hw changes

[lambda.git] / assignment5.mdwn

diff --git a/assignment5.mdwn b/assignment5.mdwn
index bd89880..cc714e9 100644 (file)
--- a/assignment5.mdwn
+++ b/assignment5.mdwn
@@ -1,10 +1,10 @@
 Assignment 5
 
 Assignment 5
 

-Types and OCAML
+Types and OCaml

 ---------------
 
 0. Recall that the S combinator is given by \x y z. x z (y z).
 ---------------
 
 0. Recall that the S combinator is given by \x y z. x z (y z).

-   Give two different typings for this function in OCAML.
+   Give two different typings for this function in OCaml.

    To get you started, here's one typing for K:
 
     # let k (y:'a) (n:'b) = y;;
    To get you started, here's one typing for K:
 
     # let k (y:'a) (n:'b) = y;;


@@ -13,7 +13,7 @@ Types and OCAML
     - : int = 1
 
 
     - : int = 1
 
 

-1. Which of the following expressions is well-typed in OCAML?  
+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?
 
    For those that are, give the type of the expression as a whole.
    For those that are not, why not?
 


@@ -68,11 +68,12 @@ Types and OCAML
 
     let _ = omega () in 2;;
 
 
     let _ = omega () in 2;;
 

-3. The following expression is an attempt to make explicit the
+3. This problem is to begin thinking about controlling order of evaluation.
+The following expression is an attempt to make explicit the

 behavior of `if`-`then`-`else` explored in the previous question.
 The idea is to define an `if`-`then`-`else` expression using 
 behavior of `if`-`then`-`else` explored in the previous question.
 The idea is to define an `if`-`then`-`else` expression using 

-other expression types.  So assume that "yes" is any OCAML expression,
-and "no" is any other OCAML expression (of the same type as "yes"!),
+other expression types.  So assume that "yes" is any OCaml expression,
+and "no" is any other OCaml expression (of the same type as "yes"!),

 and that "bool" is any boolean.  Then we can try the following:
 "if bool then yes else no" should be equivalent to
 
 and that "bool" is any boolean.  Then we can try the following:
 "if bool then yes else no" should be equivalent to
 


@@ -142,26 +143,27 @@ 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.
 

-    τ ::= α | τ1 → τ2 | ∀α. τ
-    e ::= x | λx:τ. e | e1 e2 | Λα. e | e [τ ]
+    τ ::= 'α | τ1 → τ2 | ∀'α. τ | c
+    e ::= x | λx:τ. e | e1 e2 | Λ'α. e | e [τ ]

 
    Recall that bool may be encoded as follows:
 
     bool := ∀α. α → α → α
     true := Λα. λt:α. λf :α. t
     false := Λα. λt:α. λf :α. f
 
    Recall that bool may be encoded as follows:
 
     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;


@@ -178,35 +180,35 @@ 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, 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 list predicates.

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

-    datatype ’a list = Nil | Cons of ’a * ’a list
+    type ’a list = Nil | Cons of ’a * ’a list

 
    We can encode τ lists, lists of elements of type τ as follows:
 
     τ list := ∀α. α → (τ → α → α) → α
     nilτ := Λα. λn:α. λc:τ → α → α. n
 
    We can encode τ lists, lists of elements of type τ as follows:
 
     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:
+   We can write functions like head, isNil, and map:

 
     map : (σ → τ ) → σ list → τ list
 
     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:
+   We've given you the type for map, you only need to give the term.

 
 

-    datatype ’a tree = Leaf | Node of ’a tree * ’a * ’a tree
+   With regard to `head`, 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. 

 
 

-   (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.
-
-   (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.
-
-   (c) 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.
+   Please provide both the terms and the types for each item.