Leaf_monad -> Tree_monad

[lambda.git] / assignment8.mdwn

1.      Complete the definitions of `move_botleft` and `move_right_or_up` from the same-fringe solution in the [[week11]] notes. **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 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 *)


                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 *)


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

         

                let make_fringe_enumerator (t: 'a tree) =
                        (* create a zipper targetting 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 currently-targetted leaf *)
                                        let Leaf current = z.filler
                                        (* 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
                        ;;

                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 ()
                        ;;


2.      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:

        *       `#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)