From: jim Date: Sat, 25 Apr 2015 03:21:08 +0000 (-0400) Subject: some refactoring, including exposition X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=6368722cc452c9a48d486a9a133d7fbb7b18bf08 some refactoring, including exposition --- diff --git a/exercises/_assignment12.mdwn b/exercises/_assignment12.mdwn index 2af65136..017a8d36 100644 --- a/exercises/_assignment12.mdwn +++ b/exercises/_assignment12.mdwn @@ -1,49 +1,133 @@ -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. +## Same-fringe using zippers ## - type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree) +Recall back in [[Assignment 4|assignment4#fringe]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees (here I *am* drawing the labels in the diagram): - 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 };; + . . + / \ / \ + . 3 1 . + / \ / \ + 1 2 2 3 - type 'a zipper = { level : 'a starred_level; filler: 'a tree };; +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. - 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 *) +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 - 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 *) +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. - + + + 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 *) + + + | Nonroot {up; left = Some left; right = None} -> + move_right_or_up {in_focus = Node(left, z.in_focus); context = up} +--> + -   +1. Your first assignment is to complete the definitions of `move_botleft` and `move_right_or_up`. + + 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 = - (* create a zipper targetting the botleft of t *) + (* 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 (* 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 + (* extract label of currently-focused leaf *) + let Leaf current = z.in_focus 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 @@ -56,6 +140,33 @@ (* return the next_leaf function and initial state *) next_leaf, initial_state + 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. + + 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, initial_state1 = make_fringe_enumerator t1 in let next2, initial_state2 = make_fringe_enumerator t2 in @@ -66,13 +177,16 @@ | _ -> false in loop initial_state1 initial_state2 + The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either 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. + +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. -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. +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 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: + 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 targetting the botleft of t *) + (* create a zipper focusing the botleft of t *) let zbotleft = move_botleft (new_zipper t) in (* create refcell, initially pointing to zbotleft *) let zcell = ref (Some zbotleft) in @@ -80,8 +194,8 @@ let next_leaf () : 'a option = match !zcell with | Some z -> ( - (* extract label of currently-targetted leaf *) - let Leaf current = z.filler in + (* extract label of currently-focused leaf *) + let Leaf current = z.in_focus in (* update zcell to point to next leaf, if there is one *) let () = zcell := match move_right_or_up z with | None -> None @@ -108,7 +222,7 @@ -3. Here's another implementation of the same-fringe 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 (`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. +--- + + +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 currently-focused 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 itself---either 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 same-fringe 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 (`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;