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diff --git a/assignment5.mdwn b/assignment5.mdwn
index 85ac9a18..fee1d833 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;;
@@ -123,77 +123,112 @@ and that "bool" is any boolean. Then we can try the following:
[[Hint assignment 5 problem 3]]
-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 λ-calculus.
- Ï ::= α | Ï1 â Ï2 | âα. Ï
- e ::= x | λx:Ï. e | e1 e2 | Îα. e | e [Ï ]
+ types Ï ::= c | 'a | Ï1 â Ï2 | â'a. Ï
+ expressions e ::= x | λx: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 := âα. α â α â α
- true := Îα. λt:α. λf :α. t
- false := Îα. λt:α. λf :α. f
+ bool := â'a. 'a â 'a â 'a
+ true := Î'a. λt:'a. λf :'a. t
+ false := Î'a. λt:'a. λf :'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 := âα. α â (α â α) â α
- zero := Îα. λz:α. λs:α â α. z
- succ := λn:nat. Îα. λz:α. λs:α â α. s (n [α] z s)
+ nat := â'a. 'a â ('a â 'a) â 'a
+ zero := Î'a. λz:'a. λs:'a â 'a. z
+ succ := λn:nat. Î'a. λz:'a. λs:'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 α, as long as you have a base element z : α and
-a function s : α â α.
+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 := âα. α â (Ï â α â α) â α
- nilÏ := Îα. λn:α. λc:Ï â α â α. n
- makeListÏ := λh:Ï. λt:Ï list. Îα. λn:α. λc:Ï â α â α. c h (t [α] n c)
+ Ï list := â'a. 'a â (Ï â 'a â 'a) â 'a
+ nil Ï := Î'a. λn:'a. λc:Ï â 'a â 'a. n
+ make_list Ï := λh:Ï. λt:Ï list. Î'a. λn:'a. λc: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
- = λf :Ï â Ï. λl:Ï list. l [Ï list] nilÏ (λx:Ï. λy: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
-----------