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@@ -1,4 +1,17 @@
+Assignment 5
+
 Types and OCAML
+---------------
+
+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.
+   To get you started, here's one typing for K:
+
+    # let k (y:'a) (n:'b) = y;;
+    val k : 'a -> 'b -> 'a = [fun]
+    # k 1 true;;
+    - : 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.
@@ -55,7 +68,8 @@ Types and OCAML
 
     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 
 other expression types.  So assume that "yes" is any OCAML expression,
@@ -108,3 +122,94 @@ 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]]
+
+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
+
+     (int -> int -> int) -> (int option) -> (int option) -> (int option)
+
+   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;;
+
+
+Booleans, Church numbers, and Church lists in OCAML
+---------------------------------------------------
+
+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.
+
+    τ ::= α | τ1 → τ2 | ∀α. τ
+    e ::= x | λx:τ. e | e1 e2 | Λα. 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
+
+   (where τ indicates the type of e1 and e2)
+
+   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;
+    (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 := ∀α. α → (α → α) → α
+    zero := Λα. λz:α. λs:α → α. z
+    succ := λn:nat. Λα. λz:α. λs:α → α. s (n [α] z s)
+
+   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 α, as long as you have a base element z : α and
+   a function s : α → α.
+
+   **Excercise**: get booleans and Church numbers working in OCAML,
+     including OCAML versions of bool, true, false, zero, succ, add.
+
+   Consider the following list type:
+
+    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
+    makeListτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
+
+   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.
+   Test with simple lists.
+
+   Consider the following simple binary tree type:
+
+    type ’a tree = Leaf | Node of ’a tree * ’a * ’a tree
+
+   **Excercise**
+   Write a function `sumLeaves` that computes the sum of all the
+   leaves in an int tree.
+
+   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.