X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=60f85e5b9a48c13801ca022bb88f299671c000ac;hp=3336a49685b38bf087eb3a9f67915f3bf8a32ea0;hb=726437adf263c434a88cfe7d56e4b04b060d70b4;hpb=f10b70d2b7589e44135dd0e47c87d3965ee08918 diff --git a/week7.mdwn b/week7.mdwn index 3336a496..60f85e5b 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -5,10 +5,10 @@ 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 +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. @@ -16,21 +16,21 @@ 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 @@ -38,58 +38,67 @@ 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 +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 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;;` + 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: +* 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`. To be official, we can define a function + called unit: - `let unit x = Some x;;` + let unit x = Some x;; - `val unit : 'a -> 'a option = ` + val unit : 'a -> 'a option = - So `unit` is a way to put something inside of a box. + So `unit` is a way to put something inside of a box. -* A bind operation (note the type): +* A bind operation (note the type): - `let bind m f = match m with None -> None | Some n -> f n;;` + let bind m f = match m with None -> None | Some n -> f n;; - `val bind : 'a option -> ('a -> 'b option) -> 'b option = ` + 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. + `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 + 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);;` + 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`. + 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. +So the "option monad" consists of the polymorphic option type, the +unit function, and the bind function. With the option monad, we can +think of the "safe division" operation -A note on notation: some people use the infix operator `>==` to stand +
+# let divide num den = if den = 0 then None else Some (num/den);;
+val divide : int -> int -> int option = 
+
+ +as basically a function from two integers to an integer, except with +this little bit of option frill, or option plumbing, on the side. + +A note on notation: Haskell uses 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, `*`. @@ -100,57 +109,82 @@ 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 
 # 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;;
+- : 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
-The parentheses is the magic for telling Ocaml that the +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 (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. - For an illustration of associativity in the option monad: + Some examples of associativity in the option monad: 
-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
- 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, 
 # Some 3 * unit;;
 - : int option = Some 3
+# None * unit;;
+- : 'a option = None
Now, if you studied algebra, you'll remember that a *monoid* is an @@ -162,9 +196,9 @@ serving as the left and right identity. That is, temporarily using 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 ). +laws (see , near the bottom). Monad outlook @@ -191,12 +225,15 @@ 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";;`. +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 @@ -217,16 +254,16 @@ 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 +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 @@ -240,20 +277,21 @@ 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. +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. 
 type 'a intension = s -> 'a;;
@@ -286,7 +324,7 @@ 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;;
+	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`).
@@ -310,16 +348,16 @@ 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 *)
+	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.