X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment5.mdwn;h=2078c1c094b59441c69a357729ee8fc505ca3fe0;hp=72361897b47b962f64de12c26f1895c916b8ac8a;hb=c45c91f3f0a5e1e1a4098d8fb610d55ea0611977;hpb=5ab9d6a956e6750c6aa46f97b724bffbb01a8d0f diff --git a/assignment5.mdwn b/assignment5.mdwn index 72361897..2078c1c0 100644 --- a/assignment5.mdwn +++ b/assignment5.mdwn @@ -1,10 +1,10 @@ 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;; @@ -13,7 +13,7 @@ Types and OCAML - : 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? @@ -68,11 +68,12 @@ Types and OCAML 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 @@ -142,12 +143,12 @@ Baby monads 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 --------------------------------------------------- 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. @@ -179,12 +180,12 @@ binary trees working in OCAML. 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: - 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: @@ -199,7 +200,7 @@ binary trees working in OCAML. 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: