Towards Monads: Safe division
-----------------------------
+[This section used to be near the end of the lecture notes for week 6]
+
+We begin by reasoning about what should happen when someone tries to
+divide by zero. This will lead us to a general programming technique
+called a *monad*, which we'll see in many guises in the weeks to come.
+
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>
let div' (u:int option) (v:int option) =
- match v with
+ match u with
None -> None
- | Some 0 -> None
- | Some y -> (match u with
- None -> None
- | Some x -> Some (x / y));;
+ | Some x -> (match v with
+ Some 0 -> None
+ | Some y -> Some (x / y));;
(*
val div' : int option -> int option -> int option = <fun>
The definition of `div'` shows exactly what extra needs to be said in
order to trigger the no-division-by-zero presupposition.
-For linguists: this is a complete theory of a particularly simply form
-of presupposition projection (every predicate is a hole).
-
-
-
-
-Monads in General
------------------
-
-Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [[Towards Monads]].
-In those notes, we saw 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 do want to 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 mapping any binary operation
-on plain integers into a lifted operation that understands how to deal
-with `int option`s in a sensible way.
-
[Linguitics 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
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.]
+sophisticated techniques than the super-simple Option monad.]
+
+
+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:
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.
+ 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
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
+ 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.
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`. This is another
+* 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
+ 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;;
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
+ 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`.
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
+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`. Chris really hates that symbol. Following Wadler, he prefers to
-use an infix five-pointed star ⋆, or on a keyboard, `*`. Jim on the other hand
-thinks `>>=` is what the literature uses and students won't be able to
-avoid it. Moreover, although ⋆ is OK (though not a convention that's been picked up), overloading the multiplication symbol invites its own confusion
-and Jim feels very uneasy about that. If not `>>=` then we should use
-some other unfamiliar infix symbol (but `>>=` already is such...)
+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).
-In any case, the course leaders will work this out somehow. In the meantime,
-as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
-if you ever see this notation:
+Also, if you ever see this notation:
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.
+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. It won't work in
+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:
+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
# 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
+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
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 -> 'a m`, where `'a m` is a monadic
- type, we have `(unit x) * f == f x`. For instance, `unit` is itself
+ 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>
+ # 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 `( * ) u f` instead of `* u f`. Now:
+ 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;;
+ # 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;;
+ # unit 2 >>= divide 6;;
- : int option = Some 3
# divide 6 0;;
- : int option = None
- # unit 0 * divide 6;;
+ # 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)
+ (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`.
+ 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`.
- Some examples of associativity in the option monad:
+ 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;;
+ # Some 3 >>= unit >>= unit;;
- : int option = Some 3
- # Some 3 * (fun x -> unit x * unit);;
+ # Some 3 >>= (fun x -> unit x >>= unit);;
- : int option = Some 3
- # Some 3 * divide 6 * divide 2;;
+ # Some 3 >>= divide 6 >>= divide 2;;
- : int option = Some 1
- # Some 3 * (fun x -> divide 6 x * divide 2);;
+ # Some 3 >>= (fun x -> divide 6 x >>= divide 2);;
- : int option = Some 1
- # Some 3 * divide 2 * divide 6;;
+ # Some 3 >>= divide 2 >>= divide 6;;
- : int option = None
- # Some 3 * (fun x -> divide 2 x * divide 6);;
+ # Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
- : int option = None
Of course, associativity must hold for *arbitrary* functions of
-type `'a -> 'a m`, where `m` is the monad type. It's easy to
-convince yourself that the `bind` operation for the option monad
+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
to `g y`.
* **Right identity: unit is a right identity for bind.** That is,
- `u * unit == u` for all monad objects `u`. For instance,
+ `u >>= unit == u` for all monad objects `u`. For instance,
- # Some 3 * unit;;
+ # Some 3 >>= unit;;
- : int option = Some 3
- # None * unit;;
+ # None >>= unit;;
- : 'a option = None
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, temporarily using
-`*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
+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_notes/monads_in_category_theory).
-See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
+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.
+* [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. This paper has a great first
-line: **Shall I be pure, or impure?**
+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.-->
-* [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.-->
-
-* [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from <https://www.cs.indiana.edu/cgi-pub/c311/doku.php?id=home> but the link above is to a local copy.
-There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+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_notes/monads_in_category_theory) notes do so, for example.
+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 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:
+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?
+`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>
`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 `<$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) 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.
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
+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]]
-------------
We're going to be using monads for a number of different things in the
-weeks to come. The first main application will be the State monad,
+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.
-In the meantime, we'll look at several linguistic applications for monads, based
-on what's called the *reader monad*.
+But first, we'll look at several linguistic applications for monads, based
+on what's called the *Reader monad*.
-##[[Reader monad]]##
+##[[Reader Monad for Variable Binding]]##
-##[[Intensionality monad]]##
+##[[Reader Monad for Intensionality]]##