X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=inline;f=week7.mdwn;h=857a636bad6dad17cdda8689905be7dd934174c9;hb=115b2bc2456adb01295e074ccc825942e45c46c0;hp=c81ff1253ceca9ee8cb724163cee5e139bcfe234;hpb=1713e01a3a0982e0f8fc68ed93035cea6ca8f46e;p=lambda.git diff --git a/week7.mdwn b/week7.mdwn index c81ff125..857a636b 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -1,28 +1,156 @@ [[!toc]] -Monads ------- - -Start by (re)reading the discussion of monads in the lecture notes for -week 6 [[Towards Monads]]. -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` 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. +Towards Monads: Safe division +----------------------------- -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. +[This section used to be near the end of the lecture notes for week 6] + +We begin by reasoning about what should happen when someone tries to +divide by zero. This will lead us to a general programming technique +called a *monad*, which we'll see in many guises in the weeks to come. + +Integer division presupposes that its second argument +(the divisor) is not zero, upon pain of presupposition failure. +Here's what my OCaml interpreter says: + + # 12/0;; + Exception: Division_by_zero. + +So we want to explicitly allow for the possibility that +division will return something other than a number. +We'll use OCaml's `option` type, which works like this: + + # type 'a option = None | Some of 'a;; + # None;; + - : 'a option = None + # Some 3;; + - : int option = Some 3 + +So if a division is normal, we return some number, but if the divisor is +zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function. + +
+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 @@ -39,6 +167,29 @@ 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.] + +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: @@ -59,7 +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 @@ -119,6 +275,8 @@ that provides at least the following three elements: be defined so as to make sure that the result of `f x` was also a singing box. If `f` also wanted to insert a song, you'd have to decide whether both songs would be carried through, or only one of them. + (Are you beginning to realize how wierd and wonderful monads + can be?) There is no single `bind` function that dictates how this must go. For each new monadic type, this has to be worked out in an @@ -128,17 +286,11 @@ So the "option/maybe monad" consists of the polymorphic `option` type, the `unit`/return function, and the `bind` function. -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 @@ -152,9 +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 last expression. +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 @@ -210,59 +367,63 @@ 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 - type, we have `(unit x) * f == f x`. For instance, `unit` is itself + type, we have `(unit x) >>= f == f x`. For instance, `unit` is itself a function of type `'a -> 'a m`, so we can use it for `f`: # let unit x = Some x;; val unit : 'a -> 'a option =
-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 =