- : 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.
- For those that are not, why not?
+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 rec f x = f x;;
or of `match`. That is, you must keep the `let` statements, though
you're allowed to adjust what `b`, `y`, and `n` get assigned to.
- [[Hint assignment 5 problem 3]]
+ [[hints/assignment 5 hint 1]]
Booleans, Church numerals, and v3 lists in OCaml
------------------------------------------------
-(These questions adapted from web materials by Umut Acar. See <http://www.mpi-sws.org/~umut/>.)
+(These questions adapted from web materials by Umut Acar. See
+<http://www.mpi-sws.org/~umut/>.)
-Let's think about the encodings of booleans, numerals and lists in System F, and get datastructures with the same explicit form working in OCaml. (The point... so we won't rely on OCaml's native booleans, integers, or lists.)
+Let's think about the encodings of booleans, numerals and lists in System F,
+and get data-structures with the same form working in OCaml. (Of course, OCaml
+has *native* versions of these datas-structures: its `true`, `1`, and `[1;2;3]`.
+But the point of our exercise requires that we ignore those.)
Recall from class System F, or the polymorphic λ-calculus.
where b is a boolean value, and τ is the shared type of e1 and e2.
-**Exercise**. How should we implement the following terms. Note that the result of applying them to the appropriate arguments should also give us a term of type bool.
+**Exercise**. How should we implement the following terms. Note that the result
+of applying them to the appropriate arguments should also give us a term 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;
times. In the polymorphic encoding above, the result of that iteration can be
any type 'a, as long as you have a base element z : 'a and a function s : 'a → 'a.
-**Excercise**: get booleans and Church numbers working in OCaml,
-including OCaml versions of bool, true, false, zero, iszero, succ, and **pred**.
+**Exercise**: get booleans and Church numbers working in OCaml,
+including OCaml versions of bool, true, false, zero, iszero, 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 integers.
Consider the following list type:
nil τ := Λ'a. λn:'a. λc:τ → 'a → 'a. n
make_list τ := λh:τ. λt:τ list. Λ'a. λn:'a. λc:τ → 'a → 'a. c h (t ['a] n c)
+More generally, the polymorphic list type is:
+
+ list := ∀'b. ∀'a. 'a → ('b → 'a → 'a) → 'a
+
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
-**Excercise** convert this function to OCaml. Also write an `append` function.
-Also write a **head** function. Also write nil??? Test with simple lists.
+<!--
+ = λf :σ → τ. λl:σ list. l [τ list] nil τ (λx:σ. λy:τ list. make_list τ (f x) y
+-->
+
+**Excercise** convert this function to OCaml. We've given you the type; you
+only need to give the term.
+Also give us the type and definition for a `head` function. 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.
+Be sure to test your proposals with simple lists. (You'll have to `make_list`
+the lists yourself; don't expect OCaml to magically translate between its
+native lists and the ones you buil.d)
+
+
+<!--
Consider the following simple binary tree type:
type 'a tree = Leaf | Node of 'a tree * 'a * 'a tree
**Excercise**
-Write a function `sum_leaves` that computes the sum of all the
-leaves in an int tree.
+Write a function `sum_leaves` that computes the sum of all the leaves in an int
+tree.
+
+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.
+-->
-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.
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
+Read the material on dividing by zero/towards monads from <strike>the end of lecture
+notes for week 6</strike> the start of lecture notes for week 7, 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`.
+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;;
+ let bind' (u: int option) (f: int -> (int option)) =
+ match u with None -> None | Some x -> f x;;