13 ## Option / Maybe Types ##
15 You've already defined and worked with `map` as a function on lists. Now we're going to work instead with the type OCaml defines like this:
17 type ('a) option = None | Some of 'a
19 and Haskell defines like this:
21 data Maybe a = Nothing | Just a
23 That is, instances of this type are either some `'a` (this can be any type), wrapped up in a `Some` or `Just` box, or they are a separate value representing a failure. This is sort of like working with a list guaranteed to have a length ≤ 1.
25 In one of the homework sessions, Chris posed the challenge: you know those dividers they use in checkout lines to separate your purchases from the next person's? What if you wanted to buy one of those dividers? How could they tell whether it belonged to your purchases or was separating them from others?
27 The OCaml and Haskell solution is to use not supermarket dividers but instead those gray bins from airport security. If you want to buy something, it goes into a bin. (OCaml's `Some`, Haskell's `Just`). If you want to separate your stuff from the next person, you send an empty bin (OCaml's `None`, Haskell's `Nothing`). If you happen to be buying a bin, OK, you put that into a bin. In OCaml it'd be `Some None` (or `Some (Some stuff)` if the bin you're buying itself contains some stuff); in Haskell `Just Nothing`. We won't confuse a bin that contains a bin with an empty bin. (Not even if the contained bin is itself empty.)
29 1. Your first problem is to write a `maybe_map` function for these types. Here is the type of the function you should write:
32 maybe_map : ('a -> 'b) -> ('a) option -> ('b) option
35 maybe_map :: (a -> b) -> Maybe a -> Maybe b
37 If your `maybe_map` function is given a `None` or `Nothing` as its second argument, that should be what it returns. Otherwise, it should apply the function it got as its first argument to the contents of the `Some` or `Just` bin that it got as its second, and return the result, wrapped back up in a `Some` or `Just`.
39 One way to extract the contents of a option or Maybe value is to pattern match on that value, as you did with lists. In OCaml:
52 Some other tips: In OCaml you write recursive functions using `let rec`, in Haskell you just use `let` (it's already assumed to be recursive). In OCaml when you finish typing something and want the interpreter to parse it, check and display its type, and evaluate it, type `;;` and then return. You may want to review the [[Rosetta pages here]] and also read some of the tutorials we linked to [[for OCaml]] or [[for Haskell]]. [WHERE]
55 2. Next write a `maybe_map2` function. Its type should be:
58 maybe_map2 ('a -> 'b -> 'c) -> ('a) option -> ('b) option -> ('c) option
61 maybe_map2 :: (a -> b -> c) -> Maybe a -> Maybe b -> Maybe c
67 (The questions on Color and Search Trees are adapted from homeworks in Chapters 1 and 2 of Friedman and Wand, *Essentials of Programming Languages*.)
69 Here are type definitions for one kind of binary tree:
72 type color = Red | Green | Blue | ... (* you can add as many as you like *)
73 type ('a) color_tree = Leaf of 'a | Branch of 'a color_tree * color * 'a color_tree
76 data Color = Red | Green | Blue | ... deriving (Eq, Show)
77 data Color_tree a = Leaf a | Branch (Color_tree a) Color (Color_tree a) deriving (Show)
79 These trees always have colors labeling their inner branching nodes, and will have elements of some type 'a labeling their leaves. `(int) color_tree`s will have `int`s there, `(bool) color_tree`s will have `bool`s there, and so on. The `deriving (Eq, Show)` part at the end of the Haskell declarations is boilerplate to tell Haskell you want to be able to compare the colors for equality, and also that you want the Haskell interpreter to display colors and lists to you when they are the result of evaluating an expression.
81 Here's how you create an instance of such a tree:
84 let t1 = Branch (Leaf 1, Red, Branch (Leaf 2, Green, Leaf 0))
87 let t1 = Branch (Leaf 1) Red (Branch (Leaf 2) Green (Leaf 0))
89 Here's how you pattern match such a tree, binding variables to its components:
94 | Branch (_, c, _) -> c = Red
99 Branch _ c _ -> c == Red
102 These expressions query whether `t` is a branching `color_tree` (not a leaf) whose root is labeled `Red`.
104 Choose one of these languages and write the following functions.
107 3. Define a function `tree_map` whose type is (as shown by OCaml): `('a -> 'b) -> ('a) color_tree -> ('b) color_tree`. It expects a function `f` and an `('a) color_tree`, and returns a new tree with the same structure and inner branch labels as the original, but with all of its leaves now having had `f` applied to their original value. So for example, `map (2*) t1` would return `t1` with all of its leaf values doubled.
109 4. Define a function `tree_foldleft` that accepts an argument `g : 'z -> 'a -> 'z` and a seed value `z : 'z` and a tree `t : ('a) color_tree`, and returns the result of applying `g` first to `z` and `t`'s leftmost leaf, and then applying `g` to *that result* and `t`'s second-leftmost leaf, and so on, all the way across `t`'s fringe.
