edits

diff --git a/assignment5.mdwn b/assignment5.mdwn
index bd89880..7236189 100644 (file)
--- a/assignment5.mdwn
+++ b/assignment5.mdwn
@@ -142,10 +142,12 @@ Baby monads
       match x with None -> None | Some n -> f n;;
 
 
       match x with None -> None | Some n -> f n;;
 
 

-Booleans, Church numbers, and Church lists in System F
-------------------------------------------------------
+Booleans, Church numbers, and Church lists in OCAML
+---------------------------------------------------

 
 These questions adapted from web materials written by some smart dude named Acar.
 
 These questions adapted from web materials written by some smart dude named Acar.

+The idea is to get booleans, Church numbers, "Church" lists, and
+binary trees working in OCAML.

 
    Recall from class System F, or the polymorphic λ-calculus.
 
 
    Recall from class System F, or the polymorphic λ-calculus.
 


@@ -157,11 +159,10 @@ These questions adapted from web materials written by some smart dude named Acar
     bool := ∀α. α → α → α
     true := Λα. λt:α. λf :α. t
     false := Λα. λt:α. λf :α. f
     bool := ∀α. α → α → α
     true := Λα. λt:α. λf :α. t
     false := Λα. λt:α. λf :α. f

-    ifτ e then e1 else e2 := e [τ ] e1 e2

 
    (where τ indicates the type of e1 and e2)
 
 
    (where τ indicates the type of e1 and e2)
 

-   Exercise 1. Show how to encode the following terms. Note that each of these terms, when applied to the
+   Note that each of the following terms, when applied to the

    appropriate arguments, return a result of type bool.
 
     (a) the term not that takes an argument of type bool and computes its negation;
    appropriate arguments, return a result of type bool.
 
     (a) the term not that takes an argument of type bool and computes its negation;


@@ -178,8 +179,8 @@ These questions adapted from web materials written by some smart dude named Acar
    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 → α.
 
    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 → α.
 

-   Exercise 2. Verify that these encodings (zero, succ , rec) typecheck in System F.
-   (Draw a type tree for each term.)
+   **Excercise**: get booleans and Church numbers working in OCAML,
+     including OCAML versions of bool, true, false, zero, succ, add.

 
    Consider the following list type:
 
 
    Consider the following list type:
 


@@ -189,24 +190,25 @@ These questions adapted from web materials written by some smart dude named Acar
 
     τ list := ∀α. α → (τ → α → α) → α
     nilτ := Λα. λn:α. λc:τ → α → α. n
 
     τ list := ∀α. α → (τ → α → α) → α
     nilτ := Λα. λn:α. λc:τ → α → α. n

-    consτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
+    makeListτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)

 
 

-   As with nats, The τ list type’s case analyzing elimination form is just application.
+   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
 
 
    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
 

-   Exercise 3. Consider the following simple binary tree type:
+   **Excercise** convert this function to OCAML.  Also write an `append` function.
+   Test with simple lists.

 
 

-    datatype ’a tree = Leaf | Node of ’a tree * ’a * ’a tree
+   Consider the following simple binary tree type:

 
 

-   (a) Give a System F encoding of binary trees, including a definition of the type τ tree and definitions of
-   the constructors leaf : τ tree and node : τ tree → τ → τ tree → τ tree.
+    type ’a tree = Leaf | Node of ’a tree * ’a * ’a tree

 
 

-   (b) Write a function height : τ tree → nat. You may assume the above encoding of nat as well as definitions
-   of the functions plus : nat → nat → nat and max : nat → nat → nat.
+   **Excercise**
+   Write a function `sumLeaves` that computes the sum of all the
+   leaves in an int tree.

 
 

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