<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>
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
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`.
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.
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, and I'll call
-it `x`".
+`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
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).
+Theory](/advanced_topics/monads_in_category_theory).
See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
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]]##