X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipper-lists-continuations.mdwn;h=a826eed8b22de0061fee6d8fb653b5e340406c94;hp=d13dc353d71e06e2aaa4446771ddf5df21360f2f;hb=3c9d5a19d05c749c3bf4e05874bcb153db3a18c6;hpb=fa3fcd726119256a230682b5125180daefd11235 diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index d13dc353..a826eed8 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 @@ -24,58 +22,64 @@ 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: - runit x:'a -> 'a reader = fun (e:env) -> x + let r_unit (x : 'a) : 'a reader = fun (e : env) -> x -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 +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. So 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: - 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 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) ... 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) ... 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 +'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. +[This bind 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.] + The **State Monad** is similar. We somehow intuit that we want to use the following type constructor: - type 'a state = 'store -> ('a, 'store) + type 'a state = store -> ('a, store) So our unit is naturally - let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s) + let s_unit (x : 'a) : ('a state) = fun (s : store) -> (x, s) 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_bind (u:'a state) (f:'a -> ('b state)):('b state) = + let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = 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, @@ -86,8 +90,8 @@ is an `'a`. So we have to unpack the pair: 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 won't pause to explore it here, though conceptually its unit and bind @@ -97,7 +101,7 @@ 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 (x : 'a) = [x];; l_bind u f = List.concat (List.map f u);; Recall that `List.map` take a function and a list and returns the @@ -117,21 +121,21 @@ 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): @@ -149,7 +153,7 @@ types should be): Finally, we're getting consistent principle types, so we can stop. These types should remind you of the simply-typed lambda calculus types for Church numerals (`(o -> o) -> o -> o`) with one extra bit -thrown in (in this case, and int). +thrown in (in this case, an int). So here's our type constructor for our hand-rolled lists: @@ -162,22 +166,22 @@ ints), we have So an `('a, 'b) list'` is a list containing elements of type `'a`, where `'b` is the type of some part of the plumbing. This is more -general than an ordinary Ocaml list, but we'll see how to map them -into Ocaml lists soon. We don't need to grasp the role of the `'b`'s +general than an ordinary OCaml list, but we'll see how to map them +into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s in order to proceed to build a monad: - l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z + l'_unit (x : 'a) : ('a, 'b) list = fun x -> fun f z -> f x z No problem. Arriving at bind is a little more complicated, but exactly the same principles apply, you just have to be careful and systematic about it. - l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ... + l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ... Unfortunately, we'll need to spell out the types: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) : ('c -> 'd -> 'd) -> 'd -> 'd = ... It's a rookie mistake to quail before complicated types. You should @@ -190,7 +194,7 @@ This time, `u` will only deliver up its contents if we give `u` as an argument a function expecting an `'a`. Once that argument is applied to an object of type `'a`, we'll have what we need. Thus: - .... u (fun (x:'a) -> ... (f a) ... ) ... + .... u (fun (a : 'a) -> ... (f a) ... ) ... In order for `u` to have the kind of argument it needs, we have to adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in @@ -198,21 +202,21 @@ order to deliver something of type `'b -> 'b`. The easiest way is to alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c -> 'b -> 'b`. Thus: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = - .... u (fun (x:'a) -> f a k) ... + .... u (fun (a : 'a) -> f a k) ... -[Excercise: can you arrive at a fully general bind for this type +[Exercise: can you arrive at a fully general bind for this type constructor, one that does not collapse `'d`'s with `'b`'s?] As usual, we have to abstract over `k`, but this time, no further adjustments are needed: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = - fun (k:'c -> 'b -> 'b) -> u (fun (x:'a) -> f a k) + fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) -> f a k) You should carefully check to make sure that this term is consistent with the typing. @@ -226,25 +230,76 @@ replicating the behavior of the standard List monad. Let's test: l'_bind (fun f z -> f 1 (f 2 z)) (fun i -> fun f z -> f i (f (i+1) z)) ~~> -Sigh. Ocaml won't show us our own list. So we have to choose an `f` -and a `z` that will turn our hand-crafted lists into standard Ocaml +Sigh. OCaml won't show us our own list. So we have to choose an `f` +and a `z` that will turn our hand-crafted lists into standard OCaml lists, so that they will print out. -# let cons h t = h :: t;; (* Ocaml is stupid about :: *) -# l'_bind (fun f z -> f 1 (f 2 z)) - (fun i -> fun f z -> f i (f (i+1) z)) cons [];; -- : int list = [1; 2; 2; 3] + # let cons h t = h :: t;; (* OCaml is stupid about :: *) + # l'_bind (fun f z -> f 1 (f 2 z)) + (fun i -> fun f z -> f i (f (i+1) z)) cons [];; + - : int list = [1; 2; 2; 3] Ta da! -Just for mnemonic purposes (sneaking in an instance of eta reduction -to the definition of unit), we can summarize the result as follows: +To bad this digression, though it ties together various +elements of the course, has *no relevance whatsoever* to the topic of +continuations... + +Montague's PTQ treatment of DPs as generalized quantifiers +---------------------------------------------------------- + +We've hinted that Montague's treatment of DPs as generalized +quantifiers embodies the spirit of continuations (see de Groote 2001, +Barker 2002 for lengthy discussion). Let's see why. + +First, we'll need a type constructor. As you probably know, +Montague replaced individual-denoting determiner phrases (with type `e`) +with generalized quantifiers (with [extensional] type `(e -> t) -> t`. +In particular, the denotation of a proper name like *John*, which +might originally denote a object `j` of type `e`, came to denote a +generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`. +Let's write a general function that will map individuals into their +corresponding generalized quantifier: + + gqize (x : e) = fun (p : e -> t) -> p x + +This function wraps up an individual in a fancy box. That is to say, +we are in the presence of a monad. The type constructor, the unit and +the bind follow naturally. We've done this enough times that we won't +belabor the construction of the bind function, the derivation is +similar to the List monad just given: + + type 'a continuation = ('a -> 'b) -> 'b + c_unit (x : 'a) = fun (p : 'a -> 'b) -> p x + c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd = + fun (k : 'a -> 'b) -> u (fun (x : 'a) -> f x k) + +How similar is it to the List monad? Let's examine the type +constructor and the terms from the list monad derived above: type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b - l'_unit x = fun f -> f x + l'_unit x = fun f -> f x l'_bind u f = fun k -> u (fun x -> f x k) -To bad this digression, though it ties together various -elements of the course, has *no relevance whatsoever* to the topic of -continuations. +(We performed a sneaky but valid eta reduction in the unit term.) + +The unit and the bind for the Montague continuation monad and the +homemade List monad are the same terms! In other words, the behavior +of the List monad and the behavior of the continuations monad are +parallel in a deep sense. To emphasize the parallel, we can +instantiate the type of the list' monad using the OCaml list type: + + type 'a c_list = ('a -> 'a list) -> 'a list + +Have we really discovered that lists are secretly continuations? Or +have we merely found a way of simulating lists using list +continuations? Both perspectives are valid, and we can use our +intuitions about the list monad to understand continuations, and vice +versa (not to mention our intuitions about primitive recursion in +Church numerals too). The connections will be expecially relevant +when we consider indefinites and Hamblin semantics on the linguistic +side, and non-determinism on the list monad side. + +Refunctionalizing zippers +-------------------------