111 5. How would you use the function defined in problem 4 (the previous problem) to sum up the values labeling the leaves of an `(int) color_tree`?
113 6. How would you use the function defined in problem 4 to enumerate a tree's fringe? (Don't worry about whether it comes out left-to-right or right-to-left.)
115 7. How would you use the function defined in problem 4 to make a copy of a tree with the same structure and inner node labels, but where the leftmost leaf is now labeled `0`, the second-leftmost leaf is now labeled `1`, and so on.
117 8. (More challenging.) Write a recursive function that makes a copy of a tree with the same structure and inner node labels, but replaces each leaf label with the int that reports how many of that leaf's ancestors are labeled `Red`. For example, if we give your function a tree:
130 (for any leaf values `a` through `e`), it should return:
143 9. (More challenging.) Assume you have a `color_tree` whose leaves are labeled with `int`s (which might be negative). For this problem, assume also that the the same color never labels multiple inner nodes. Write a recursive function that reports which color has the greatest score when you sum up all the values of its descendent leaves. Since some leaves may have negative values, the answer won't always be the color at the tree root. In the case of ties, you can return whichever of the highest scoring colors you like.
148 (More challenging.) For the next problem, assume the following type definition:
151 type search_tree = Nil | Inner of search_tree * int * search_tree
154 data Search_tree = Nil | Inner Search_tree Int Search_tree deriving (Show)
156 That is, its leaves have no labels and its inner nodes are labeled with `int`s. Additionally, assume that all the `int`s in nodes descending to the left from a given node will be less than the `int` of that parent node, and all the `int`s in nodes descending to the right will be greater. We can't straightforwardly specify this constraint in OCaml's or Haskell's type definitions. We just have to be sure to maintain it by hand.
158 10. Write a function `search_for` with the following type, as displayed by OCaml:
160 type direction = Left | Right
161 search_for : int -> search_tree -> direction list option
163 Haskell would say instead:
165 data Direction = Left | Right deriving (Eq, Show)
166 search_for :: Int -> Search_tree -> Maybe [Direction]
168 Your function should search through the tree for the specified `int`. If it's never found, it should return the value OCaml calls `None` and Haskell calls `Nothing`. If it finds the `int` right at the root of the search_tree, it should return the value OCaml calls `Some []` and Haskell calls `Just []`. If it finds the `int` by first going down the left branch from the tree root, and then going right twice, it should return `Some [Left; Right; Right]` or `Just [Left, Right, Right]`.
173 Above, you defined `maybe_map2` [WHERE]. Before we encountered `map2` for lists. There are in fact several different approaches to mapping two lists together.
175 11. One approach is to apply the supplied function to the first element of each list, and then to the second element of each list, and so on, until the lists are exhausted. If the lists are of different lengths, you might stop with the shortest, or you might raise an error. Different implementations make different choices about that. Let's call this function:
178 map2_zip : ('a -> 'b -> 'c) -> ('a) list -> ('b) list -> ('c) list
180 Write a recursive function that implements this, in Haskell or OCaml. Let's say you can stop when the shorter list runs out, if they're of different lengths. (OCaml and Haskell each already have functions in their standard libraries that do this. This also corresponds to what you can write as a list comprehension in Haskell like this:
182 :set -XParallelListComp
183 [ f x y | x <- xs | y <- ys ]
185 But we want you to write this function from scratch.)
187 12. What is the relation between the function you just wrote, and the `maybe_map2` function you wrote for an earlier problem?
189 13. Another strategy is to take the *cross product* of the two lists. If the function:
192 map2_cross : ('a -> 'b -> 'c) -> ('a) list -> ('b) list -> ('c) list list
194 is applied to the arguments `f`, `[x0, x1, x2]`, and `[y0, y1]`, then the result should be: `[[f x0 y0, f x0 y1], [f x1 y0, f x1 y1], [f x2 y0, f x2 y1]]`. Write this function.
196 A similar choice between "zipping" and "crossing" could be made when `map2`-ing two trees. For example, the trees:
206 could be "zipped" like this (ignoring any parts of branches on the one tree that extend farther than the corresponding branch on the other):
214 14. You can try defining that if you like, for extra credit.
216 "Crossing" the trees would instead add copies of the second tree as subtrees replacing each leaf of the original tree, with the leaves of that larger tree labeled with `f` applied to `3` and `6`, then `f` applied to `3` and `8`, and so on across the fringe of the second tree; then beginning again (in the subtree that replaces the `4` leaf) with `f` applied to `4` and `6`, and so on.
218 * In all the plain `map` functions, whether for lists, or for option/Maybes, or for trees, the structure of the result exactly matched the structure of the argument.
