Start by (re)reading the discussion of monads in the lecture notes for
week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
-In those notes, we saw a way to separate thining about error
+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 monad: in each place where we had something of type `int`, we
+option monad: 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 just an integer, or else some special value which
we could interpret as signaling that something had 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
-ints and int options. We tried to accomplish this by defining a
-`bind` operator, which enabled us to peel away the option husk to get
+`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 bindary operation
+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 options in a sensible way.
+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
to do several things. First, we would have to make use of the
polymorphism of the `option` type. In the arithmetic example, we only
-made use of int options, but when we're composing natural language
-expression meanings, we'll need to use types like `N int`, `Det Int`,
-`VP int`, and so on. But that works automatically, because we can use
+made use of `int option`s, but when we're composing natural language
+expression meanings, we'll need to use types like `N option`, `Det option`,
+`VP option`, and so on. But that works automatically, because we can use
any type for the `'a` in `'a option`. Ultimately, we'd want to have a
theory of accommodation, and a theory of the situations in which
material within the sentence can satisfy presuppositions for other
not surprisingly, these refinements will require some more
sophisticated techniques than the super-simple option monad.]
-So what examctly is a monad? As usual, we're not going to be pedantic
+So what exactly is a monad? As usual, we're not going to be pedantic
about it, but for our purposes, we can consider a monad to be a system
that provides at least the following three elements:
-* A way to build a complex type from some basic type. In the 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 `x` has type `int option`, then `x` is
- a box that (may) contain an integer.
+* A complex type that's built around some more basic type. Usually
+ it 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 `x` has type `int option`,
+ then `x` 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 `n` by mapping it to
+ the option `Some n`. 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 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`. 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 m f = match m with None -> None | Some n -> f n;;
+ 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 options
+ and end with bool options).
+
+ Intuitively, the interpretation of what `bind` does is like this:
+ the first argument is a monadic value m, 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 m f = (let x = unbox m in f x);;
+
+ The guts of the definition of the `bind` operation amount to
+ specifying how to unbox the monadic value `m`. In the bind
+ opertor for the option monad, we unboxed the option monad by
+ matching the monadic value `m` with `Some n`---whenever `m`
+ happened to be a box containing an integer `n`, this allowed us to
+ get our hands on that `n` and feed it to `f`.
+
+ If the monadic box didn't contain any ordinary value, then
+ we just 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 `n`, then the `bind` operation would probably
+ be defined so as to make sure that the result of `f n` was also
+ a singing box. If `f` also inserted a song, you'd have to decide
+ whether both songs would be carried through, or only one of them.
+
+ 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. With the option monad, we can
+think of the "safe division" operation
- `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 n by mapping an arbitrary integer `n` to
- the option `Some n`. To be official, we can define a function
- called unit:
-
- `let unit x = Some x;;`
-
- `val unit : 'a -> 'a option = <fun>`
-
- So `unit` is a way to put something inside of a box.
-
-* A bind operation (note the type):
+<pre>
+# let divide' num den = if den = 0 then None else Some (num/den);;
+val divide' : int -> int -> int option = <fun>
+</pre>
- `let bind m f = match m with None -> None | Some n -> f n;;`
+as basically a function from two integers to an integer, except with
+this little bit of option plumbing on the side.
- `val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>`
+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...)
- `bind` takes two arguments (a monadic object and a function from
- ordinary objects to monadic objects), and returns a monadic
- object.
+In any case, the course leaders will work this out somehow. In the meantime,
+as you read around, wherever you see `m >>= f`, that means `bind m f`. Also,
+if you ever see this notation:
- Intuitively, the interpretation of what `bind` does is like this:
- the first argument computes a monadic object m, which will
- evaluate to a box containing some ordinary value, call it `x`.
- Then the second argument uses `x` to compute a new monadic
- value. Conceptually, then, we have
+ do
+ x <- m
+ f x
- `let bind m f = (let x = unwrap m in f x);;`
+That's a Haskell shorthand for `m >>= (\x -> f x)`, that is, `bind m f`.
+Similarly:
- The guts of the definition of the `bind` operation amount to
- specifying how to unwrap the monadic object `m`. In the bind
- opertor for the option monad, we unwraped the option monad by
- matching the monadic object `m` with `Some n`--whenever `m`
- happend to be a box containing an integer `n`, this allowed us to
- get our hands on that `n` and feed it to `f`.
+ do
+ x <- m
+ y <- n
+ f x y
-So the "Option monad" consists of the polymorphic option type, the
-unit function, and the bind function.
+is shorthand for `m >>= (\x -> n >>= (\y -> f x y))`, that is, `bind m (fun x
+-> bind n (fun y -> f x y))`. Those who did last week's homework may recognize
+this.
-A note on notation: some people use the infix operator `>==` to stand
-for `bind`. I really hate that symbol. Following Wadler, I prefer to
-infix five-pointed star, or on a keyboard, `*`.
+(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
+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.)
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 -> 'a M`, where `'a M` is a monadic
- object, we have `(unit x) * f == f x`. For instance, `unit` is a
- function of type `'a -> 'a option`, so we have
+* **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
+ object, we have `(unit x) * f == f x`. For instance, `unit` is a
+ function of type `'a -> 'a option`, so we have
<pre>
# let ( * ) m f = match m with None -> None | Some n -> f n;;
val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
# let unit x = Some x;;
val unit : 'a -> 'a option = <fun>
+
+# unit 2;;
+- : int option = Some 2
# unit 2 * unit;;
- : int option = Some 2
+
+# 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
</pre>
- 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
- `m * f` or `( * ) m f` instead of `* m f`.
