6 Start by (re)reading the discussion of monads in the lecture notes for
7 week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
8 In those notes, we saw a way to separate thining about error
9 conditions (such as trying to divide by zero) from thinking about
10 normal arithmetic computations. We did this by making use of the
11 Option monad: in each place where we had something of type `int`, we
12 put instead something of type `int option`, which is a sum type
13 consisting either of just an integer, or else some special value which
14 we could interpret as signaling that something had gone wrong.
16 The goal was to make normal computing as convenient as possible: when
17 we're adding or multiplying, we don't have to worry about generating
18 any new errors, so we do want to think about the difference between
19 ints and int options. We tried to accomplish this by defining a
20 `bind` operator, which enabled us to peel away the option husk to get
21 at the delicious integer inside. There was also a homework problem
22 which made this even more convenient by mapping any bindary operation
23 on plain integers into a lifted operation that understands how to deal
24 with int options in a sensible way.
26 [Linguitics note: Dividing by zero is supposed to feel like a kind of
27 presupposition failure. If we wanted to adapt this approach to
28 building a simple account of presupposition projection, we would have
29 to do several things. First, we would have to make use of the
30 polymorphism of the `option` type. In the arithmetic example, we only
31 made use of int options, but when we're composing natural language
32 expression meanings, we'll need to use types like `N int`, `Det Int`,
33 `VP int`, and so on. But that works automatically, because we can use
34 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
35 theory of accommodation, and a theory of the situations in which
36 material within the sentence can satisfy presuppositions for other
37 material that otherwise would trigger a presupposition violation; but,
38 not surprisingly, these refinements will require some more
39 sophisticated techniques than the super-simple option monad.]
41 So what examctly is a monad? As usual, we're not going to be pedantic
42 about it, but for our purposes, we can consider a monad to be a system
43 that provides at least the following three elements:
45 * A way to build a complex type from some basic type. In the division
46 example, the polymorphism of the `'a option` type provides a way of
47 building an option out of any other type of object. People often
48 use a container metaphor: if `x` has type `int option`, then `x` is
49 a box that (may) contain an integer.
51 type 'a option = None | Some of 'a;;
53 * A way to turn an ordinary value into a monadic value. In Ocaml, we
54 did this for any integer n by mapping an arbitrary integer `n` to
55 the option `Some n`. To be official, we can define a function
59 val unit : 'a -> 'a option = <fun>
61 So `unit` is a way to put something inside of a box.
63 * A bind operation (note the type):
66 let bind m f = match m with None -> None | Some n -> f n;;
67 val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
70 `bind` takes two arguments (a monadic object and a function from
71 ordinary objects to monadic objects), and returns a monadic
74 Intuitively, the interpretation of what `bind` does is like this:
75 the first argument computes a monadic object m, which will
76 evaluate to a box containing some ordinary value, call it `x`.
77 Then the second argument uses `x` to compute a new monadic
78 value. Conceptually, then, we have
80 let bind m f = (let x = unwrap m in f x);;
82 The guts of the definition of the `bind` operation amount to
83 specifying how to unwrap the monadic object `m`. In the bind
84 opertor for the option monad, we unwraped the option monad by
85 matching the monadic object `m` with `Some n`--whenever `m`
86 happend to be a box containing an integer `n`, this allowed us to
87 get our hands on that `n` and feed it to `f`.
89 So the "Option monad" consists of the polymorphic option type, the
90 unit function, and the bind function.
92 A note on notation: some people use the infix operator `>==` to stand
93 for `bind`. I really hate that symbol. Following Wadler, I prefer to
94 infix five-pointed star, or on a keyboard, `*`.
100 Just like good robots, monads must obey three laws designed to prevent
101 them from hurting the people that use them or themselves.
103 * Left identity: unit is a left identity for the bind operation.
104 That is, for all `f:'a -> 'a M`, where `'a M` is a monadic
105 object, we have `(unit x) * f == f x`. For instance, `unit` is a
106 function of type `'a -> 'a option`, so we have
109 # let ( * ) m f = match m with None -> None | Some n -> f n;;
110 val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
111 # let unit x = Some x;;
112 val unit : 'a -> 'a option = <fun>
114 - : int option = Some 2
117 The parentheses is the magic for telling Ocaml that the
118 function to be defined (in this case, the name of the function
119 is `*`, pronounced "bind") is an infix operator, so we write
120 `m * f` or `( * ) m f` instead of `* m f`.
122 * Associativity: bind obeys a kind of associativity, like this:
124 (m * f) * g == m * (fun x -> f x * g)
126 If you don't understand why the lambda form is necessary, you need
127 to look again at the type of bind. This is important.
