X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment5.mdwn;h=f402ec61a70bbe9ebe4e5f7c2f4a2f1ecc853ede;hp=85ac9a180a6d29cba48e9200aa68d507cd476df5;hb=88f7b0444146d9379521cc31ad9ef87b909e0b3b;hpb=ae46ef3c2cb5de4b994fc7df4a95055d03c6b867
diff --git a/assignment5.mdwn b/assignment5.mdwn
index 85ac9a18..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;;
@@ -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]]
-Booleans, Church numbers, and Church lists in OCaml
----------------------------------------------------
+Booleans, Church numerals, and v3 lists in OCaml
+------------------------------------------------
-(These questions adapted from web materials by Umut Acar. See .)
+(These questions adapted from web materials by Umut Acar. See
+.)
-The idea is to get booleans, Church numbers, v3 lists, and
-binary trees working in OCaml.
+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.)
Recall from class System F, or the polymorphic Î»-calculus.
- Ï ::= Î± | Ï1 â Ï2 | âÎ±. Ï
- e ::= x | Î»x:Ï. e | e1 e2 | ÎÎ±. e | e [Ï ]
+ types Ï ::= c | 'a | Ï1 â Ï2 | â'a. Ï
+ expressions e ::= x | Î»x:Ï. e | e1 e2 | Î'a. e | e [Ï]
-Recall that bool may be encoded as follows:
+The boolean type, and its two values, may be encoded as follows:
- bool := âÎ±. Î± â Î± â Î±
- true := ÎÎ±. Î»t:Î±. Î»f :Î±. t
- false := ÎÎ±. Î»t:Î±. Î»f :Î±. f
+ 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
-(where Ï indicates the type of e1 and e2)
+It's used like this:
-Note that each of the following terms, when applied to the
-appropriate arguments, return a result of type bool.
+ b [Ï] e1 e2
+
+where b is a boolean value, and Ï is the shared type of e1 and e2.
+
+**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 := âÎ±. Î± â (Î± â Î±) â Î±
- zero := ÎÎ±. Î»z:Î±. Î»s:Î± â Î±. z
- succ := Î»n:nat. ÎÎ±. 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 âa list = Nil | Cons of âa * âa 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 := âÎ±. Î± â (Ï â Î± â Î±) â Î±
- nilÏ := ÎÎ±. Î»n:Î±. Î»c:Ï â Î± â Î±. n
- makeListÏ := Î»h:Ï. Î»t:Ï list. ÎÎ±. 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)
+
+
+
-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.
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
+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`.
+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' (x: int option) (f: int -> (int option)) =
- match x with None -> None | Some n -> f n;;
+ let bind' (u: int option) (f: int -> (int option)) =
+ match u with None -> None | Some x -> f x;;