From 50a08096ab665e3fb7a7ee67e88875c248530bcb Mon Sep 17 00:00:00 2001 From: jim Date: Sun, 22 Mar 2015 10:26:09 -0400 Subject: [PATCH] removed --- topics/_week8_scratch.mdwn | 434 --------------------------------------------- 1 file changed, 434 deletions(-) delete mode 100644 topics/_week8_scratch.mdwn diff --git a/topics/_week8_scratch.mdwn b/topics/_week8_scratch.mdwn deleted file mode 100644 index 6dfbdeb8..00000000 --- a/topics/_week8_scratch.mdwn +++ /dev/null @@ -1,434 +0,0 @@ -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 - 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 `u` has type `int option`, - then `u` 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 `x` by mapping it to - the option `Some x`. In the general case, this operation is - known as `unit` or `return.` Both of those names are terrible. This - operation is only very loosely connected to the `unit` type we were - 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. [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 - of the singleton function as an example: it takes an ordinary object - and returns a set containing that object. In the example we've been - considering: - - let unit x = Some x;; - val unit : 'a -> 'a option = - - So `unit` is a way to put something inside of a monadic box. It's crucial - to the usefulness of monads that there will be monadic boxes that - aren't the result of that operation. In the Option/Maybe monad, for - instance, there's also the empty box `None`. In another (whimsical) - example, you might have, in addition to boxes merely containing integers, - special boxes that contain integers and also sing a song when they're opened. - - The unit/return operation will always be the simplest, conceptually - most straightforward way to lift an ordinary value into a monadic value - of the monadic type in question. - -* Thirdly, an operation that's often called `bind`. As we said before, this is another - unfortunate name: this operation is only very loosely connected to - what linguists usually mean by "binding." In our Option/Maybe monad, the - bind operation is: - - let bind u f = match u with None -> None | Some x -> f x;; - val bind : 'a option -> ('a -> 'b option) -> 'b option = - - Note the type: `bind` takes two arguments: first, a monadic box - (in this case, an `'a option`); and second, a function from - ordinary objects to monadic boxes. `bind` then returns a monadic - value: in this case, a `'b option` (you can start with, e.g., `int option`s - and end with `bool option`s). - - Intuitively, the interpretation of what `bind` does is this: - the first argument is a monadic value `u`, which - evaluates to a box that (maybe) contains some ordinary value, call it `x`. - Then the second argument uses `x` to compute a new monadic - value. Conceptually, then, we have - - let bind u f = (let x = unbox u in f x);; - - The guts of the definition of the `bind` operation amount to - specifying how to unbox the monadic value `u`. In the `bind` - operator for the Option monad, we unboxed the monadic value by - matching it with the pattern `Some x`---whenever `u` - happened to be a box containing an integer `x`, this allowed us to - get our hands on that `x` and feed it to `f`. - - If the monadic box didn't contain any ordinary value, - we instead pass through the empty box unaltered. - - In a more complicated case, like our whimsical "singing box" example - from before, if the monadic value happened to be a singing box - containing an integer `x`, then the `bind` operation would probably - 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 - useful way. - -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`: wherever you see `u >>= f`, that means `bind u f`. -Wadler uses ⋆, but that hasn't been widely adopted (unfortunately). - -Also, if you ever see this notation: - - do - x <- u - f x - -That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`. -Similarly: - - do - x <- u - 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. 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. (See our page on [[Translating between OCaml Scheme and Haskell]].) It won't work in -OCaml. In fact, the `<-` symbol already means something different in OCaml, -having to do with mutable record fields. We'll be discussing mutation someday -soon.) - -As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is: - - # type 'a list - -The `unit`/return operation is: - - # let unit x = [x];; - val unit : 'a -> 'a list = - -That is, the simplest way to lift an `'a` into an `'a list` is just to make a -singleton list of that `'a`. Finally, the `bind` operation is: - - # let bind u f = List.concat (List.map f u);; - val bind : 'a list -> ('a -> 'b list) -> 'b list = - -What's going on here? Well, consider `List.map f u` first. This goes through all -the members of the list `u`. There may be just a single member, if `u = unit x` -for some `x`. Or on the other hand, there may be no members, or many members. In -any case, we go through them in turn and feed them to `f`. Anything that gets fed -to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`. -For example, it might return a list of all that value's divisors. Then we'll -have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch -of `'b list`s into a single `'b list`: - - # List.concat [[1]; [1;2]; [1;3]; [1;2;4]] - - : int list = [1; 1; 2; 1; 3; 1; 2; 4] - -So now we've seen two monads: the Option/Maybe monad, and the List monad. For any -monadic system, there has to be a specification of the complex monad type, -which will be parameterized on some simpler type `'a`, and the `unit`/return -operation, and the `bind` operation. These will be different for different -monadic systems. - -Many monadic systems will also define special-purpose operations that only make -sense for that system. - -Although the `unit` and `bind` operation are defined