[lambda.git] / exercises / _assignment12.mdwn

1.  Complete the definitions of `move_botleft` and `move_right_or_up` from the same-fringe solution in [[this week's notes|/topics/week12_list_and_tree_zippers]]. **Test your attempts** 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.

        type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)

        type 'a starred_level = Root | Starring_left of 'a starred_nonroot | Starring_right of 'a starred_nonroot
        and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;

        type 'a zipper = { level : 'a starred_level; filler: 'a tree };;

        let new_zipper (t : 'a tree) : 'a zipper =
          {level = Root; filler = t}

        let rec move_botleft (z : 'a zipper) : 'a zipper =
          (* returns z if the targetted node in z has no children *)
          (* else returns move_botleft (zipper which results from moving down from z to the leftmost child) *)
          _____ (* YOU SUPPLY THE DEFINITION *)

    <!--
    match z.filler with Leaf _ -> z | Node (l, r) -> move_botleft { level = Starring_left { parent = z.level; sibling = r }; filler = l }
    -->

        let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
          (* if it's possible to move right in z, returns Some (the result of doing so) *)
          (* else if it's not possible to move any further up in z, returns None *)
          (* else returns move_right_or_up (result of moving up in z) *)
          _____ (* YOU SUPPLY THE DEFINITION *)

    <!--
    match z.level with
    | Starring_left { parent = p; sibling = right } -> Some { level = Starring_right { parent = p; sibling = z.filler }; filler = right }
    | Starring_right { parent = p; sibling = left } -> let new_tree = Node (left, z.filler) in move_right_or_up { level = p; filler = new_tree}
    | Root -> None
    -->

    &nbsp;

        let make_fringe_enumerator (t: 'a tree) : 'b * 'a zipper option =
          (* create a zipper targetting the botleft of t *)
          let zbotleft = move_botleft (new_zipper t) in
          (* create initial state, pointing to zbotleft *)
          let initial_state = Some zbotleft in
          (* construct the next_leaf function *)
          let next_leaf : 'a zipper option -> ('a * 'a zipper option) option = function
            | Some z -> (
              (* extract label of currently-targetted leaf *)
              let Leaf current = z.filler in
              (* create next_state pointing to next leaf, if there is one *)
              let next_state : 'a zipper option = match move_right_or_up z with
                | None -> None
                | Some z' -> Some (move_botleft z') in
              (* return saved label and next_state *)
              Some (current, next_state)
              )
            | None -> (* we've finished enumerating the fringe *)
              None in
          (* return the next_leaf function and initial state *)
          next_leaf, initial_state

        let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
          let next1, initial_state1 = make_fringe_enumerator t1 in
          let next2, initial_state2 = make_fringe_enumerator t2 in
          let rec loop state1 state2 : bool =
            match next1 state1, next2 state2 with
            | Some (a, state1'), Some (b, state2') when a = b -> loop state1' state2'
            | None, None -> true
            | _ -> false in
          loop initial_state1 initial_state2


2.  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 `process` function and an initial state, and each time you want to advance the `process` 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 `process` 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 `process` 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 back-tracking. 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 `process` 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 targetting the botleft of t *)
          let zbotleft = move_botleft (new_zipper t) in
          (* create refcell, initially pointing to zbotleft *)
          let zcell = ref (Some zbotleft) in
          (* construct the next_leaf function *)
          let next_leaf () : 'a option =
            match !zcell with
            | Some z -> (
              (* extract label of currently-targetted leaf *)
              let Leaf current = z.filler in
              (* update zcell to point to next leaf, if there is one *)
              let () = zcell := match move_right_or_up z with
                | None -> None
                | Some z' -> _____ in
              (* return saved label *)
              _____
              )
            | None -> (* we've finished enumerating the fringe *)
              None in
          (* return the next_leaf function *)
          next_leaf

        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 _____, _____ with
            | Some a, Some b when a = b -> loop ()
            | None, None -> true
            | _ -> false in
          loop ()

    You should fill in the blanks.

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

        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 ()
    -->


3.  Here's another implementation of the same-fringe function, in Scheme. It's taken from <http://c2.com/cgi/wiki?SameFringeProblem>. 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 (`tail-thunk` 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
    *   non-empty 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 non-empty 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:

        (define (lazy-flatten tree)
          (letrec ([helper (lambda (tree tail-thunk)
                             (cond
                               [(pair? tree)
                                 (helper (car tree) (lambda () (helper _____ tail-thunk)))]
                               [else (cons tree tail-thunk)]))])
                  (helper tree (lambda () _____))))

        (define (stream-equal? stream1 stream2)
          (cond
            [(and (null? stream1) (null? stream2)) _____]
            [(and (pair? stream1) (pair? stream2))
              (and (equal? (car stream1) (car stream2))
                _____)]
            [else #f]))

        (define (same-fringe? tree1 tree2)
          (stream-equal? (lazy-flatten tree1) (lazy-flatten tree2)))

        (define tree1 '(((1 . 2) . (3 . 4)) . (5 . 6)))
        (define tree2 '(1 . (((2 . 3) . (4 . 5)) . 6)))

        (same-fringe? 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".