X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment5.mdwn;h=f402ec61a70bbe9ebe4e5f7c2f4a2f1ecc853ede;hp=61096c4e7a2b34a344e02d6755fe1e53f8bd30b1;hb=a9fc616a72a86be53a9ce7289fa3608799b44956;hpb=40b7072437e3fe1c9fecdfec65063aecfbf85cd4 diff --git a/assignment5.mdwn b/assignment5.mdwn index 61096c4e..f402ec61 100644 --- a/assignment5.mdwn +++ b/assignment5.mdwn @@ -13,9 +13,9 @@ Types and OCaml - : int = 1 -1. 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? +1. 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? let rec f x = f x;; @@ -35,38 +35,38 @@ Types and OCaml 2. Throughout this problem, assume that we have - let rec omega x = omega x;; + let rec blackhole x = blackhole x;; All of the following are well-typed. Which ones terminate? What are the generalizations? - omega;; + blackhole;; - omega ();; + blackhole ();; - fun () -> omega ();; + fun () -> blackhole ();; - (fun () -> omega ()) ();; + (fun () -> blackhole ()) ();; - if true then omega else omega;; + if true then blackhole else blackhole;; - if false then omega else omega;; + if false then blackhole else blackhole;; - if true then omega else omega ();; + if true then blackhole else blackhole ();; - if false then omega else omega ();; + if false then blackhole else blackhole ();; - if true then omega () else omega;; + if true then blackhole () else blackhole;; - if false then omega () else omega;; + if false then blackhole () else blackhole;; - if true then omega () else omega ();; + if true then blackhole () else blackhole ();; - if false then omega () else omega ();; + if false then blackhole () else blackhole ();; - let _ = omega in 2;; + let _ = blackhole in 2;; - let _ = omega () in 2;; + let _ = blackhole () in 2;; 3. This problem is to begin thinking about controlling order of evaluation. The following expression is an attempt to make explicit the @@ -104,15 +104,15 @@ and that "bool" is any boolean. Then we can try the following: However, - let rec omega x = omega x in - if true then omega else omega ();; + let rec blackhole x = blackhole x in + if true then blackhole else blackhole ();; terminates, but - let rec omega x = omega x in + let rec blackhole x = blackhole x in let b = true in - let y = omega in - let n = omega () in + let y = blackhole in + let n = blackhole () in match b with true -> y | false -> n;; does not terminate. Incidentally, `match bool with true -> yes | @@ -121,97 +121,132 @@ and that "bool" is any boolean. Then we can try the following: or of `match`. That is, you must keep the `let` statements, though you're allowed to adjust what `b`, `y`, and `n` get assigned to. - [[Hint assignment 5 problem 3]] + [[hints/assignment 5 hint 1]] -Baby monads ------------ - -Read the lecture notes for week 6, then write a -function `lift` that generalized the correspondence between + and -`add`: that is, `lift` takes any two-place operation on integers -and returns a version that takes arguments of type `int option` -instead, returning a result of `int option`. In other words, -`lift` will have type - - (int -> int -> int) -> (int option) -> (int option) -> (int option) - -so that `lift (+) (Some 3) (Some 4)` will evalute to `Some 7`. -Don't worry about why you need to put `+` inside of parentheses. -You should make use of `bind` in your definition of `lift`: +Booleans, Church numerals, and v3 lists in OCaml +------------------------------------------------ - let bind (x: int option) (f: int -> (int option)) = - match x with None -> None | Some n -> f n;; +(These questions adapted from web materials by Umut Acar. See +.) +Let's think about the encodings of booleans, numerals and lists in System F, +and get data-structures with the same form working in OCaml. (Of course, OCaml +has *native* versions of these datas-structures: its `true`, `1`, and `[1;2;3]`. +But the point of our exercise requires that we ignore those.) -Booleans, Church numbers, and Church lists in OCaml ---------------------------------------------------- - -(These questions adapted from web materials by Umut Acar. See .) +Recall from class System F, or the polymorphic Î»-calculus. -The idea is to get booleans, Church numbers, "Church" lists, and -binary trees working in OCaml. + types Ï ::= c | 'a | Ï1 â Ï2 | â'a. Ï + expressions e ::= x | Î»x:Ï. e | e1 e2 | Î'a. e | e [Ï] -Recall from class System F, or the polymorphic Î»-calculus. +The boolean type, and its two values, may be encoded as follows: - Ï ::= Î± | Ï1 â Ï2 | âÎ±. Ï - e ::= x | Î»x:Ï. e | e1 e2 | ÎÎ±. e | e [Ï ] + bool := â'a. 