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.
+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.
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
+ 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;
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:
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
- 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.