ass5: merged Chris' changes

[lambda.git] / assignment5.mdwn

diff --git a/assignment5.mdwn b/assignment5.mdwn
index a79df5f..fee1d83 100644 (file)
--- a/assignment5.mdwn
+++ b/assignment5.mdwn
@@ -13,9 +13,9 @@ Types and OCaml
                - : int = 1
 
 
                - : 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;;
 
 
                let rec f x = f x;;
 


@@ -126,9 +126,13 @@ and that "bool" is any boolean.  Then we can try the following:
 Booleans, Church numerals, and v3 lists in OCaml
 ------------------------------------------------
 
 Booleans, Church numerals, and v3 lists in OCaml
 ------------------------------------------------
 

-(These questions adapted from web materials by Umut Acar. See <http://www.mpi-sws.org/~umut/>.)
+(These questions adapted from web materials by Umut Acar. See
+<http://www.mpi-sws.org/~umut/>.)

 
 

-Let's think about the encodings of booleans, numerals and lists in System F, and get datastructures with the same explicit form working in OCaml. (The point... so we won't rely on OCaml's native booleans, integers, or lists.)
+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.
 
 
 Recall from class System F, or the polymorphic λ-calculus.
 


@@ -147,7 +151,9 @@ It's used like this:
 
 where b is a boolean value, and τ is the shared type of e1 and e2.
 
 
 where b is a boolean value, and τ 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.
+**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;
 
 (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;


@@ -164,8 +170,13 @@ A nat n is defined 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 → 'a.
 
 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 → 'a.
 

-**Excercise**: get booleans and Church numbers working in OCaml,
-including OCaml versions of bool, true, false, zero, iszero, succ, and **pred**.
+**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:
 
 
 Consider the following list type:
 


@@ -177,27 +188,47 @@ We can encode τ lists, lists of elements of type τ as follows:
        nil τ := Λ'a. λn:'a. λc:τ → 'a → 'a. n
        make_list τ := λh:τ. λt:τ list. Λ'a. λn:'a. λc:τ → 'a → 'a. c h (t ['a] n c)
 
        nil τ := Λ'a. λn:'a. λc:τ → 'a → 'a. n
        make_list τ := λh:τ. λt:τ list. Λ'a. λn:'a. λc:τ → 'a → 'a. c h (t ['a] n c)
 

+More generally, the polymorphic list type is:
+
+       list := ∀'b. ∀'a. 'a → ('b → 'a → 'a) → 'a
+

 As with nats, recursion is built into the datatype.
 
 We can write functions like map:
 
        map : (σ → τ ) → σ list → τ list
 As with nats, recursion is built into the datatype.
 
 We can write functions like map:
 
        map : (σ → τ ) → σ list → τ list

-               = λf :σ → τ. λl:σ list. l [τ list] nilτ (λx:σ. λy:τ list. consτ (f x) y

 
 

-**Excercise** convert this function to OCaml.  Also write an `append` function.
-Also write a **head** function. Also write nil??? Test with simple lists.
+<!--
+               = λf :σ → τ. λl:σ list. l [τ list] nil τ (λx:σ. λy:τ list. make_list τ (f x) y
+-->
+
+**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)
+
+
+<!--

 Consider the following simple binary tree type:
 
        type 'a tree = Leaf | Node of 'a tree * 'a * 'a tree
 
 **Excercise**
 Consider the following simple binary tree type:
 
        type 'a tree = Leaf | Node of 'a tree * 'a * 'a tree
 
 **Excercise**

-Write a function `sum_leaves` that computes the sum of all the
-leaves in an int tree.
+Write a function `sum_leaves` that computes the sum of all the leaves in an int
+tree.
+
+Write a function `in_order` : τ tree → τ list that computes the in-order
+traversal of a binary tree. You may assume the above encoding of lists; define
+any auxiliary functions you need.
+-->

 
 

-Write a function `in_order` : τ tree → τ list that computes the in-order traversal of a binary tree. You
-may assume the above encoding of lists; define any auxiliary functions you need.

 
 Baby monads
 -----------
 
 Baby monads
 -----------