220 [LOOKUP in APPLICATIVE]
222 * In the `map2` functions, whether for lists or for option/Maybes or for trees, and whether done in the "zipping" style or in the "crossing" style, the structure of the result may be a bit different from the structure of the arguments. But the *structure* of the arguments is enough to determine the structure of the result; you don't have to look at the specific list elements or labels on a tree's leaves or nodes to know what the *structure* of the result will be.
224 * We can imagine more radical transformations, where the structure of the result *does* depend on what specific elements the original structure(s) had. For example, what if we had to transform a tree by turning every leaf into a subtree that contained all of those leaf's prime factors. Or consider our problem from last week [WHERE] where you converted `[3, 2, 0, 1]` not into `[[3,3,3], [2,2], [], [1]]` --- which still has the same structure, that is length, as the original --- but rather into `[3, 3, 3, 2, 2, 1]` --- which doesn't.
226 These three levels of how radical a transformation you are making to a structure, and the parallels between the transformations to lists, to option/Maybes, and to trees, will be ideas we build on in coming weeks.
232 ## Untyped Lambda Terms ##
234 In OCaml, you can define some datatypes that represent terms in the untyped Lambda Calculus like this:
236 type identifier = string
237 type lambda_term = Var of identifier | Abstract of identifier * _____ | App of _____ ;;
239 We've left some gaps.
241 In Haskell, you'd define it instead like this:
243 type Identifier = String
244 data Lambda_term = Var Identifier | Abstract Identifier _____ | App ________
246 15. Again, we've left some gaps. Choose one of these languages and fill in the gaps to complete the definition.
248 16. Write a function `occurs_free` that has the following type:
250 occurs_free : identifier -> lambda_term -> bool
252 That's how OCaml would show it. Haskell would use double colons `::` instead, and would also capitalize all the type names. Your function should tell us whether the supplied identifier ever occurs free in the supplied `lambda_term`.
257 ## Encoding Booleans, Church numerals, and Right-Fold Lists in System F ##
259 (These questions are adapted from web materials by Umut Acar. See
260 <http://www.mpi-sws.org/~umut/>.)
263 (For the System F questions, you can either download and compile Pierce's evaluator for system F to test your work [WHERE], or work on paper.)
268 Let's think about the encodings of booleans, numerals and lists in System F,
269 and get data-structures with the same form working in OCaml or Haskell. (Of course, OCaml and Haskell
270 have *native* versions of these datas-structures: OCaml's `true`, `1`, and `[1;2;3]`.
271 But the point of our exercise requires that we ignore those.)
273 Recall from class System F, or the polymorphic λ-calculus, with this grammar:
275 types ::= type_constants | α ... | type1 -> type2 | ∀α. type
276 expressions ::= x ... | λx:type. expr | expr1 expr2 | Λα. expr | expr [type]
278 The boolean type, and its two values, may be encoded as follows:
280 bool ≡ ∀α. α -> α -> α
281 true ≡ Λα. λy:α. λn:α. y
282 false ≡ Λα. λy:α. λn:α. n
288 where `b` is a boolean value, and `T` is the shared type of `res1` and `res2`.
291 17. How should we implement the following terms? Note that the result
292 of applying them to the appropriate arguments should also give us a term of
295 (a) the term `not` that takes an argument of type `bool` and computes its negation
296 (b) the term `and` that takes two arguments of type `bool` and computes their conjunction
297 (c) the term `or` that takes two arguments of type `bool` and computes their disjunction
299 The type `nat` (for "natural number") may be encoded as follows:
301 nat ≡ ∀α. (α -> α) -> α -> α
302 zero ≡ Λα. λs:α -> α. λz:α. z
303 succ ≡ λn:nat. Λα. λs:α -> α. λz:α. s (n [α] s z)
305 A number `n` is defined by what it can do, which is to compute a function iterated n
306 times. In the polymorphic encoding above, the result of that iteration can be
307 any type `α`, as long as you have a function `s : α -> α` and a base element `z : α`.
309 18. Get booleans and Church numbers working in OCaml or Haskell,
310 including versions of `bool`, `true`, `false`, `zero`, `zero?` (though this is not a legal function name in either of those languages, use something else), `succ`, and `pred`.
311 It's especially useful to do a version of `pred`, starting with one of the (untyped) versions available in the lambda library accessible from the main wiki page. [WHERE] The point of the exercise is to do these things on your own, so avoid using the built-in OCaml or Haskell booleans and integers.
