+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.
evaluates to 1, and
let b = true in let y = 1 in let n = 2 in
- match b with true -> 1 | false -> 2;;
+ match b with true -> y | false -> n;;
also evaluates to 1. Likewise,
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:
+
+ datatype ’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.