lists-monad tweaks

[lambda.git] / list_monad_as_continuation_monad.mdwn

diff --git a/list_monad_as_continuation_monad.mdwn b/list_monad_as_continuation_monad.mdwn
index acc0509..81a5be3 100644 (file)
--- a/list_monad_as_continuation_monad.mdwn
+++ b/list_monad_as_continuation_monad.mdwn
@@ -10,7 +10,7 @@ Rethinking the list monad
 To construct a monad, the key element is to settle on a type
 constructor, and the monad more or less naturally follows from that.
 We'll remind you of some examples of how monads follow from the type
 To construct a monad, the key element is to settle on a type
 constructor, and the monad more or less 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
+constructor in a moment.  This will involve some review of familiar

 material, but it's worth doing for two reasons: it will set up a
 pattern for the new discussion further below, and it will tie together
 some previously unconnected elements of the course (more specifically,
 material, but it's worth doing for two reasons: it will set up a
 pattern for the new discussion further below, and it will tie together
 some previously unconnected elements of the course (more specifically,


@@ -19,11 +19,11 @@ version 3 lists and monads).
 For instance, take the **Reader Monad**.  Once we decide that the type
 constructor is
 
 For instance, take the **Reader Monad**.  Once we decide that the type
 constructor is
 

-    type 'a reader = env -> 'a
+       type 'a reader = env -> 'a

 
 then the choice of unit and bind is natural:
 
 
 then the choice of unit and bind is natural:
 

-    let r_unit (a : 'a) : 'a reader = fun (e : env) -> a
+       let r_unit (a : 'a) : 'a reader = fun (e : env) -> a

 
 The reason this is a fairly natural choice is that because the type of
 an `'a reader` is `env -> 'a` (by definition), the type of the
 
 The reason this is a fairly natural choice is that because the type of
 an `'a reader` is `env -> 'a` (by definition), the type of the


@@ -33,13 +33,13 @@ for the reader monad.
 
 Since the type of the `bind` operator is required to be
 
 
 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 reason our way to the traditional reader `bind` function as
 follows. We start by declaring the types determined by the definition
 of a bind operation:
 
 
 We can reason our way to the traditional reader `bind` function as
 follows. We start by declaring the types determined by the definition
 of a bind operation:
 

-    let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...
+       let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...

 
 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
 
 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


@@ -72,15 +72,15 @@ monads.]
 
 The **State Monad** is similar.  Once we've decided to use the following type constructor:
 
 
 The **State Monad** is similar.  Once we've decided to use the following type constructor:
 

-    type 'a state = store -> ('a, store)
+       type 'a state = store -> ('a, store)

 
 Then our unit is naturally:
 
 
 Then our unit is naturally:
 

-    let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
+       let s_unit (a : 'a) : 'a state = fun (s : store) -> (a, s)

 
 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:
 
 
 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 =
+       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
            ... f (...) ...
 
 But unlocking the `u` box is a little more complicated.  As before, we


@@ -102,9 +102,9 @@ follow just as naturally from its type constructor.
 Our other familiar monad is the **List Monad**, which we were told
 looks like this:
 
 Our other familiar monad is the **List Monad**, which we were told
 looks like this:
 

-    type 'a list = ['a];;
-    l_unit (a : 'a) = [a];;
-    l_bind u f = List.concat (List.map f u);;
+       type 'a list = ['a];;
+       l_unit (a : 'a) = [a];;
+       l_bind u f = List.concat (List.map f u);;

 
 Thinking through the list monad will take a little time, but doing so
 will provide a connection with continuations.
 
 Thinking through the list monad will take a little time, but doing so
 will provide a connection with continuations.


@@ -112,21 +112,21 @@ will provide a connection with continuations.
 Recall that `List.map` takes a function and a list and returns the
 result to applying the function to the elements of the list:
 
 Recall that `List.map` takes a function and a list and returns the
 result to applying the function to the elements of the list:
 

-    List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
+       List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]

 
 

-and List.concat takes a list of lists and erases the embdded list
+and `List.concat` takes a list of lists and erases the embedded list

 boundaries:
 
 boundaries:
 

-    List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
+       List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]

 
 And sure enough,
 
 
 And sure enough,
 

-    l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
+       l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]

 
 Now, why this unit, and why this bind?  Well, ideally a unit should
 not throw away information, so we can rule out `fun x -> []` as an
 ideal unit.  And units should not add more information than required,
 
 Now, why this unit, and why this bind?  Well, ideally a unit should
 not throw away information, so we can rule out `fun x -> []` as an
 ideal unit.  And units should not add more information than required,

-so there's no obvious reason to prefer `fun x -> [x,x]`.  In other
+so there's no obvious reason to prefer `fun x -> [x;x]`.  In other

 words, `fun x -> [x]` is a reasonable choice for a unit.
 
 As for bind, an `'a list` monadic object contains a lot of objects of
 words, `fun x -> [x]` is a reasonable choice for a unit.
 
 As for bind, an `'a list` monadic object contains a lot of objects of


@@ -138,25 +138,25 @@ have a collection of lists, one for each of the `'a`'s.  One
 possibility is that we could gather them all up in a list, so that
 `bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`.  But that restricts
 the object returned by the second argument of `bind` to always be of
 possibility is that we could gather them all up in a list, so that
 `bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`.  But that restricts
 the object returned by the second argument of `bind` to always be of

-type `'b list list`.  We can elimiate that restriction by flattening
+type `'b list list`.  We can eliminate that restriction by flattening

 the list of lists into a single list: this is
 the list of lists into a single list: this is

-just List.concat applied to the output of List.map.  So there is some logic to the
+just `List.concat` applied to the output of `List.map`.  So there is some logic to the

 choice of unit and bind for the list monad.
 
 Yet we can still desire to go deeper, and see if the appropriate bind
 behavior emerges from the types, as it did for the previously
 considered monads.  But we can't do that if we leave the list type as
 choice of unit and bind for the list monad.
 
 Yet we can still desire to go deeper, and see if the appropriate bind
 behavior emerges from the types, as it did for the previously
 considered monads.  But we can't do that if we leave the list type as

-a primitive Ocaml type.  However, we know several ways of implementing
-lists using just functions.  In what follows, we're going to use type
+a primitive OCaml type.  However, we know several ways of implementing
+lists using just functions.  In what follows, we're going to use version

 3 lists, the right fold implementation (though it's important and
 intriguing to wonder how things would change if we used some other
 3 lists, the right fold implementation (though it's important and
 intriguing to wonder how things would change if we used some other

-strategy for implementating lists).  These were the lists that made
-lists look like Church numerals with extra bits embdded in them:
+strategy for implementing lists).  These were the lists that made
+lists look like Church numerals with extra bits embedded 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))
+       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
 principle types of these functions (rather than inferring what the
 
 and so on.  To save time, we'll let the OCaml interpreter infer the
 principle types of these functions (rather than inferring what the


@@ -174,41 +174,41 @@ types should be ourselves):
 We can see what the consistent, general principle types are at the end, 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 type
 We can see what the consistent, general principle types are at the end, 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 type

-thrown in, the type of the element a the head of the list
+thrown in, the type of the element at the head of the list

 (in this case, an int).
 
 So here's our type constructor for our hand-rolled lists:
 
 (in this case, an int).
 
 So here's our type constructor for our hand-rolled lists:
 

-    type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
+       type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b

 
 Generalizing to lists that contain any kind of element (not just
 
 Generalizing to lists that contain any kind of element (not just

-ints), we have
+`int`s), we have

 
 

-    type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
+       type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b

 
 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
 
 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 fully grasp the role of the `'b`'s
+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:
 
 in order to proceed to build a monad:
 

-    l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z
+       l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun k z -> k a 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.
+Take an `'a` and return its v3-style singleton. 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'  = ...

 
 Unpacking the types gives:
 
 
 Unpacking the types gives:
 

-    l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
-            (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
-            : ('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 = ...

 
 

-Perhaps a bit intimiating.
+Perhaps a bit intimidating.

 But it's a rookie mistake to quail before complicated types. You should
 But it's a rookie mistake to quail before complicated types. You should

-be no more intimiated by complex types than by a linguistic tree with
+be no more intimidated by complex types than by a linguistic tree with

 deeply embedded branches: complex structure created by repeated
 application of simple rules.
 
 deeply embedded branches: complex structure created by repeated
 application of simple rules.
 


@@ -237,16 +237,16 @@ Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over
 
 This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that our bind is:
 
 
 This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that our bind is:
 

-    l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
-            (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
-            : ('c -> 'b -> 'b) -> 'b -> 'b =
-      fun k -> u (fun a b -> f a k b)
+       l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
+               (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
+                  : ('c -> 'b -> 'b) -> 'b -> 'b =
+           fun k -> u (fun a b -> f a k b)

 
 That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.
 
 Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:
 
 
 That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.
 
 Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:
 

-      fun k z -> u (fun a b -> f a k b) z
+       fun k z -> u (fun a b -> f a k b) z

 
 Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?
 
 
 Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?
 


@@ -258,28 +258,28 @@ Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' w
 
 Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
 
 
 Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
 

-      fun k z -> u (fun a b -> f a k b) z
+       fun k z -> u (fun a b -> f a k b) z

 
 do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
 
 
 do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
 

-       []
+       []        ; result of applying f to leftmost a

        [2]
        [2; 4]
        [2]
        [2; 4]

-       [2; 4; 8]
+       [2; 4; 8] ; result of applying f to rightmost a

 
 (or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.
 
 So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
 
        0 ==>
 
 (or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.
 
 So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
 
        0 ==>

-       right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
-       right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
-       right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
-       right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0
+       right-fold + and 0 over [2; 4; 8] = ((2+4+8+0) ==>
+       right-fold + and 2+4+8+0 over [2; 4] = 2+4+(2+4+8+0) ==>
+       right-fold + and 2+4+2+4+8+0 over [2] = 2+(2+4+(2+4+8+0)) ==>
+       right-fold + and 2+2+4+2+4+8+0 over [] = 2+(2+4+(2+4+8+0))

 
 which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
 
 
 which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
 

-      fun k z -> u (fun a b -> f a k b) z
+       fun k z -> u (fun a b -> f a k b) z

 
 will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as
 
 
 will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as
 


@@ -289,15 +289,14 @@ would.
 
 For future reference, we might make two eta-reductions to our formula, so that we have instead:
 
 
 For future reference, we might make two eta-reductions to our formula, so that we have instead:
 

-      let l'_bind = fun k -> u (fun a -> f a k);;
+       let l'_bind = fun k -> u (fun a -> f a k);;

 
 Let's make some more tests:
 
 
 Let's make some more tests:
 

-
-    l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
-
-    l'_bind (fun f z -> f 1 (f 2 z))
-            (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
+       l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
+       
+       l'_bind (fun f z -> f 1 (f 2 z))
+           (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>

 
 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


@@ -318,7 +317,7 @@ 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.
 
 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,
+First, we'll need a type constructor.  As we've said,

 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
 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


@@ -342,7 +341,7 @@ highly similar to the List monad just given:
        type 'a continuation = ('a -> 'b) -> 'b
        c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
        c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
        type 'a continuation = ('a -> 'b) -> 'b
        c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
        c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =

-         fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
+           fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)

 
 Note that `c_unit` is exactly the `gqize` function that Montague used
 to lift individuals into the continuation monad.
 
 Note that `c_unit` is exactly the `gqize` function that Montague used
 to lift individuals into the continuation monad.


@@ -352,15 +351,20 @@ it in the List monad.  How similar is it to the List monad?  Let's
 examine the type constructor and the terms from the list monad derived
 above:
 
 examine the type constructor and the terms from the list monad derived
 above:
 

-    type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
-    l'_unit a = fun f -> f a
-    l'_bind u f = fun k -> u (fun a -> f a k)
+       type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
+       let l'_unit a = fun k z -> k a z
+
+This can be eta-reduced to:
+
+       let l'_unit a = fun k -> k a

 
 

-(We performed a sneaky but valid eta reduction in the unit term.)
+       let l'_bind u f =
+           (* we mentioned three versions of this, the fully eta-expanded: *)
+           fun k z -> u (fun a b -> f a k b) z
+               (* an intermediate version, and the fully eta-reduced: *)
+           fun k -> u (fun a -> f a k)

 
 

-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
+Consider the most eta-reduced versions of `l'_unit` and `l'_bind`. They're the same as the unit and bind for the Montague continuation monad! In other words, the behavior of our v3-list monad and the behavior of the continuations monad are

 parallel in a deep sense.
 
 Have we really discovered that lists are secretly continuations?  Or
 parallel in a deep sense.
 
 Have we really discovered that lists are secretly continuations?  Or