313 Consider the following list type, specified using OCaml or Haskell datatypes:
316 type ('a) my_list = Nil | Cons of 'a * 'a my_list
319 data My_list a = Nil | Cons a (My_list a)
321 We can encode that type into System F as a right-fold, just as we did in the untyped Lambda Calculus, like this:
323 list_T ≡ ∀α. (T -> α -> α) -> α -> α
324 nil_T ≡ Λα. λc:T -> α -> α. λn:α. n
325 cons_T ≡ λx:T. λxs:list_T. Λα. λc:T -> α -> α. λn:α. c x (xs [α] c n)
327 A more general polymorphic list type is:
329 list ≡ ∀β. ∀α. (β -> α -> α) -> α -> α
331 As with nats, the natural recursion is built into our encoding of this datatype. So we can write functions like `map`:
333 map : (S -> T) -> list_S -> list_T
336 = λf:S -> T. λxs:list. xs [S] [list [T]] (λx:S. λys:list [T]. cons [T] (f x) ys) (nil [T])
339 19. Convert this list encoding and the `map` function to OCaml or Haskell. Call it `sysf_list`, `sysf_nil` and so on, to avoid collision with the names for native lists in these languages.
341 20. Also give us the type and definition for a `sysf_head` function. Think about what value to give back if the argument is the empty list. Ultimately, we might want to make use of the option/Maybe technique explored in questions 1--2, but for this assignment, just pick a strategy, no matter how clunky.
343 Be sure to test your proposals with simple lists. (You'll have to `sysf_cons` up a few sample lists yourself; don't expect OCaml or Haskell to magically translate between their native lists and the ones you've just defined.)
346 21. Modify the implementation of the predecessor function [[given in the class notes|topics/week5_system_f]] [WHERE] to implement a `sysf_tail` function for your lists.
357 22. Recall that the S combinator is given by `\f g x. f x (g x)`. Give two different typings for this term in OCaml. To get you started, here's one typing for **K**:
359 # let k (y:'a) (n:'b) = y;;
360 val k : 'a -> 'b -> 'a = [fun]
364 If you can't understand how one term can have several types, recall our discussion in this week's notes [WHERE] of "principal types".
369 ## Evaluation Order ##
371 Do these last three problems specifically with OCaml in mind, not Haskell. Analogues of the questions exist in Haskell, but because the default evaluation rules for these languages are different, it's too complicated to look at how these questions should be translated into the Haskell setting.
374 23. Which of the following expressions is well-typed in OCaml? For those that are, give the type of the expression as a whole. For those that are not, why not?
380 let rec f x = f x in f f;;
382 let rec f x = f x in f ();;
386 let rec f () = f ();;
388 let rec f () = f () in f f;;
390 let rec f () = f () in f ();;
392 24. Throughout this problem, assume that we have:
394 let rec blackhole x = blackhole x;;
396 <!-- Haskell could say: `let blackhole = \x -> fix (\f -> f)` -->
398 All of the following are well-typed. Which ones terminate? What are the generalizations?
404 fun () -> blackhole ();;
406 (fun () -> blackhole ()) ();;
408 if true then blackhole else blackhole;;
410 if false then blackhole else blackhole;;
412 if true then blackhole else blackhole ();;
414 if false then blackhole else blackhole ();;
416 if true then blackhole () else blackhole;;
418 if false then blackhole () else blackhole;;
420 if true then blackhole () else blackhole ();;
422 if false then blackhole () else blackhole ();;
424 let _ = blackhole in 2;;
426 let _ = blackhole () in 2;;
428 25. This problem is to think about how to control order of evaluation.
429 The following expression is an attempt to make explicit the
430 behavior of `if ... then ... else ...` explored in the previous question.
431 The idea is to define an `if ... then ... else ...` expression using
432 other expression types. So assume that `yes` is any (possibly complex) OCaml expression,
433 and `no` is any other OCaml expression (of the same type as `yes`!),
434 and that `bool` is any boolean expression. Then we can try the following:
435 `if bool then yes else no` should be equivalent to
440 match b with true -> y | false -> n
442 This almost works. For instance,
444 if true then 1 else 2;;
448 let b = true in let y = 1 in let n = 2 in
449 match b with true -> y | false -> n;;
451 also evaluates to 1. Likewise,
453 if false then 1 else 2;;
457 let b = false in let y = 1 in let n = 2 in
458 match b with true -> y | false -> n;;
464 let rec blackhole x = blackhole x in
465 if true then blackhole else blackhole ();;
469 let rec blackhole x = blackhole x in
472 let n = blackhole () in
473 match b with true -> y | false -> n;;
475 does not terminate. Incidentally, using the shorter `match bool with true -> yes | false -> no;;` rather than the longer `let b = bool ... in match b with ...` *would* work as we desire. But your assignment is to control the evaluation order without using the special evaluation order properties of OCaml's native `if` or of its `match`. That is, you must keep the `let b = ... in match b with ...` structure in your answer, though you are allowed to adjust what `b`, `y`, and `n` get assigned to.
477 [[hints/assignment 5 hint 1]] WHERE