X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=inline;f=zipper-lists-continuations.mdwn;h=8858a8930699f29335825e27600ff8c2f6bbb253;hb=9273d85724c80393c8ec4dea26705c47ccdf10c8;hp=d13dc353d71e06e2aaa4446771ddf5df21360f2f;hpb=b078fa71d02d362b0b9b763abc77f75d8f19e22e;p=lambda.git diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index d13dc353..8858a893 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -5,15 +5,13 @@ continuation monad. The three approches are: -* Rethinking the list monad; -* Montague's PTQ treatment of DPs as generalized quantifiers; and -* Refunctionalizing zippers (Shan: zippers are defunctionalized continuations); +[[!toc]] Rethinking the list monad ------------------------- To construct a monad, the key element is to settle on a type -constructor, and the monad naturally follows from that. I'll remind +constructor, and the monad naturally follows from that. We'll remind you of some examples of how monads follow from the type constructor in a moment. This will involve some review of familair material, but it's worth doing for two reasons: it will set up a pattern for the new @@ -24,72 +22,77 @@ and monads). For instance, take the **Reader Monad**. Once we decide that the type constructor is - type 'a reader = fun e:env -> 'a + type 'a reader = env -> 'a -then we can deduce the unit and the bind: +then the choice of unit and bind is natural: - runit x:'a -> 'a reader = fun (e:env) -> x + let r_unit (a : 'a) : 'a reader = fun (e : env) -> a -Since the type of an `'a reader` is `fun e:env -> 'a` (by definition), -the type of the `runit` function is `'a -> e:env -> 'a`, which is a -specific case of the type of the *K* combinator. So it makes sense +Since the type of an `'a reader` is `env -> 'a` (by definition), +the type of the `r_unit` function is `'a -> env -> 'a`, which is a +specific case of the type of the *K* combinator. It makes sense that *K* is the unit for the reader monad. Since the type of the `bind` operator is required to be - r_bind:('a reader) -> ('a -> 'b reader) -> ('b reader) + r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader) -We can deduce the correct `bind` function as follows: +We can reason our way to the correct `bind` function as follows. We start by declaring the type: - r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) = + let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = -We have to open up the `u` box and get out the `'a` object in order to +Now we have to open up the `u` box and get out the `'a` object in order to feed it to `f`. Since `u` is a function from environments to -objects of type `'a`, we'll have +objects of type `'a`, the way we open a box in this monad is +by applying it to an environment: - .... f (u e) ... + ... f (u e) ... This subexpression types to `'b reader`, which is good. The only -problem is that we don't have an `e`, so we have to abstract over that -variable: +problem is that we invented an environment `e` that we didn't already have , +so we have to abstract over that variable to balance the books: - fun e -> f (u e) ... + fun e -> f (u e) ... This types to `env -> 'b reader`, but we want to end up with `env -> -'b`. The easiest way to turn a 'b reader into a 'b is to apply it to -an environment. So we end up as follows: +'b`. Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to an environment. So we end up as follows: - r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) = f (u e) e + r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = + f (u e) e -And we're done. +And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does. -The **State Monad** is similar. We somehow intuit that we want to use -the following type constructor: +[The bind we cite here is a condensed version of the careful `let a = u e in ...` +constructions we provided in earlier lectures. We use the condensed +version here in order to emphasize similarities of structure across +monads.] - type 'a state = 'store -> ('a, 'store) +The **State Monad** is similar. Once we've decided to use the following type constructor: -So our unit is naturally + type 'a state = store -> ('a, store) - let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s) +Then our unit is naturally: -And we deduce the bind in a way similar to the reasoning given above. -First, we need to apply `f` to the contents of the `u` box: + let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s) - let s_bind (u:'a state) (f:'a -> ('b state)):('b state) = +And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box: + + let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = + ... f (...) ... But unlocking the `u` box is a little more complicated. As before, we need to posit a state `s` that we can apply `u` to. Once we do so, however, we won't have an `'a`, we'll have a pair whose first element is an `'a`. So we have to unpack the pair: - ... let (a, s') = u s in ... (f a) ... + ... let (a, s') = u s in ... (f a) ... Abstracting over the `s` and adjusting the types gives the result: - let s_bind (u:'a state) (f:'a -> ('b state)):('b state) = - fun (s:state) -> let (a, s') = u s in f a s' + let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = + fun (s : store) -> let (a, s') = u s in f a s' -The **Option Monad** doesn't follow the same pattern so closely, so we +The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we won't pause to explore it here, though conceptually its unit and bind follow just as naturally from its type constructor. @@ -97,7 +100,7 @@ Our other familiar monad is the **List Monad**, which we were told looks like this: type 'a list = ['a];; - l_unit (x:'a) = [x];; + l_unit (a : 'a) = [a];; l_bind u f = List.concat (List.map f u);; Recall that `List.map` take a function and a list and returns the @@ -117,43 +120,41 @@ And sure enough, But where is the reasoning that led us to this unit and bind? And what is the type `['a]`? Magic. -So let's take a *completely useless digressing* and see if we can -gain some insight into the details of the List monad. Let's choose -type constructor that we can peer into, using some of the technology -we built up so laboriously during the first half of the course. I'm -going to use type 3 lists, partly because I know they'll give the -result I want, but also because they're my favorite. These were the -lists that made lists look like Church numerals with extra bits -embdded in them: +So let's indulge ourselves in a completely useless digression and see +if we can gain some insight into the details of the List monad. Let's +choose type constructor that we can peer into, using some of the +technology we built up so laboriously during the first half of the +course. We're going to use type 3 lists, partly because we know +they'll give the result we want, but also because they're the coolest. +These were the lists that made lists look like Church numerals with +extra bits embdded in them: empty list: fun f z -> z list with one element: fun f z -> f 1 z list with two elements: fun f z -> f 2 (f 1 z) list with three elements: fun f z -> f 3 (f 2 (f 1 z)) -and so on. To save time, we'll let the Ocaml interpreter infer the +and so on. To save time, we'll let the OCaml interpreter infer the principle types of these functions (rather than deducing what the types should be): -
-# fun f z -> z;; -- : 'a -> 'b -> 'b =+ # fun f z -> z;; + - : 'a -> 'b -> 'b =-# fun f z -> f 1 z;; -- : (int -> 'a -> 'b) -> 'a -> 'b = -# fun f z -> f 2 (f 1 z);; -- : (int -> 'a -> 'a) -> 'a -> 'a = -# fun f z -> f 3 (f 2 (f 1 z)) -- : (int -> 'a -> 'a) -> 'a -> 'a = -
+let t1 = Node ((Node ((Leaf 2), (Leaf 3))), + (Node ((Leaf 5),(Node ((Leaf 7), + (Leaf 11)))))) + + . + ___|___ + | | + . . +_|__ _|__ +| | | | +2 3 5 . + _|__ + | | + 7 11 ++ +Our first task will be to replace each leaf with its double: + +
+let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) = + match t with Leaf x -> Leaf (newleaf x) + | Node (l, r) -> Node ((treemap newleaf l), + (treemap newleaf r));; ++`treemap` takes a function that transforms old leaves into new leaves, +and maps that function over all the leaves in the tree, leaving the +structure of the tree unchanged. For instance: + +
+let double i = i + i;; +treemap double t1;; +- : int tree = +Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) + + . + ___|____ + | | + . . +_|__ __|__ +| | | | +4 6 10 . + __|___ + | | + 14 22 ++ +We could have built the doubling operation right into the `treemap` +code. However, because what to do to each leaf is a parameter, we can +decide to do something else to the leaves without needing to rewrite +`treemap`. For instance, we can easily square each leaf instead by +supplying the appropriate `int -> int` operation in place of `double`: + +
+let square x = x * x;; +treemap square t1;; +- : int tree =ppp +Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) ++ +Note that what `treemap` does is take some global, contextual +information---what to do to each leaf---and supplies that information +to each subpart of the computation. In other words, `treemap` has the +behavior of a reader monad. Let's make that explicit. + +In general, we're on a journey of making our treemap function more and +more flexible. So the next step---combining the tree transducer with +a reader monad---is to have the treemap function return a (monadized) +tree that is ready to accept any `int->int` function and produce the +updated tree. + +\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11)))) +
+\f . + ____|____ + | | + . . +__|__ __|__ +| | | | +f2 f3 f5 . + __|___ + | | + f7 f11 ++ +That is, we want to transform the ordinary tree `t1` (of type `int +tree`) into a reader object of type `(int->int)-> int tree`: something +that, when you apply it to an `int->int` function returns an `int +tree` in which each leaf `x` has been replaced with `(f x)`. + +With previous readers, we always knew which kind of environment to +expect: either an assignment function (the original calculator +simulation), a world (the intensionality monad), an integer (the +Jacobson-inspired link monad), etc. In this situation, it will be +enough for now to expect that our reader will expect a function of +type `int->int`. + +
+type 'a reader = (int->int) -> 'a;; (* mnemonic: e for environment *) +let reader_unit (x:'a): 'a reader = fun _ -> x;; +let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;; ++ +It's easy to figure out how to turn an `int` into an `int reader`: + +