129 For an illustration of associativity in the option monad:
132 Some 3 * unit * unit;;
133 - : int option = Some 3
134 Some 3 * (fun x -> unit x * unit);;
135 - : int option = Some 3
138 Of course, associativity must hold for arbitrary functions of
139 type `'a -> M 'a`, where `M` is the monad type. It's easy to
140 convince yourself that the bind operation for the option monad
141 obeys associativity by dividing the inputs into cases: if `m`
142 matches `None`, both computations will result in `None`; if
143 `m` matches `Some n`, and `f n` evalutes to `None`, then both
144 computations will again result in `None`; and if the value of
145 `f n` matches `Some r`, then both computations will evaluate
148 * Right identity: unit is a right identity for bind. That is,
149 `m * unit == m` for all monad objects `m`. For instance,
153 - : int option = Some 3
156 Now, if you studied algebra, you'll remember that a *monoid* is an
157 associative operation with a left and right identity. For instance,
158 the natural numbers along with multiplication form a monoid with 1
159 serving as the left and right identity. That is, temporarily using
160 `*` to mean arithmetic multiplication, `1 * n == n == n * 1` for all
161 `n`, and `(a * b) * c == a * (b * c)` for all `a`, `b`, and `c`. As
162 presented here, a monad is not exactly a monoid, because (unlike the
163 arguments of a monoid operation) the two arguments of the bind are of
164 different types. But if we generalize bind so that both arguments are
165 of type `'a -> M 'a`, then we get plain identity laws and
166 associativity laws, and the monad laws are exactly like the monoid
167 laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>).
173 We're going to be using monads for a number of different things in the
174 weeks to come. The first main application will be the State monad,
175 which will enable us to model mutation: variables whose values appear
176 to change as the computation progresses. Later, we will study the
179 The intensionality monad
180 ------------------------
182 In the meantime, we'll see a linguistic application for monads:
183 intensional function application. In Shan (2001) [Monads for natural
184 language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that
185 making expressions sensitive to the world of evaluation is
186 conceptually the same thing as making use of a *reader monad* (which
187 we'll see again soon). This technique was beautifully re-invented
188 by Ben-Avi and Winter (2007) in their paper [A modular
190 intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
191 though without explicitly using monads.
193 All of the code in the discussion below can be found here: [[intensionality-monad.ml]].
194 To run it, download the file, start Ocaml, and say `# #use
195 "intensionality-monad.ml";;`.
197 Here's the idea: since people can have different attitudes towards
198 different propositions that happen to have the same truth value, we
199 can't have sentences denoting simple truth values. Then if John
200 believed that the earth was round, it would force him to believe
201 Fermat's last theorem holds, since both propositions are equally true.
202 The traditional solution is to allow sentences to denote a function
203 from worlds to truth values, what Montague called an intension.
204 So if `s` is the type of possible worlds, we have the following
209 Extensional types Intensional types Examples
210 -------------------------------------------------------------------
212 S s->t s->t John left
214 VP s->e->t s->(s->e)->t left
215 Vt s->e->e->t s->(s->e)->(s->e)->t saw
216 Vs s->t->e->t s->(s->t)->(s->e)->t thought
219 This system is modeled on the way Montague arranged his grammar.
220 (There are significant simplifications: for instance, determiner
221 phrases are thought of as corresponding to individuals rather than to
222 generalized quantifiers.) If you're curious about the initial `s`'s
223 in the extensional types, they're there because the behavior of these
224 expressions depends on which world they're evaluated at. If you are
225 in a situation in which you can hold the evaluation world constant,
226 you can further simplify the extensional types. (Usually, the
227 dependence of the extension of an expression on the evaluation world
228 is hidden in a superscript, or built into the lexical interpretation
231 The main difference between the intensional types and the extensional
232 types is that in the intensional types, the arguments are functions
233 from worlds to extensions: intransitive verb phrases like "left" now
234 take intensional concepts as arguments (type s->e) rather than plain
235 individuals (type e), and attitude verbs like "think" now take
236 propositions (type s->t) rather than truth values (type t).
238 The intenstional types are more complicated than the intensional
239 types. Wouldn't it be nice to keep the complicated types to just
240 those attitude verbs that need to worry about intensions, and keep the
241 rest of the grammar as extensional as possible? This desire is
242 parallel to our earlier desire to limit the concern about division by
243 zero to the division function, and let the other functions ignore
244 division-by-zero problems as much as possible.
