- : 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 numbers, and Church lists in OCaml
----------------------------------------------------
+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/>.)
-The idea is to get booleans, Church numbers, v3 lists, and
-binary trees working in OCaml.
+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.
- τ ::= α | τ1 → τ2 | ∀α. τ
- e ::= x | λx:τ. e | e1 e2 | Λα. e | e [τ ]
+ types τ ::= c | 'a | τ1 → τ2 | ∀'a. τ
+ expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
-Recall that bool may be encoded as follows:
+The boolean type, and its two values, may be encoded as follows:
- bool := ∀α. α → α → α
- true := Λα. λt:α. λf :α. t
- false := Λα. λt:α. λf :α. f
+ bool := ∀'a. 'a → 'a → 'a
+ true := Λ'a. λt:'a. λf :'a. t
+ false := Λ'a. λt:'a. λf :'a. f
-(where τ indicates the type of e1 and e2)
+It's used like this:
-Note that each of the following terms, when applied to the
-appropriate arguments, return a result of type bool.
+ b [τ] e1 e2
+
+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.
(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;
(c) the term or that takes two arguments of type bool and computes their disjunction.
+
The type nat (for "natural number") may be encoded as follows:
- nat := ∀α. α → (α → α) → α
- zero := Λα. λz:α. λs:α → α. z
- succ := λn:nat. Λα. λz:α. λs:α → α. s (n [α] z s)
+ nat := ∀'a. 'a → ('a → 'a) → 'a
+ zero := Λ'a. λz:'a. λs:'a → 'a. z
+ succ := λn:nat. Λ'a. λz:'a. λs:'a → 'a. s (n ['a] z s)
-A nat n is defined by what it can do, which is to compute a function iterated n times. In the polymorphic
-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.
+A nat n is defined by what it can do, which is to compute a function iterated n
+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, succ, add.
+**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:
- type ’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
- makeListτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
+ τ list := ∀'a. 'a → (τ → 'a → 'a) → 'a
+ 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.
-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
+ type 'a tree = Leaf | Node of 'a tree * 'a * 'a tree
**Excercise**
-Write a function `sumLeaves` 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 `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.
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;;