means `bind u f`.
-Wadler uses ⋆, but that hasn't been widely adopted (unfortunately).
-
-Also, if you ever see this notation:
-
- do
- x <- u
- f x
-
-That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
-Similarly:
-
- do
- x <- u
- y <- v
- f x y
-
-is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u
-(fun x -> bind v (fun y -> f x y))`. Those who did last week's
-homework may recognize this last expression. You can think of the
-notation like this: take the singing box `u` and evaluate it (which
-includes listening to the song). Take the int contained in the
-singing box (the end result of evaluting `u`) and bind the variable
-`x` to that int. So `x <- u` means "Sing me up an int, which I'll call
-`x`".
-
-(Note that the above "do" notation comes from Haskell. We're mentioning it here
-because you're likely to see it when reading about monads. (See our page on [[Translating between OCaml Scheme and Haskell]].) It won't work in
-OCaml. In fact, the `<-` symbol already means something different in OCaml,
-having to do with mutable record fields. We'll be discussing mutation someday
-soon.)
-
-As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:
-
- # type 'a list
-
-The `unit`/return operation is:
-
- # let unit x = [x];;
- val unit : 'a -> 'a list = <fun>
-
-That is, the simplest way to lift an `'a` into an `'a list` is just to make a
-singleton list of that `'a`. Finally, the `bind` operation is:
-
- # let bind u f = List.concat (List.map f u);;
- val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
-
-What's going on here? Well, consider `List.map f u` first. This goes through all
-the members of the list `u`. There may be just a single member, if `u = unit x`
-for some `x`. Or on the other hand, there may be no members, or many members. In
-any case, we go through them in turn and feed them to `f`. Anything that gets fed
-to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`.
-For example, it might return a list of all that value's divisors. Then we'll
-have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch
-of `'b list`s into a single `'b list`:
-
- # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
- - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
-
-So now we've seen two monads: the Option/Maybe monad, and the List monad. For any
-monadic system, there has to be a specification of the complex monad type,
-which will be parameterized on some simpler type `'a`, and the `unit`/return
-operation, and the `bind` operation. These will be different for different
-monadic systems.
-
-Many monadic systems will also define special-purpose operations that only make
-sense for that system.
-
-Although the `unit` and `bind` operation are defined differently for different
-monadic systems, there are some general rules they always have to follow.
-
-
-The Monad Laws
---------------
-
-Just like good robots, monads must obey three laws designed to prevent
-them from hurting the people that use them or themselves.
-
-* **Left identity: unit is a left identity for the bind operation.**
- That is, for all `f:'a -> 'b m`, where `'b m` is a monadic
- type, we have `(unit x) >>= f == f x`. For instance, `unit` is itself
- a function of type `'a -> 'a m`, so we can use it for `f`:
-
- # let unit x = Some x;;
- val unit : 'a -> 'a option = <fun>
- # let ( >>= ) u f = match u with None -> None | Some x -> f x;;
- val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
-
- The parentheses is the magic for telling OCaml that the
- function to be defined (in this case, the name of the function
- is `>>=`, pronounced "bind") is an infix operator, so we write
- `u >>= f` or equivalently `( >>= ) u f` instead of `>>= u
- f`.
-
- # unit 2;;
- - : int option = Some 2
- # unit 2 >>= unit;;
- - : int option = Some 2
-
- Now, for a less trivial instance of a function from `int`s to `int option`s:
-
- # let divide x y = if 0 = y then None else Some (x/y);;
- val divide : int -> int -> int option = <fun>
- # divide 6 2;;
- - : int option = Some 3
- # unit 2 >>= divide 6;;
- - : int option = Some 3
-
- # divide 6 0;;
- - : int option = None
- # unit 0 >>= divide 6;;
- - : int option = None
-
-
-* **Associativity: bind obeys a kind of associativity**. Like this:
-
- (u >>= f) >>= g == u >>= (fun x -> f x >>= g)
-
- If you don't understand why the lambda form is necessary (the
- "fun x -> ..." part), you need to look again at the type of `bind`.
