+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]]
+
+4. 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;;
+
+
+Church lists in System F
+------------------------
+
+These questions adapted from web materials written by some dude named Acar.
+
+ 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 ]
+ Despite its simplicity, System F is quite expressive. As discussed in class, it has sufficient expressive power
+ to be able to encode many datatypes found in other programming languages, including products, sums, and
+ inductive datatypes.
+ For example, 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)
+ Exercise 1. Show how to encode the following terms. Note that each of these 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 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.
+ Conveniently, this encoding “is” its own elimination form, in a sense:
+ rec(e, e0, x:τ. e1) := e [τ ] e0 (λx:τ. e1)
+ The case analysis is baked into the very definition of the type.
+ Exercise 2. Verify that these encodings (zero, succ , rec) typecheck in System F. Write down the typing
+ derivations for the terms.
+ 1
+
+ ══════════════════════════════════════════════════════════════════════════
+
+ As mentioned in class, System F can express any inductive datatype. 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:1
+ 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)
+ As with nats, The τ list type’s case analyzing elimination form is just application. 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:
+ datatype ’a tree =
+ Leaf
+ | Node of ’a tree * ’a * ’a tree
+ (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.
+
+--
+Jim Pryor
+jim@jimpryor.net