+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
+`m * f` or `( * ) m f` instead of `* m f`.
-* Associativity: bind obeys a kind of associativity, like this:
+* **Associativity: bind obeys a kind of associativity**. Like this:
- (m * f) * g == m * (fun x -> f x * g)
+ (m * f) * g == m * (fun x -> f x * g)
- If you don't understand why the lambda form is necessary, you need
- to look again at the type of bind. This is important.
+ 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.
- For an illustration of associativity in the option monad:
+ Some examples of associativity in the option monad:
<pre>
-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;;
+- : 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
</pre>
- Of course, associativity must hold for arbitrary functions of
- type `'a -> M 'a`, 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 `m`
- matches `None`, both computations will result in `None`; if
- `m` matches `Some n`, and `f n` evalutes to `None`, then both
- computations will again result in `None`; and if the value of
- `f n` matches `Some r`, then both computations will evaluate
- to `g r`.
+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
+obeys associativity by dividing the inputs into cases: if `m`
+matches `None`, both computations will result in `None`; if
+`m` matches `Some n`, and `f n` evalutes to `None`, then both
+computations will again result in `None`; and if the value of
+`f n` matches `Some r`, then both computations will evaluate
+to `g r`.
-* Right identity: unit is a right identity for bind. That is,
- `m * unit == m` for all monad objects `m`. For instance,
+* **Right identity: unit is a right identity for bind.** That is,
+ `m * unit == m` for all monad objects `m`. For instance,
<pre>
# Some 3 * unit;;
- : int option = Some 3
+# None * unit;;
+- : 'a option = None
</pre>
Now, if you studied algebra, you'll remember that a *monoid* is an
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 if we generalize bind so that both arguments are
-of type `'a -> M 'a`, then we get plain identity laws and
+of type `'a -> 'a m`, then we get plain identity laws and
associativity laws, and the monad laws are exactly like the monoid
-laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>).
+laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>, near the bottom).
Monad outlook
though without explicitly using monads.
All of the code in the discussion below can be found here: [[intensionality-monad.ml]].
-To run it, download the file, start Ocaml, and say `# #use
-"intensionality-monad.ml";;`.
+To run it, download the file, start OCaml, and say
+
+ # #use "intensionality-monad.ml";;
+
+Note the extra `#` attached to the directive `use`.
Here's the idea: since people can have different attitudes towards
different propositions that happen to have the same truth value, we
-can't have sentences denoting simple truth values. Then if John
+can't have sentences denoting simple truth values. If we did, then if John
believed that the earth was round, it would force him to believe
Fermat's last theorem holds, since both propositions are equally true.
The traditional solution is to allow sentences to denote a function
</pre>
This system is modeled on the way Montague arranged his grammar.
-(There are significant simplifications: for instance, determiner
+There are significant simplifications: for instance, determiner
phrases are thought of as corresponding to individuals rather than to
-generalized quantifiers.) If you're curious about the initial `s`'s
+generalized quantifiers. If you're curious about the initial `s`'s
in the extensional types, they're there because the behavior of these
expressions depends on which world they're evaluated at. If you are
in a situation in which you can hold the evaluation world constant,
-you can further simplify the extensional types. (Usually, the
+you can further simplify the extensional types. Usually, the
dependence of the extension of an expression on the evaluation world
is hidden in a superscript, or built into the lexical interpretation
-function.)
+function.
The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
those attitude verbs that need to worry about intensions, and keep the
rest of the grammar as extensional as possible? This desire is
parallel to our earlier desire to limit the concern about division by
-zero to the division function, and let the other functions ignore
-division-by-zero problems as much as possible.
+zero to the division function, and let the other functions, like
+addition or multiplication, ignore division-by-zero problems as much
+as possible.
So here's what we do:
-In Ocaml, we'll use integers to model possible worlds:
+In OCaml, we'll use integers to model possible worlds:
- type s = int;;
- type e = char;;
- type t = bool;;
+ type s = int;;
+ type e = char;;
+ type t = bool;;
Characters (characters in the computational sense, i.e., letters like
`'a'` and `'b'`, not Kaplanian characters) will model individuals, and
-Ocaml booleans will serve for truth values.
+OCaml booleans will serve for truth values.
<pre>
type 'a intension = s -> 'a;;
We now build up some extensional meanings:
- let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+ let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
This function says that everyone always left, except for Cam in world
2 (i.e., `left 2 'c' == false`).
A transitive predicate:
- let saw w x y = (w < 2) && (y < x);;
- extapp (extapp saw bill) ann 1;; (* true *)
- extapp (extapp saw bill) ann 2;; (* false *)
+ let saw w x y = (w < 2) && (y < x);;
+ extapp (extapp saw bill) ann 1;; (* true *)
+ extapp (extapp saw bill) ann 2;; (* false *)
In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone
in world two.
Good. Now for intensions:
- let intapp fn arg w = fn w arg;;
+ let intapp fn arg w = fn w arg;;
The only difference between intensional application and extensional
application is that we don't feed the evaluation world to the argument.