-
- Wadler and others try to make this look nicer by phrasing it like this,
- where U, V, and W are schematic for any expressions with the relevant monadic type:
-
- (U >>= fun x -> V) >>= fun y -> W == U >>= fun x -> (V >>= fun y -> W)
-
- Some examples of associativity in the Option monad (bear in
- mind that in the Ocaml implementation of integer division, 2/3
- evaluates to zero, throwing away the remainder):
-
- # Some 3 >>= unit >>= unit;;
- - : int option = Some 3
- # Some 3 >>= (fun x -> unit x >>= unit);;
- - : int option = Some 3
-
- # Some 3 >>= divide 6 >>= divide 2;;
- - : int option = Some 1
- # Some 3 >>= (fun x -> divide 6 x >>= divide 2);;
- - : int option = Some 1
-
- # Some 3 >>= divide 2 >>= divide 6;;
- - : int option = None
- # Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
- - : int option = None
-
- Of course, associativity must hold for *arbitrary* functions of
- type `'a -> 'b m`, where `m` is the monad type. It's easy to
- convince yourself that the `bind` operation for the Option monad
- obeys associativity by dividing the inputs into cases: if `u`
- matches `None`, both computations will result in `None`; if
- `u` matches `Some x`, and `f x` evalutes to `None`, then both
- computations will again result in `None`; and if the value of
- `f x` matches `Some y`, then both computations will evaluate
- to `g y`.
-
-* **Right identity: unit is a right identity for bind.** That is,
- `u >>= unit == u` for all monad objects `u`. For instance,
-
- # Some 3 >>= unit;;
- - : int option = Some 3
- # None >>= unit;;
- - : 'a option = None
-
-
-More details about monads
--------------------------
-
-If you studied algebra, you'll remember that a *monoid* is an
-associative operation with a left and right identity. For instance,
-the natural numbers along with multiplication form a monoid with 1
-serving as the left and right identity. That is, `1 * u == u == u * 1` for all
-`u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`. As
-presented here, a monad is not exactly a monoid, because (unlike the
-arguments of a monoid operation) the two arguments of the bind are of
-different types. But it's possible to make the connection between
-monads and monoids much closer. This is discussed in [Monads in Category
-Theory](/advanced_topics/monads_in_category_theory).
-
-See also:
-
-* [Haskell wikibook on Monad Laws](http://www.haskell.org/haskellwiki/Monad_Laws).
-* [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 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)
-
-
-Here are some papers that introduced monads into functional programming:
-
-* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following 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.
-<!-- This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
- Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
- The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.-->
-
-* [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.
-<!-- The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.-->
-
-
-There's a long list of monad tutorials on the [[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.
-
-In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category
-Theory](/advanced_topics/monads_in_category_theory) notes do so, for example.
-
-Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
-
-You may sometimes see:
-
- u >> v
-
-That just means:
-
- u >>= fun _ -> v
-
-that is:
-
- bind u (fun _ -> v)
-
-You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
-
-The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example:
-
- # let even x = (x mod 2 = 0);;
- val g : int -> bool = <fun>
-
-`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad?
-
- # let lift g = fun u -> bind u (fun x -> Some (g x));;
- val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
-
-`lift even` will now be a function from `int option`s to `bool option`s. We can
-also define a lift operation for binary functions:
-
- # let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));;
- val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = <fun>
-
-`lift2 (+)` will now be a function from `int option`s and `int option`s to `int option`s. This should look familiar to those who did the homework.
-
-The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`!
