Assignment 5
-Types and OCAML
+Types and OCaml
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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?
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 OCAML
+Booleans, Church numbers, and Church lists in OCaml
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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.
+binary trees working in OCaml.
Recall from class System F, or the polymorphic λ-calculus.
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.
+ **Excercise**: get booleans and Church numbers working in OCaml,
+ including OCaml versions of bool, true, false, zero, succ, add.
Consider the following list type:
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.
+ **Excercise** convert this function to OCaml. Also write an `append` function.
Test with simple lists.
Consider the following simple binary tree type: