From f98f15f0f1b9ec1307688682f74fd5a724a4ddf3 Mon Sep 17 00:00:00 2001
From: jim
Date: Sat, 25 Apr 2015 05:51:44 0400
Subject: [PATCH] overhaul

exercises/_assignment12.mdwn  316 +++++++++++++++++++++
1 file changed, 155 insertions(+), 161 deletions()
diff git a/exercises/_assignment12.mdwn b/exercises/_assignment12.mdwn
index 017a8d36..58068c4f 100644
 a/exercises/_assignment12.mdwn
+++ b/exercises/_assignment12.mdwn
@@ 2,22 +2,22 @@
Recall back in [[Assignment 4assignment4#fringe]], we asked you to enumerate the "fringe" of a leaflabeled tree. Both of these trees (here I *am* drawing the labels in the diagram):
 . .
 / \ / \
 . 3 1 .
 / \ / \
 1 2 2 3
+ . .
+ / \ / \
+ . 3 1 .
+ / \ / \
+ 1 2 2 3
have the same fringe: `[1; 2; 3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter new ways to approach the problem of determining when two trees have the same fringe.
Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the focused leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost leaf in a subtree), move upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be focused on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:
 . .
 / \ / \
 . 3 1 .
 / \ / \
 1 2 2 3
+ . .
+ / \ / \
+ . 3 1 .
+ / \ / \
+ 1 2 2 3
you won't move upwards at the same steps. Keep comparing "the next leaves" until they are different, or you exhaust the leaves of only one of the trees (then again the trees have different fringes), or you exhaust the leaves of both trees at the same time, without having found leaves with different labels. In this last case, the trees have the same fringe.
@@ 27,7 +27,7 @@ Let's sketch the implementation of this. We won't provide all the details for an
In these exercises, we'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:
 # type blah = Blah of (int * int * (char > bool));;
+ # type blah = Blah of int * int * (char > bool);;
and then having to remember which element in the triple was which:
@@ 48,16 +48,16 @@ records let you attach descriptive labels to the components of the tuple:
These were the strategies to extract the components of an unlabeled tuple:
 let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
+ let h = fst some_pair (* accessor functions fst and snd are only predefined for pairs *)
 let (h, w, test) = b1;; (* works for arbitrary tuples *)
+ let (h, w, test) = b1 (* works for arbitrary tuples *)
match b1 with
  (h, w, test) > ...;; (* same as preceding *)
+  (h, w, test) > ... (* same as preceding *)
Here is how you can extract the components of a labeled record:
 let h = b2.height;; (* handy! *)
+ let h = b2.height (* handy! *)
let {height = h; weight = w; char_tester = test} = b2 in
(* go on to use h, w, and test ... *)
@@ 68,7 +68,7 @@ Here is how you can extract the components of a labeled record:
Anyway, using record types, we might define the tree zipper interface like so. First, we define a type for leaflabeled, binary trees:
 type 'a tree = Leaf of 'a  Node of ('a tree * 'a tree)
+ type 'a tree = Leaf of 'a  Node of 'a tree * 'a tree
Next, the types for our tree zippers:
@@ 111,20 +111,22 @@ Here are the beginnings of functions to move from one focused tree to another:
 Root > None
 Nonroot {up; left = Some left; right = None} >
move_right_or_up {in_focus = Node(left, z.in_focus); context = up}
+  _ > assert false
>
1. Your first assignment is to complete the definitions of `move_botleft` and `move_right_or_up`.
+1. Your first assignment is to complete the definitions of `move_botleft` and `move_right_or_up`. (Really it should be `move_right_or_up_..._and_right`.)
Having completed that, we can use define a function that enumerates a tree's fringe, step by step, until it's exhausted:
 let make_fringe_enumerator (t: 'a tree) : 'b * 'a zipper option =
+ let make_fringe_enumerator (t: 'a tree) =
(* create a zipper focusing the botleft of t *)
let zbotleft = move_botleft (new_zipper t) in
(* create initial state, pointing to zbotleft *)
 let initial_state = Some zbotleft in
+ let initial_state : 'a zipper option = Some zbotleft in
(* construct the next_leaf function *)
 let next_leaf : 'a zipper option > ('a * 'a zipper option) option = function
+ let next_leaf : 'a zipper option > ('a * 'a zipper option) option =
+ fun state > match state with
 Some z > (
(* extract label of currentlyfocused leaf *)
let Leaf current = z.in_focus in
@@ 143,27 +145,33 @@ Here are the beginnings of functions to move from one focused tree to another:
Here's an example of `make_fringe_enumerator` in action:
# let tree1 = Leaf 1;;
 val tree1 : int tree = Leaf 1
 # let next1, state1 = make_fringe_enumerator tree1;;
 val next1 : unit > int option =
 # let res1, state1' = next1 state1;;
  : int option = Some 1
 # next1 state1';;
  : int option = None
 # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
 val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
 # let next2, state2 = make_fringe_enumerator tree2;;
 val next2 : unit > int option =
 # let res2, state2' = next2 state2;;
  : int option = Some 1
 # let res2, state2'' = next2 state2';;
  : int option = Some 2
 # let res2, state2''' = next2 state2'';;
  : int option = Some 3
 # let res2, state2'''' = next2 state2''';;
  : int option = None

 You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will return `None`. The subprogram's memory of where it is and what steps to perform next are stored in the `state` variables that are part of its input and output.
+ val tree1 : int tree = Leaf 1
+ # let next1, state1 = make_fringe_enumerator tree1;;
+ val next1 : int zipper option > (int * int zipper option) option =
+ val state1 : int zipper option = Some ...
+ # let Some (res1, state1') = next1 state1;;
+ val res1 : int = 1
+ val state1' : int zipper option = None
+ # next1 state1';;
+  : (int * int zipper option) option = None
+ # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
+ val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
+ # let next2, state2 = make_fringe_enumerator tree2;;
+ val next2 : int zipper option > (int * int zipper option) option =
+ val state2 : int zipper option = Some ...
+ # let Some (res2, state2') = next2 state2;;
+ val res2 : int = 1
+ val state2' : int zipper option = Some ...
+ # let Some (res2, state2'') = next2 state2';;
+ val res2 : int = 2
+ val state2'' : int zipper option = Some ...
+ # let Some (res2, state2''') = next2 state2'';;
+ val res2 : int = 3
+ val state2''' : int zipper option = None
+ # next2 state2''';;
+  : (int * int zipper option) option = None
+
+ You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will return `None`. The subprogram's memory of where it is and what steps to perform next are stored in the `next_state` variables that are part of its input and output.
Using these fringe enumerators, we can write our `same_fringe` function like this:
@@ 177,21 +185,21 @@ Here are the beginnings of functions to move from one focused tree to another:
 _ > false in
loop initial_state1 initial_state2
 The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itselfeither because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.
+ The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itselfeither because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 state1, next2 state2` matches the pattern `None, None`) then we've established that the trees do have the same fringe.
2. Test your implementations of `move_botleft` and `move_right_or_up` against some example trees to see if the resulting `make_fringe_enumerator` and `same_fringe` functions work as expected. Show us some of your tests.
3. Now we'll talk about another way to implement the `make_fringe_enumerator` function above (and so too the `same_fringe` function which uses it). Notice that the pattern given above is that the `make_fringe_enumerator` creates a `next_leaf` function and an initial state, and each time you want to advance the `next_leaf` by one step, you do so by calling it with the current state. It will return a result plus a modified state, which you can use when you want to call it again and take another step. All of the `next_leaf` function's memory about where it is in the enumeration is contained in the state. If you saved an old state, took three steps, and then called the `next_leaf` function again with the saved old state, it would be back where it was three steps ago. But in fact, the way we use the process and state above, there is no backtracking. Neither do we "fork" any of the states and pursue different forward paths. Their progress is deterministic, and fixed independently of anything that `same_fringe` might do. All that's up to `same_fringe` is to take the decision of when (and whether) to take another step forward.
+3. Now we'll talk about another way to implement the `make_fringe_enumerator` function above (and so too the `same_fringe` function which uses it). Notice that the pattern given above is that the `make_fringe_enumerator` creates a `next_leaf` function and an initial state, and each time you want to advance the `next_leaf` by one step, you do so by calling it with the current state. It will return a leaf label plus a modified state, which you can use when you want to call it again and take another step. All of the `next_leaf` function's memory about where it is in the enumeration is contained in the state. If you saved an old state, took three steps, and then called the `next_leaf` function again with the saved old state, it would be back where it was three steps ago. But in fact, the way we use the `next_leaf` function and state above, there is no backtracking. Neither do we "fork" any of the states and pursue different forward paths. Their progress is deterministic, and fixed independently of anything that `same_fringe` might do. All that's up to `same_fringe` is to take the decision of when (and whether) to take another step forward.
Given that usage pattern, it would be appropriate and convenient to make the `next_leaf` function remember its state itself, in a mutable variable. The client function `same_fringe` doesn't need to do anything with, or even be given access to, this variable. Here's how we might write `make_fringe_enumerator` according to this plan:
let make_fringe_enumerator (t: 'a tree) =
(* create a zipper focusing the botleft of t *)
let zbotleft = move_botleft (new_zipper t) in
 (* create refcell, initially pointing to zbotleft *)
+ (* create a refcell, initially pointing to zbotleft *)
let zcell = ref (Some zbotleft) in
(* construct the next_leaf function *)
 let next_leaf () : 'a option =
+ let next_leaf : unit > 'a option = fun () >
match !zcell with
 Some z > (
(* extract label of currentlyfocused leaf *)
@@ 227,7 +235,7 @@ Here are the beginnings of functions to move from one focused tree to another:
(* create refcell, initially pointing to zbotleft *)
let zcell = ref (Some zbotleft) in
(* construct the next_leaf function *)
 let next_leaf () : 'a option =
+ let next_leaf : unit > 'a option = fun () >
match !zcell with
 Some z > (
(* extract label of currentlyfocused leaf *)
@@ 255,99 +263,108 @@ Here are the beginnings of functions to move from one focused tree to another:
loop ()
>
+ Here's an example of our new `make_fringe_enumerator` in action:
+
+ # let tree1 = Leaf 1;;
+ val tree1 : int tree = Leaf 1
+ # let next1 = make_fringe_enumerator tree1;;
+ val next1 : unit > int option =
+ # next1 ();;
+  : int option = Some 1
+ # next1 ();;
+  : int option = None
+ # next1 ();;
+  : int option = None
+ # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
+ val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
+ # let next2 = make_fringe_enumerator tree2;;
+ val next2 : unit > int option =
+ # next2 ();;
+  : int option = Some 1
+ # next2 ();;
+  : int option = Some 2
+ # next2 ();;
+  : int option = Some 3
+ # next2 ();;
+  : int option = None
+ # next2 ();;
+  : int option = None
+
+## Samefringe using streams ##
+
+Now we'll describe a different way to create "the little subprograms" that we built above with `make_fringe_enumerator`. This code will make use of a data structure called a "stream". A stream is like a list in that it wraps a series of elements of a single type. It differs from a list in that the tail of the series is left uncomputed until needed. We turn the stream off and on by thunking it, nad by forcing the thunk.
+
+We'll first show how to implement streams in OCaml, so that the types are manifest. But then we'll switch to a Scheme version to do the samefringe problem. In part because that's how we have the code already written; but also because some later discussion will use the Scheme code as a starting point. In principle, though, we could have used OCaml throughout.
+
+So here is a natural OCaml type for a stream. (You could also do things differently.)
+
+ type 'a stream = End  Next of 'a * (unit > 'a stream);;
+
+We have a special variant called `End` that encodes a stream that contains no (more) elements, analogous to the empty list `[]`. Streams that are not empty contain a first element, paired with a thunked stream representing the rest of the stream. In order to get access to the next element in the stream, we must force the thunk by applying it to `()`. Watch the behavior of this stream in detail. This stream delivers the natural numbers, in order: `1, 2, 3, ...`
+
+ # let rec make_int_stream i = Next (i, fun () > make_int_stream (i + 1));;
+ val make_int_stream : int > int stream =
+
+ # let int_stream = make_int_stream 1;;
+ val int_stream : int stream = Next (1, ) (* First element: 1 *)
+
+ # let tail = match int_stream with Next (i, rest) > rest;;
+ val tail : unit > int stream = (* Tail: a thunk *)
+
+ (* Force the thunk to compute the second element *)
+ # tail ();;
+  : int stream = Next (2, ) (* Second element: 2 *)
+
+ # match tail () with Next (_, rest) > rest ();;
+  : int stream = Next (3, ) (* Third element: 3 *)
+
+You can think of `int_stream` as a functional object that provides access to an infinite sequence of integers, one at a time. It's as if we had written `[1;2;...]` where `...` meant "continue for as long as some other process needs new integers".



Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:

 let make_fringe_enumerator (t: 'a tree) =
 (* create a zipper focusing the botleft of t *)
 let zbotleft = move_botleft (new_zipper t)
 (* create a refcell initially pointing to zbotleft *)
 in let zcell = ref (Some zbotleft)
 (* construct the next_leaf function *)
 in let next_leaf () : 'a option =
 match !zcell with
  Some z > (
 (* extract label of currentlyfocused leaf *)
 let Leaf current = z.in_focus
 (* update zcell to point to next leaf, if there is one *)
 in let () = zcell := match move_right_or_up z with
  None > None
  Some z' > Some (move_botleft z')
 (* return saved label *)
 in Some current
  None > (* we've finished enumerating the fringe *)
 None
 )
 (* return the next_leaf function *)
 in next_leaf
 ;;


Using these fringe enumerators, we can write our `same_fringe` function like this:

 let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
 let next1 = make_fringe_enumerator t1
 in let next2 = make_fringe_enumerator t2
 in let rec loop () : bool =
 match next1 (), next2 () with
  Some a, Some b when a = b > loop ()
  None, None > true
  _ > false
 in loop ()
 ;;

The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itselfeither because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.





4. Here's another implementation of the samefringe function, in Scheme. It's taken from . It uses thunks to delay the evaluation of code that computes the tail of a list of a tree's fringe. It also involves passing "the rest of the enumeration of the fringe" as a thunk argument (`tailthunk` below). Your assignment is to fill in the blanks in the code, **and also to supply comments to the code,** to explain what every significant piece is doing. Don't forget to supply the comments, this is an important part of the assignment.

 This code uses Scheme's `cond` construct. That works like this;

 (cond
 ((test1 argument argument) result1)
 ((test2 argument argument) result2)
 ((test3 argument argument) result3)
 (else result4))

 is equivalent to:

 (if (test1 argument argument)
 ; then
 result1
 ; else
 (if (test2 argument argument)
 ; then
 result2
 ; else
 (if (test3 argument argument)
 ; then
 result3
 ; else
 result4)))

 Some other Scheme details or reminders:

 * `#t` is true and `#f` is false
 * `(lambda () ...)` constructs a thunk
 * there is no difference in meaning between `[...]` and `(...)`; we just sometimes use the square brackets for clarity
 * `'(1 . 2)` and `(cons 1 2)` are pairs (the same pair)
 * `(list)` and `'()` both evaluate to the empty list
 * `(null? lst)` tests whether `lst` is the empty list
 * nonempty lists are implemented as pairs whose second member is a list
 * `'()` `'(1)` `'(1 2)` `'(1 2 3)` are all lists
 * `(list)` `(list 1)` `(list 1 2)` `(list 1 2 3)` are the same lists as the preceding
 * `'(1 2 3)` and `(cons 1 '(2 3))` are both pairs and lists (the same list)
 * `(pair? lst)` tests whether `lst` is a pair; if `lst` is a nonempty list, it will also pass this test; if `lst` fails this test, it may be because `lst` is the empty list, or because it's not a list or pair at all
 * `(car lst)` extracts the first member of a pair / head of a list
 * `(cdr lst)` extracts the second member of a pair / tail of a list

 Here is the implementation:
+Okay, now armed with the idea of a stream, let's use a Scheme version of them to handle the samefringe problem. This code is taken from . It uses thunks to delay the evaluation of code that computes the tail of a list of a tree's fringe. It also involves passing "the rest of the enumeration of the fringe" as a thunk argument (`tailthunk` below). Your assignment is to fill in the blanks in the code, **and also to supply comments to the code,** to explain what every significant piece is doing. Don't forget to supply the comments, this is an important part of the assignment.
+
+This code uses Scheme's `cond` construct. That works like this;
+
+ (cond
+ ((test1 argument argument) result1)
+ ((test2 argument argument) result2)
+ ((test3 argument argument) result3)
+ (else result4))
+
+is equivalent to:
+
+ (if (test1 argument argument)
+ ; then
+ result1
+ ; else
+ (if (test2 argument argument)
+ ; then
+ result2
+ ; else
+ (if (test3 argument argument)
+ ; then
+ result3
+ ; else
+ result4)))
+
+Some other Scheme details or reminders:
+
+* `#t` is true and `#f` is false
+* `(lambda () ...)` constructs a thunk
+* there is no difference in meaning between `[...]` and `(...)`; we just sometimes use the square brackets for clarity
+* `'(1 . 2)` and `(cons 1 2)` are pairs (the same pair)
+* `(list)` and `'()` both evaluate to the empty list
+* `(null? lst)` tests whether `lst` is the empty list
+* nonempty lists are implemented as pairs whose second member is a list
+* `'()` `'(1)` `'(1 2)` `'(1 2 3)` are all lists
+* `(list)` `(list 1)` `(list 1 2)` `(list 1 2 3)` are the same lists as the preceding
+* `'(1 2 3)` and `(cons 1 '(2 3))` are both pairs and lists (the same list)
+* `(pair? lst)` tests whether `lst` is a pair; if `lst` is a nonempty list, it will also pass this test; if `lst` fails this test, it may be because `lst` is the empty list, or because it's not a list or pair at all
+* `(car lst)` extracts the first member of a pair / head of a list
+* `(cdr lst)` extracts the second member of a pair / tail of a list
+
+
+
+4. Here is the Scheme code handling the samefringe problem. You should fill in the blanks:
(define (lazyflatten tree)
(letrec ([helper (lambda (tree tailthunk)
@@ 373,26 +390,3 @@ The auxiliary `loop` function will keep calling itself recursively until a diffe
(samefringe? tree1 tree2)
The previous problem implemented a "stream" in Scheme. A stream is like a list in that it wraps a series of elements of a single type. It differs from a list in that the tail of the series is left uncomputed until needed. We will turn the stream on and off by thunking it. Here is one way to implement streams in OCaml:

 type 'a stream = End  Next of 'a * (unit > 'a stream);;

There is a special stream called End that represents a stream that contains no (more) elements, analogous to the empty list []. Streams that are not empty contain a first object, paired with a thunked stream representing the rest of the series. In order to get access to the next element in the stream, we must force the thunk by applying it to the unit. Watch the behavior of this stream in detail. This stream delivers the natural numbers, in order: 1, 2, 3, ...

 # let rec make_int_stream i = Next (i, fun () > make_int_stream (i + 1));;
 val make_int_stream : int > int stream = [fun]

 # let int_stream = make_int_stream 1;;
 val int_stream : int stream = Next (1, [fun]) (* First element: 1 *)

 # let tail = match int_stream with Next (i, rest) > rest;;
 val tail : unit > int stream = [fun] (* Tail: a thunk *)

 (* Force the thunk to compute the second element *)
 # tail ();;
  : int stream = Next (2, [fun]) (* Second element: 2 *)

 # match tail () with Next (_, rest) > rest ();;
  : int stream = Next (3, [fun]) (* Third element: 3 *)

You can think of `int_stream` as a functional object that provides access to an infinite sequence of integers, one at a time. It's as if we had written `[1;2;...]` where `...` meant "continue for as long as some other process needs new integers".

2.11.0