-
-Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
-
-In general, any lift/map operation can be relied on to satisfy these laws:
-
- * lift id = id
- * lift (compose f g) = compose (lift f) (lift g)
-
-where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these
-two computations to give the same result:
-
- List.map (fun x -> f (g x)) lst
- List.map f (List.map g lst)
-
-Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
-
- ap [f] [x; y] = [f x; f y]
- ap (Some f) (Some x) = Some (f x)
-
-and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
-
- ap (unit id) u = u
- ap (ap (ap (unit compose) u) v) w = ap u (ap v w)
- ap (unit f) (unit x) = unit (f x)
- ap u (unit x) = ap (unit (fun f -> f x)) u
-
-Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where
-we went from:
-
- [[1]; [1;2]; [1;3]; [1;2;4]]
-
-to:
-
- [1; 1; 2; 1; 3; 1; 2; 4]
-
-That is the `join` operation.
-
-All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions:
-
- lift f u = u >>= compose unit f
- lift f u = ap (unit f) u
- lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y)))
- lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v
- ap u v = u >>= (fun f -> lift f v)
- ap u v = lift2 id u v
- join m2 = m2 >>= id
- u >>= f = join (lift f u)
- u >> v = u >>= (fun _ -> v)
- u >> v = lift2 (fun _ -> id) u v
-
-
-
-Monad outlook
--------------
-
-We're going to be using monads for a number of different things in the
-weeks to come. One major application will be the State monad,
-which will enable us to model mutation: variables whose values appear
-to change as the computation progresses. Later, we will study the
-Continuation monad.
-
-But first, we'll look at several linguistic applications for monads, based
-on what's called the *Reader monad*.
-
-##[[Reader Monad for Variable Binding]]##
-
-##[[Reader Monad for Intensionality]]##
-
<!-- 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
-==================
The [[tradition in the functional programming
literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
of monads, followed in coming weeks by instances of monads from the philosophical and
linguistics literature.
-> <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>
+> <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 `map`s. Their "monoidal" character is captured in the Monad Laws, for which see below.</small>
+
+[[!toc levels=2]]
+
## Box types: type expressions with one free type variable ##
<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`.
+> 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".)
<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$` `â\97\8f`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`.
+> We'll use `m$` as a left-associative infix operator, reminiscent of (the right-associative) `$` which is just ordinary function application (also expressed by mere left-associative juxtaposition). In the class presentation Jim called `m$` `â\9a«`. 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 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.
+> 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.
<code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
+> 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 `★`.
+
<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>
> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
-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".
-
-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*
k >=> mid == k
mid >=> k == 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 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.
+ 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 <code>P -> <u>P</u></code>. But they needn't be. Their types only need to "cohere" in the sense that the output type of the one arrow is a boxing of the input type of the next.
- 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:
+ 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.
+ (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.
+
> <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>
> <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>
+ > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre>
> 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.</small>
+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 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`).
<!--
-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`.
+Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the general Monad operations. For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`. That would instead be defined as a special-purpose operation, specific to some restricted class of Monads.
-->
* 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]`. 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.
+* 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. I thought they used to use `<<` for `flip (>>)` instead, but now I can't confirm that this was ever the case. -->
* <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 ()`.
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:
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:
-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.
+ 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 (Haskell) `concat (map k cs)` or (OCaml) `List.flatten (List.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.
+
+<pre>
+[3, 2, 0, 1] >>=<sub>α list</sub> (\a -> dup a a) ==> [3, 3, 3, 2, 2, 1]
+
+ Some a >>=<sub>α option</sub> (\a -> Some 0) ==> Some 0
+ None >>=<sub>α option</sub> (\a -> Some 0) ==> None
+ Some a >>=<sub>α option</sub> (\a -> None ) ==> None
+
+ .
+ / \
+ . / \
+ / \ . . \
+ . 3 >>=<sub>(α,unit) tree</sub> (\a -> / \ ) ==> / \ .
+ / \ a a / \ / \
+1 2 . . 3 3
+ / \ / \
+ 1 1 2 2
+</pre>
+
+
+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 <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.)
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) <small>(where you should follow the further links discussing Functors, Natural Transformations, and Monads)</small>
+[5](http://www.stephendiehl.com/posts/monads.html)
+
+
+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. <!-- http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf --> 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.
+<!-- This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
+ Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
+ The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell. -->
+
+* [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.
+<!-- The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers. -->
+
+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.