[[!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 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 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 `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 binary operation on plain integers into a lifted operation that understands how to deal 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 option`s, 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 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. `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`. 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. With the option monad, we can think of the "safe division" operation
# 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, `*`. 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;;
- : 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 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 (the "fun x" part), you need to look again at the type of bind. Some examples 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

# 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`. * 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 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 , near the bottom). 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";; 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. 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 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, 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: 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