246 So here's what we do:
248 In Ocaml, we'll use integers to model possible worlds:
254 Characters (characters in the computational sense, i.e., letters like
255 `'a'` and `'b'`, not Kaplanian characters) will model individuals, and
256 Ocaml booleans will serve for truth values.
259 type 'a intension = s -> 'a;;
260 let unit x (w:s) = x;;
263 let bill = unit 'b';;
267 In our monad, the intension of an extensional type `'a` is `s -> 'a`,
268 a function from worlds to extensions. Our unit will be the constant
269 function (an instance of the K combinator) that returns the same
270 individual at each world.
272 Then `ann = unit 'a'` is a rigid designator: a constant function from
273 worlds to individuals that returns `'a'` no matter which world is used
276 Let's test compliance with the left identity law:
279 # let bind m f (w:s) = f (m w) w;;
280 val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = <fun>
281 # bind (unit 'a') unit 1;;
285 We'll assume that this and the other laws always hold.
287 We now build up some extensional meanings:
289 let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
291 This function says that everyone always left, except for Cam in world
292 2 (i.e., `left 2 'c' == false`).
294 Then the way to evaluate an extensional sentence is to determine the
295 extension of the verb phrase, and then apply that extension to the
296 extension of the subject:
299 let extapp fn arg w = fn w (arg w);;
308 `extapp` stands for "extensional function application".
309 So Ann left in world 1, but Cam didn't leave in world 2.
311 A transitive predicate:
313 let saw w x y = (w < 2) && (y < x);;
314 extapp (extapp saw bill) ann 1;; (* true *)
315 extapp (extapp saw bill) ann 2;; (* false *)
317 In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone
320 Good. Now for intensions:
322 let intapp fn arg w = fn w arg;;
324 The only difference between intensional application and extensional
325 application is that we don't feed the evaluation world to the argument.
326 (See Montague's rules of (intensional) functional application, T4 -- T10.)
327 In other words, instead of taking an extension as an argument,
328 Montague's predicates take a full-blown intension.
330 But for so-called extensional predicates like "left" and "saw",
331 the extra power is not used. We'd like to define intensional versions
332 of these predicates that depend only on their extensional essence.
333 Just as we used bind to define a version of addition that interacted
334 with the option monad, we now use bind to intensionalize an
338 let lift pred w arg = bind arg (fun x w -> pred w x) w;;
340 intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
341 intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
344 Because `bind` unwraps the intensionality of the argument, when the
345 lifted "left" receives an individual concept (e.g., `unit 'a'`) as
346 argument, it's the extension of the individual concept (i.e., `'a'`)
347 that gets fed to the basic extensional version of "left". (For those
348 of you who know Montague's PTQ, this use of bind captures Montague's
349 third meaning postulate.)
351 Likewise for extensional transitive predicates like "saw":
354 let lift2 pred w arg1 arg2 =
355 bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
356 intapp (intapp (lift2 saw) bill) ann 1;; (* true: Ann saw Bill in world 1 *)
357 intapp (intapp (lift2 saw) bill) ann 2;; (* false: No one saw anyone in world 2 *)
360 Crucially, an intensional predicate does not use `bind` to consume its
361 arguments. Attitude verbs like "thought" are intensional with respect
362 to their sentential complement, but extensional with respect to their
363 subject (as Montague noticed, almost all verbs in English are
364 extensional with respect to their subject; a possible exception is "appear"):
367 let think (w:s) (p:s->t) (x:e) =
368 match (x, p 2) with ('a', false) -> false | _ -> p w;;
371 Ann disbelieves any proposition that is false in world 2. Apparently,
372 she firmly believes we're in world 2. Everyone else believes a
373 proposition iff that proposition is true in the world of evaluation.
376 intapp (lift (intapp think
383 So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
385 The `lift` is there because "think Bill left" is extensional wrt its
386 subject. The important bit is that "think" takes the intension of
387 "Bill left" as its first argument.
390 intapp (lift (intapp think
397 But even in world 1, Ann doesn't believe that Cam left (even though he
398 did: `intapp (lift left) cam 1 == true`). Ann's thoughts are hung up
399 on what is happening in world 2, where Cam doesn't leave.
401 *Small project*: add intersective ("red") and non-intersective
402 adjectives ("good") to the fragment. The intersective adjectives
403 will be extensional with respect to the nominal they combine with
404 (using bind), and the non-intersective adjectives will take
405 intensional arguments.
407 Finally, note that within an intensional grammar, extensional funtion
408 application is essentially just bind:
411 # let swap f x y = f y x;;
412 # bind cam (swap left) 2;;