Assignment 5
-Types and OCAML
+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.
+ 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;;
- : 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?
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,
-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
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.
- 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
- 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, 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:
- 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
- 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
- := 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.