+let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;; +int2int_reader 2 (fun i -> i + i);; +- : int = 4 ++ +But what do we do when the integers are scattered over the leaves of a +tree? A binary tree is not the kind of thing that we can apply a +function of type `int->int` to. + +
+let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader = + match t with Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x')) + | Node (l, r) -> reader_bind (treemonadizer f l) (fun x -> + reader_bind (treemonadizer f r) (fun y -> + reader_unit (Node (x, y))));; ++ +This function says: give me a function `f` that knows how to turn +something of type `'a` into an `'b reader`, and I'll show you how to +turn an `'a tree` into an `'a tree reader`. In more fanciful terms, +the `treemonadizer` function builds plumbing that connects all of the +leaves of a tree into one connected monadic network; it threads the +monad through the leaves. + +
+# treemonadizer int2int_reader t1 (fun i -> i + i);; +- : int tree = +Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) ++ +Here, our environment is the doubling function (`fun i -> i + i`). If +we apply the very same `int tree reader` (namely, `treemonadizer +int2int_reader t1`) to a different `int->int` function---say, the +squaring function, `fun i -> i * i`---we get an entirely different +result: + +
+# treemonadizer int2int_reader t1 (fun i -> i * i);; +- : int tree = +Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) ++ +Now that we have a tree transducer that accepts a monad as a +parameter, we can see what it would take to swap in a different monad. +For instance, we can use a state monad to count the number of nodes in +the tree. + +
+type 'a state = int -> 'a * int;; +let state_unit x i = (x, i+.5);; +let state_bind u f i = let (a, i') = u i in f a (i'+.5);; ++ +Gratifyingly, we can use the `treemonadizer` function without any +modification whatsoever, except for replacing the (parametric) type +`reader` with `state`: + +
+let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state = + match t with Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x')) + | Node (l, r) -> state_bind (treemonadizer f l) (fun x -> + state_bind (treemonadizer f r) (fun y -> + state_unit (Node (x, y))));; ++ +Then we can count the number of nodes in the tree: + +
+# treemonadizer state_unit t1 0;; +- : int tree * int = +(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13) + + . + ___|___ + | | + . . +_|__ _|__ +| | | | +2 3 5 . + _|__ + | | + 7 11 ++ +Notice that we've counted each internal node twice---it's a good +exercise to adjust the code to count each node once. + +One more revealing example before getting down to business: replacing +`state` everywhere in `treemonadizer` with `list` gives us + +
+# treemonadizer (fun x -> [ [x; square x] ]) t1;; +- : int list tree list = +[Node + (Node (Leaf [2; 4], Leaf [3; 9]), + Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))] ++ +Unlike the previous cases, instead of turning a tree into a function +from some input to a result, this transformer replaces each `int` with +a list of `int`'s. + +Now for the main point. What if we wanted to convert a tree to a list +of leaves? + +
+type ('a, 'r) continuation = ('a -> 'r) -> 'r;; +let continuation_unit x c = c x;; +let continuation_bind u f c = u (fun a -> f a c);; + +let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation = + match t with Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x')) + | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x -> + continuation_bind (treemonadizer f r) (fun y -> + continuation_unit (Node (x, y))));; ++ +We use the continuation monad described above, and insert the +`continuation` type in the appropriate place in the `treemonadizer` code. +We then compute: + +
+# treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);; +- : int list = [2; 3; 5; 7; 11] ++ +We have found a way of collapsing a tree into a list of its leaves. + +The continuation monad is amazingly flexible; we can use it to +simulate some of the computations performed above. To see how, first +note that an interestingly uninteresting thing happens if we use the +continuation unit as our first argument to `treemonadizer`, and then +apply the result to the identity function: + +
+# treemonadizer continuation_unit t1 (fun x -> x);; +- : int tree = +Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) ++ +That is, nothing happens. But we can begin to substitute more +interesting functions for the first argument of `treemonadizer`: + +
+(* Simulating the tree reader: distributing a operation over the leaves *) +# treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);; +- : int tree = +Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) + +(* Simulating the int list tree list *) +# treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);; +- : int list tree = +Node + (Node (Leaf [2; 4], Leaf [3; 9]), + Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) + +(* Counting leaves *) +# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; +- : int = 5 ++ +We could simulate the tree state example too, but it would require +generalizing the type of the continuation monad to + + type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;