X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=7544425c4bbcca2c3a1b72cf0cf088199a18586f;hp=62ef89f0ab8b7a2faa8fecf83aa141963d523a54;hb=HEAD;hpb=7834e35d4a2de390ce6a1e9113186dcbb9c07a6f diff --git a/week7.mdwn b/week7.mdwn deleted file mode 100644 index 62ef89f0..00000000 --- a/week7.mdwn +++ /dev/null @@ -1,412 +0,0 @@ -[[!toc]] - -Monads ------- - -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 -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 -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 -at the delicious integer inside. There was also a homework problem -which made this even more convenient by mapping any bindary operation -on plain integers into a lifted operation that understands how to deal -with int options 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 -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 -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.] - -So what examctly 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. - - 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 = - - So `unit` is a way to put something inside of a box. - -* A bind operation (note the type): - - let bind m f = match m with None -> None | Some n -> f n;; - val bind : 'a option -> ('a -> 'b option) -> 'b option = - - `bind` takes two arguments (a monadic object and a function from - ordinary objects to monadic objects), and returns a monadic - object. - - 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 - - let bind m f = (let x = unwrap m in f x);; - - 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`. - -So the "Option monad" consists of the polymorphic option type, the -unit function, and the bind function. - -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, `*`. - - -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 - -
-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = 
-# let unit x = Some x;;
-val unit : 'a -> 'a option = 
-# unit 2 * unit;;
-- : int option = Some 2
-
- - 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: - - (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. - - For an illustration of associativity in the option monad: - -
-Some 3 * unit * unit;; 
-- : int option = Some 3
-Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
-
- - 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`. - -* Right identity: unit is a right identity for bind. That is, - `m * unit == m` for all monad objects `m`. For instance, - -
-# Some 3 * unit;;
-- : int option = Some 3
-
- -Now, 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 * n == n == n * 1` for all -`n`, and `(a * b) * c == a * (b * c)` for all `a`, `b`, and `c`. 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 if we generalize bind so that both arguments are -of type `'a -> M 'a`, then we get plain identity laws and -associativity laws, and the monad laws are exactly like the monoid -laws (see ). - - -Monad outlook -------------- - -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, -which will enable us to model mutation: variables whose values appear -to change as the computation progresses. Later, we will study the -Continuation monad. - -The intensionality monad ------------------------- - -In the meantime, we'll see a linguistic application for monads: -intensional function application. In Shan (2001) [Monads for natural -language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that -making expressions sensitive to the world of evaluation is -conceptually the same thing as making use of a *reader monad* (which -we'll see again soon). This technique was beautifully re-invented -by Ben-Avi and Winter (2007) in their paper [A modular -approach to -intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf), -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";;`. - -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 -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 -from worlds to truth values, what Montague called an intension. -So if `s` is the type of possible worlds, we have the following -situation: - - -
-Extensional types                 Intensional types       Examples
--------------------------------------------------------------------
-
-S         s->t                    s->t                    John left
-DP        s->e                    s->e                    John
-VP        s->e->t                 s->(s->e)->t            left
-Vt        s->e->e->t              s->(s->e)->(s->e)->t    saw
-Vs        s->t->e->t              s->(s->t)->(s->e)->t    thought
-
- -This system is modeled on the way Montague arranged his grammar. -(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 -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 -dependence of the extension of an expression on the evaluation world -is hidden in a superscript, or built into the lexical interpretation -function.) - -The main difference between the intensional types and the extensional -types is that in the intensional types, the arguments are functions -from worlds to extensions: intransitive verb phrases like "left" now -take intensional concepts as arguments (type s->e) rather than plain -individuals (type e), and attitude verbs like "think" now take -propositions (type s->t) rather than truth values (type t). - -The intenstional types are more complicated than the intensional -types. Wouldn't it be nice to keep the complicated types to just -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. - -So here's what we do: - -In Ocaml, we'll use integers to model possible worlds: - - 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. - -
-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
-
-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-
- -In our monad, the intension of an extensional type `'a` is `s -> 'a`, -a function from worlds to extensions. Our unit will be the constant -function (an instance of the K combinator) that returns the same -individual at each world. - -Then `ann = unit 'a'` is a rigid designator: a constant function from -worlds to individuals that returns `'a'` no matter which world is used -as an argument. - -Let's test compliance with the left identity law: - -
-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = 
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-
- -We'll assume that this and the other laws always hold. - -We now build up some extensional meanings: - - 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`). - -Then the way to evaluate an extensional sentence is to determine the -extension of the verb phrase, and then apply that extension to the -extension of the subject: - -
-let extapp fn arg w = fn w (arg w);;
-
-extapp left ann 1;;
-# - : bool = true
-
-extapp left cam 2;;
-# - : bool = false
-
- -`extapp` stands for "extensional function application". -So Ann left in world 1, but Cam didn't leave in world 2. - -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 *) - -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;; - -The only difference between intensional application and extensional -application is that we don't feed the evaluation world to the argument. -(See Montague's rules of (intensional) functional application, T4 -- T10.) -In other words, instead of taking an extension as an argument, -Montague's predicates take a full-blown intension. - -But for so-called extensional predicates like "left" and "saw", -the extra power is not used. We'd like to define intensional versions -of these predicates that depend only on their extensional essence. -Just as we used bind to define a version of addition that interacted -with the option monad, we now use bind to intensionalize an -extensional verb: - -
-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
-
-intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
-intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
-
- -Because `bind` unwraps the intensionality of the argument, when the -lifted "left" receives an individual concept (e.g., `unit 'a'`) as -argument, it's the extension of the individual concept (i.e., `'a'`) -that gets fed to the basic extensional version of "left". (For those -of you who know Montague's PTQ, this use of bind captures Montague's -third meaning postulate.) - -Likewise for extensional transitive predicates like "saw": - -
-let lift2 pred w arg1 arg2 = 
-  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
-intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
-intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
-
- -Crucially, an intensional predicate does not use `bind` to consume its -arguments. Attitude verbs like "thought" are intensional with respect -to their sentential complement, but extensional with respect to their -subject (as Montague noticed, almost all verbs in English are -extensional with respect to their subject; a possible exception is "appear"): - -
-let think (w:s) (p:s->t) (x:e) = 
-  match (x, p 2) with ('a', false) -> false | _ -> p w;;
-
- -Ann disbelieves any proposition that is false in world 2. Apparently, -she firmly believes we're in world 2. Everyone else believes a -proposition iff that proposition is true in the world of evaluation. - -
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'b'))))
-       (unit 'a') 
-1;; (* true *)
-
- -So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave). - -The `lift` is there because "think Bill left" is extensional wrt its -subject. The important bit is that "think" takes the intension of -"Bill left" as its first argument. - -
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'c'))))
-       (unit 'a') 
-1;; (* false *)
-
- -But even in world 1, Ann doesn't believe that Cam left (even though he -did: `intapp (lift left) cam 1 == true`). Ann's thoughts are hung up -on what is happening in world 2, where Cam doesn't leave. - -*Small project*: add intersective ("red") and non-intersective - adjectives ("good") to the fragment. The intersective adjectives - will be extensional with respect to the nominal they combine with - (using bind), and the non-intersective adjectives will take - intensional arguments. - -Finally, note that within an intensional grammar, extensional funtion -application is essentially just bind: - -
-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-