X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=0e9f25ef8455d508c33571e9734d644e59e50d94;hp=78a41506cc48e7388941bb4a28554d501107d6c0;hb=b0139eb9a0b1865ca04f76157ce8dcfbd203374b;hpb=5e53f11286531aec59bd3c665522ad5776d4188d;ds=sidebyside diff --git a/week7.mdwn b/week7.mdwn index 78a41506..0e9f25ef 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -6,6 +6,10 @@ Towards Monads: Safe division [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: @@ -148,35 +152,6 @@ None objects or real numbers---those details are hidden inside of the The definition of `div'` shows exactly what extra needs to be said in order to trigger the no-division-by-zero presupposition. -For linguists: this is a complete theory of a particularly simply form -of presupposition projection (every predicate is a hole). - - - - -Monads in General ------------------ - -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. - -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 @@ -192,6 +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: @@ -212,7 +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 @@ -272,6 +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 @@ -281,17 +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 @@ -305,9 +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 @@ -363,59 +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 = - # let ( * ) u f = match u with None -> None | Some x -> f x;; - val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = + # let ( >>= ) u f = match u with None -> None | Some x -> f x;; + val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = 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: + is `>>=`, pronounced "bind") is an infix operator, so we write + `u >>= f` or equivalently `( >>= ) u f` instead of `>>= u + f`. Now, for a less trivial instance of a function from `int`s + to `int option`s: # unit 2;; - : int option = Some 2 - # unit 2 * unit;; + # unit 2 >>= unit;; - : int option = Some 2 # let divide x y = if 0 = y then None else Some (x/y);; val divide : int -> int -> int option = # divide 6 2;; - : int option = Some 3 - # unit 2 * divide 6;; + # unit 2 >>= divide 6;; - : int option = Some 3 # divide 6 0;; - : int option = None - # unit 0 * divide 6;; + # unit 0 >>= divide 6;; - : int option = None * **Associativity: bind obeys a kind of associativity**. Like this: - (u * f) * g == u * (fun x -> f x * g) + (u >>= f) >>= g == u >>= (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 examples of associativity in the option monad (bear in + mind that in the Ocaml implementation of integer division, 2/3 + evaluates to zero, throwing away the remainder): - # 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;; + # Some 3 >>= divide 6 >>= divide 2;; - : int option = Some 1 - # Some 3 * (fun x -> divide 6 x * divide 2);; + # Some 3 >>= (fun x -> divide 6 x >>= divide 2);; - : int option = Some 1 - # Some 3 * divide 2 * divide 6;; + # Some 3 >>= divide 2 >>= divide 6;; - : int option = None - # Some 3 * (fun x -> divide 2 x * divide 6);; + # Some 3 >>= (fun x -> divide 2 x >>= divide 6);; - : int option = None Of course, associativity must hold for *arbitrary* functions of @@ -429,11 +437,11 @@ 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;; + # Some 3 >>= unit;; - : int option = Some 3 - # None * unit;; + # None >>= unit;; - : 'a option = None @@ -443,14 +451,13 @@ More details about monads 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 * u == u == u * 1` for all +serving as the left and right identity. That is, `1 * u == u == u * 1` for all `u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`. 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 it's possible to make the connection between monads and monoids much closer. This is discussed in [Monads in Category -Theory](/advanced_notes/monads_in_category_theory). +Theory](/advanced_topics/monads_in_category_theory). See also . Here are some papers that introduced monads into functional programming: @@ -474,7 +481,7 @@ invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Pres * [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from but the link above is to a local copy. -There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh. +There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes us laugh. In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category Theory](/advanced_notes/monads_in_category_theory) notes do so, for example. @@ -495,7 +502,7 @@ that is: You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used. -The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example: +The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example: # let even x = (x mod 2 = 0);; val g : int -> bool =