<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>
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
bind operation is:
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;;
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;;
- : 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;;
(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;;
- : int option = Some 3
- : 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
+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
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>
-------------
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]]##