'a â 'a â 'a + true := Î'a. Î»t:'a. Î»f :'a. t + false := Î'a. Î»t:'a. Î»f :'a. f -Recall that bool may be encoded as follows: +It's used like this: - bool := âÎ±. Î± â Î± â Î± - true := ÎÎ±. Î»t:Î±. Î»f :Î±. t - false := ÎÎ±. Î»t:Î±. Î»f :Î±. f + b [Ï] e1 e2 -(where Ï indicates the type of e1 and e2) +where b is a boolean value, and Ï is the shared type of e1 and e2. -Note that each of the following terms, when applied to the -appropriate arguments, return a result of type bool. +**Exercise**. How should we implement the following terms. Note that the result +of applying them to the appropriate arguments should also give us a term of +type bool. (a) the term not that takes an argument of type bool and computes its negation; (b) the term and that takes two arguments of type bool and computes their conjunction; (c) the term or that takes two arguments of type bool and computes their disjunction. + The type nat (for "natural number") may be encoded as follows: - nat := âÎ±. Î± â (Î± â Î±) â Î± - zero := ÎÎ±. Î»z:Î±. Î»s:Î± â Î±. z - succ := Î»n:nat. ÎÎ±. Î»z:Î±. Î»s:Î± â Î±. s (n [Î±] z s) + nat := â'a. 'a â ('a â 'a) â 'a + zero := Î'a. Î»z:'a. Î»s:'a â 'a. z + succ := Î»n:nat. Î'a. Î»z:'a. Î»s:'a â 'a. s (n ['a] z s) -A nat n is deï¬ned by what it can do, which is to compute a function iterated n times. In the polymorphic -encoding above, the result of that iteration can be any type Î±, as long as you have a base element z : Î± and -a function s : Î± â Î±. +A nat n is deï¬ned by what it can do, which is to compute a function iterated n +times. In the polymorphic encoding above, the result of that iteration can be +any type 'a, as long as you have a base element z : 'a and a function s : 'a â 'a. -**Excercise**: get booleans and Church numbers working in OCaml, -including OCaml versions of bool, true, false, zero, succ, add. +**Exercise**: get booleans and Church numbers working in OCaml, +including OCaml versions of bool, true, false, zero, iszero, succ, and pred. +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. The point of the excercise +is to do these things on your own, so avoid using the built-in +OCaml booleans and integers. Consider the following list type: - type âa list = Nil | Cons of âa * âa list + type 'a list = Nil | Cons of 'a * 'a list We can encode Ï lists, lists of elements of type Ï as follows: - Ï list := âÎ±. Î± â (Ï â Î± â Î±) â Î± - nilÏ := ÎÎ±. Î»n:Î±. Î»c:Ï â Î± â Î±. n - makeListÏ := Î»h:Ï. Î»t:Ï list. ÎÎ±. Î»n:Î±. Î»c:Ï â Î± â Î±. c h (t [Î±] n c) + Ï list := â'a. 'a â (Ï â 'a â 'a) â 'a + nil Ï := Î'a. Î»n:'a. Î»c:Ï â 'a â 'a. n + make_list Ï := Î»h:Ï. Î»t:Ï list. Î'a. Î»n:'a. Î»c:Ï â 'a â 'a. c h (t ['a] n c) + +More generally, the polymorphic list type is: + + list := â'b. â'a. 'a â ('b â 'a â 'a) â 'a As with nats, recursion is built into the datatype. We can write functions like map: map : (Ï â Ï ) â Ï list â Ï list - = Î»f :Ï â Ï. Î»l:Ï list. l [Ï list] nilÏ (Î»x:Ï. Î»y:Ï list. consÏ (f x) y -**Excercise** convert this function to OCaml. Also write an `append` function. -Test with simple lists. + + +**Excercise** convert this function to OCaml. We've given you the type; you +only need to give the term. +Also give us the type and definition for a `head` function. Think about what +value to give back if the argument is the empty list. Ultimately, we might +want to make use of our `'a option` technique, but for this assignment, just +pick a strategy, no matter how clunky. + +Be sure to test your proposals with simple lists. (You'll have to `make_list` +the lists yourself; don't expect OCaml to magically translate between its +native lists and the ones you buil.d) + + + + + +Baby monads +----------- + +Read the material on dividing by zero/towards monads from the end of lecture +notes for week 6 the start of lecture notes for week 7, then write a function `lift'` that generalized the +correspondence between + and `add'`: that is, `lift'` takes any two-place +operation on integers and returns a version that takes arguments of type `int +option` instead, returning a result of `int option`. In other words, `lift'` +will have type: + + (int -> int -> int) -> (int option) -> (int option) -> (int option) + +so that `lift' (+) (Some 3) (Some 4)` will evalute to `Some 7`. +Don't worry about why you need to put `+` inside of parentheses. +You should make use of `bind'` in your definition of `lift'`: + + let bind' (u: int option) (f: int -> (int option)) = + match u with None -> None | Some x -> f x;; -Write a function `inOrder` : Ï tree â Ï list that computes the in-order traversal of a binary tree. You -may assume the above encoding of lists; deï¬ne any auxiliary functions you need.