+let div' (x:int) (y:int) = + match y with + 0 -> None + | _ -> Some (x / y);; + +(* +val div' : int -> int -> int option = fun +# div' 12 2;; +- : int option = Some 6 +# div' 12 0;; +- : int option = None +# div' (div' 12 2) 3;; +Characters 4-14: + div' (div' 12 2) 3;; + ^^^^^^^^^^ +Error: This expression has type int option + but an expression was expected of type int +*) ++ +This starts off well: dividing 12 by 2, no problem; dividing 12 by 0, +just the behavior we were hoping for. But we want to be able to use +the output of the safe-division function as input for further division +operations. So we have to jack up the types of the inputs: + +

+let div' (u:int option) (v:int option) = + match v with + None -> None + | Some 0 -> None + | Some y -> (match u with + None -> None + | Some x -> Some (x / y));; + +(* +val div' : int option -> int option -> int option =+ +Beautiful, just what we need: now we can try to divide by anything we +want, without fear that we're going to trigger any system errors. + +I prefer to line up the `match` alternatives by using OCaml's +built-in tuple type: + ++# div' (Some 12) (Some 2);; +- : int option = Some 6 +# div' (Some 12) (Some 0);; +- : int option = None +# div' (div' (Some 12) (Some 0)) (Some 3);; +- : int option = None +*) +

+let div' (u:int option) (v:int option) = + match (u, v) with + (None, _) -> None + | (_, None) -> None + | (_, Some 0) -> None + | (Some x, Some y) -> Some (x / y);; ++ +So far so good. But what if we want to combine division with +other arithmetic operations? We need to make those other operations +aware of the possibility that one of their arguments has triggered a +presupposition failure: + +

+let add' (u:int option) (v:int option) = + match (u, v) with + (None, _) -> None + | (_, None) -> None + | (Some x, Some y) -> Some (x + y);; + +(* +val add' : int option -> int option -> int option =+ +This works, but is somewhat disappointing: the `add'` operation +doesn't trigger any presupposition of its own, so it is a shame that +it needs to be adjusted because someone else might make trouble. + +But we can automate the adjustment. The standard way in OCaml, +Haskell, etc., is to define a `bind` operator (the name `bind` is not +well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well. + ++# add' (Some 12) (Some 4);; +- : int option = Some 16 +# add' (div' (Some 12) (Some 0)) (Some 4);; +- : int option = None +*) +

+let bind' (u: int option) (f: int -> (int option)) = + match u with + None -> None + | Some x -> f x;; + +let add' (u: int option) (v: int option) = + bind' u (fun x -> bind' v (fun y -> Some (x + y)));; + +let div' (u: int option) (v: int option) = + bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));; + +(* +# div' (div' (Some 12) (Some 2)) (Some 3);; +- : int option = Some 2 +# div' (div' (Some 12) (Some 0)) (Some 3);; +- : int option = None +# add' (div' (Some 12) (Some 0)) (Some 3);; +- : int option = None +*) ++ +Compare the new definitions of `add'` and `div'` closely: the definition +for `add'` shows what it looks like to equip an ordinary operation to +survive in dangerous presupposition-filled world. Note that the new +definition of `add'` does not need to test whether its arguments are +None objects or real numbers---those details are hidden inside of the +`bind'` function. + +The definition of `div'` shows exactly what extra needs to be said in +order to trigger the no-division-by-zero presupposition. [Linguitics note: Dividing by zero is supposed to feel like a kind of presupposition failure. If we wanted to adapt this approach to @@ -39,16 +167,38 @@ 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 + +Monads in General +----------------- + +We've just seen 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` +type: 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 one choice with an `int` payload, or else a `None` choice +which we interpret as signaling that something has 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 would rather not 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 defining a +`lift` operator that mapped any binary operation on plain integers +into a lifted operation that understands how to deal with `int +option`s in a sensible way. + +So what exactly is a monad? We can consider a monad to be a system that provides at least the following three elements: * A complex type that's built around some more basic type. Usually - it will be polymorphic, and so can apply to different basic types. + the complex type 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. + People often use a container metaphor: if `u` has type `int option`, + then `u` is a box that (may) contain an integer. type 'a option = None | Some of 'a;; @@ -60,7 +210,12 @@ that provides at least the following three elements: 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. + uses. [The rationale for "unit" comes from the monad laws + (see below), where the unit function serves as an identity, + just like the unit number (i.e., 1) serves as the identity + object for multiplication. The rationale for "return" comes + from a misguided desire to resonate with C programmers and + other imperative types.] 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 @@ -90,14 +245,14 @@ that provides at least the following three elements: let bind u f = match u with None -> None | Some x -> f x;; val bind : 'a option -> ('a -> 'b option) -> 'b option =