differently for different -monadic systems, there are some general rules they always have to follow. - - -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 -> 'b m`, where `'b m` is a monadic - 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 = - - 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 equivalently `( >>= ) u f` instead of `>>= u - f`. - - # unit 2;; - - : int option = Some 2 - # unit 2 >>= unit;; - - : int option = Some 2 - - Now, for a less trivial instance of a function from `int`s to `int option`s: - - # 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;; - - : int option = Some 3 - - # divide 6 0;; - - : int option = None - # 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) - - 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`. - - Wadler and others try to make this look nicer by phrasing it like this, - where U, V, and W are schematic for any expressions with the relevant monadic type: - - (U >>= fun x -> V) >>= fun y -> W == U >>= fun x -> (V >>= fun y -> W) - - 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;; - - : 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 -> 'b 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 `u` - matches `None`, both computations will result in `None`; if - `u` matches `Some x`, and `f x` evalutes to `None`, then both - computations will again result in `None`; and if the value of - `f x` matches `Some y`, then both computations will evaluate - to `g y`. - -* **Right identity: unit is a right identity for bind.** That is, - `u >>= unit == u` for all monad objects `u`. For instance, - - # Some 3 >>= unit;; - - : int option = Some 3 - # None >>= unit;; - - : 'a option = None - - -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, `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_topics/monads_in_category_theory). - -See also: - -* [Haskell wikibook on Monad Laws](http://www.haskell.org/haskellwiki/Monad_Laws). -* [Yet Another Haskell Tutorial on Monad Laws](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Definition) -* [Haskell wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads) -* [Haskell wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads) -* [Haskell wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation) -* [Yet Another Haskell Tutorial on do-notation](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Do_Notation) - - -Here are some papers that introduced monads into functional programming: - -* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible. - -* [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps): -invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992. - - -* [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf): -in M. Broy, editor, *Marktoberdorf Summer School on Program Design -Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems -sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer, -editors, *Advanced Functional Programming*, Springer Verlag, -LNCS 925, 1995. Some errata fixed August 2001. - - - -There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you. - -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_topics/monads_in_category_theory) notes do so, for example. - -Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference. - -You may sometimes see: - - u >> v - -That just means: - - u >>= fun _ -> v - -that is: - - bind u (fun _ -> v) - -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: - - # let even x = (x mod 2 = 0);; - val g : int -> bool = - -`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad? - - # let lift g = fun u -> bind u (fun x -> Some (g x));; - val lift : ('a -> 'b) -> 'a option -> 'b option = - -`lift even` will now be a function from `int option`s to `bool option`s. We can -also define a lift operation for binary functions: - - # let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));; - val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = - -`lift2 (+)` will now be a function from `int option`s and `int option`s to `int option`s. This should look familiar to those who did the homework. - -The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`! - -Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with. - -In general, any lift/map operation can be relied on to satisfy these laws: - - * lift id = id - * lift (compose f g) = compose (lift f) (lift g) - -where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these -two computations to give the same result: - - List.map (fun x -> f (g x)) lst - List.map f (List.map g lst) - -Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this: - - ap [f] [x; y] = [f x; f y] - ap (Some f) (Some x) = Some (f x) - -and so on. Here are the laws that any `ap` operation can be relied on to satisfy: - - ap (unit id) u = u - ap (ap (ap (unit compose) u) v) w = ap u (ap v w) - ap (unit f) (unit x) = unit (f x) - ap u (unit x) = ap (unit (fun f -> f x)) u - -Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where -we went from: - - [[1]; [1;2]; [1;3]; [1;2;4]] - -to: - - [1; 1; 2; 1; 3; 1; 2; 4] - -That is the `join` operation. - -All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions: - - lift f u = u >>= compose unit f - lift f u = ap (unit f) u - lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y))) - lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v - ap u v = u >>= (fun f -> lift f v) - ap u v = lift2 id u v - join m2 = m2 >>= id - u >>= f = join (lift f u) - u >> v = u >>= (fun _ -> v) - u >> v = lift2 (fun _ -> id) u v - - - -Monad outlook -------------- - -We're going to be using monads for a number of different things in the -weeks to come. One major 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. - -But first, we'll look at several linguistic applications for monads, based -on what's called the *Reader monad*. - -##[[Reader Monad for Variable Binding]]## - -##[[Reader Monad for Intensionality]]## - -- 2.11.0