-# let divide' num den = if den = 0 then None else Some (num/den);; -val divide' : int -> int -> int option =+So the "option/maybe monad" consists of the polymorphic `option` type, the +`unit`/return function, and the `bind` function. -as basically a function from two integers to an integer, except with -this little bit of option plumbing on the side. -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...) +A note on notation: Haskell uses the infix operator `>>=` to stand for +`bind`: wherever you see `u >>= f`, that means `bind u f`. +Wadler uses ⋆, but that hasn't been widely adopted (unfortunately). -In any case, the course leaders will work this out somehow. In the meantime, -as you read around, wherever you see `u >>= f`, that means `bind u f`. Also, -if you ever see this notation: +Also, if you ever see this notation: do x <- u @@ -161,9 +304,14 @@ Similarly: y <- v f x y -is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x --> bind v (fun y -> f x y))`. Those who did last week's homework may recognize -this. +is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u +(fun x -> bind v (fun y -> f x y))`. Those who did last week's +homework may recognize this last expression. You can think of the +notation like this: take the singing box `u` and evaluate it (which +includes listening to the song). Take the int contained in the +singing box (the end result of evaluting `u`) and bind the variable +`x` to that int. So `x <- u` means "Sing me up an int, which I'll call +`x`". (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 @@ -175,39 +323,39 @@ As we proceed, we'll be seeing a variety of other monad systems. For example, an # type 'a list -The unit/return operation is: +The `unit`/return operation is: # let unit x = [x];; val unit : 'a -> 'a list =-

-# let ( * ) u f = match u with None -> None | Some x -> f x;; -val ( * ) : 'a option -> ('a -> 'b option) -> 'b option =+ # let divide x y = if 0 = y then None else Some (x/y);; + val divide : int -> int -> int option =-# let unit x = Some x;; -val unit : 'a -> 'a option = + # let unit x = Some x;; + val unit : 'a -> 'a option = + # let ( >>= ) u f = match u with None -> None | Some x -> f x;; + val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = -# unit 2;; -- : int option = Some 2 -# 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 + `u * f` or `( * ) u f` instead of `* u f`. Now: -# divide 6 2;; -- : int option = Some 3 -# unit 2 * divide 6;; -- : int option = Some 3 + # unit 2;; + - : int option = Some 2 + # unit 2 >>= unit;; + - : int option = Some 2 -# divide 6 0;; -- : int option = None -# unit 0 * divide 6;; -- : int option = None -

-# Some 3 * unit * unit;; -- : int option = Some 3 -# Some 3 * (fun x -> unit x * unit);; -- : int option = Some 3 + # 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 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 -+ # 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 +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 +convince yourself that the `bind` operation for the option monad obeys associativity by dividing the inputs into cases: if `u` matches `None`, both computations will result in `None`; if `u` matches `Some x`, and `f x` evalutes to `None`, then both @@ -286,14 +433,12 @@ computations will again result in `None`; and if the value of to `g y`. * **Right identity: unit is a right identity for bind.** That is, - `u * unit == u` for all monad objects `u`. For instance, + `u >>= unit == u` for all monad objects `u`. For instance, -

-# Some 3 * unit;; -- : int option = Some 3 -# None * unit;; -- : 'a option = None -+ # Some 3 >>= unit;; + - : int option = Some 3 + # None >>= unit;; + - : 'a option = None More details about monads @@ -308,28 +453,37 @@ 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 it's possible to make the connection between -monads and monoids much closer. This is discussed in [[Advanced Notes/Monads in Category Theory]]. See also

-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 u f (w:s) = f (u w) w;; -val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =- -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: - --# bind (unit 'a') unit 1;; -- : char = 'a' -

-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 -