-evaluates to 1, and
-
- let b = true in let y = 1 in let n = 2 in
- match b with true -> y | false -> n;;
-
-also evaluates to 1. Likewise,
-
- if false then 1 else 2;;
-
-and
-
- let b = false in let y = 1 in let n = 2 in
- match b with true -> y | false -> n;;
-
-both evaluate to 2.
-
-However,
-
- let rec omega x = omega x in
- if true then omega else omega ();;
-
-terminates, but
-
- let rec omega x = omega x in
- let b = true in
- let y = omega in
- let n = omega () in
- match b with true -> y | false -> n;;
-
-does not terminate. Incidentally, `match bool with true -> yes |
-false -> no;;` works as desired, but your assignment is to solve it
-without using the magical evaluation order properties of either `if`
-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]]
-
-4. 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
-
- (int -> int -> int) -> (int option) -> (int option) -> (int option)
-
- 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;;
-
-
-Church lists in System F
-------------------------
-
-These questions adapted from web materials written by some dude named Acar.
-
- 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 ]
- Despite its simplicity, System F is quite expressive. As discussed in class, it has sufficient expressive power
- to be able to encode many datatypes found in other programming languages, including products, sums, and
- inductive datatypes.
- For example, 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
- appropriate arguments, return a result 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 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)
- 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.
- Conveniently, this encoding “is” its own elimination form, in a sense:
- rec(e, e0, x:τ. e1) := e [τ ] e0 (λx:τ. e1)
- The case analysis is baked into the very definition of the type.
- Exercise 2. Verify that these encodings (zero, succ , rec) typecheck in System F. Write down the typing
- derivations for the terms.
- 1
-
- ══════════════════════════════════════════════════════════════════════════
-
- As mentioned in class, System F can express any inductive datatype. Consider the following list type:
- datatype ’a list =
- Nil
- | Cons of ’a * ’a list
- We can encode τ lists, lists of elements of type τ as follows:1
- 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)
- As with nats, The τ list type’s case analyzing elimination form is just application. 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
- Exercise 3. Consider the following simple binary tree type:
- datatype ’a tree =
- Leaf
- | Node of ’a tree * ’a * ’a tree
- (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.
-
---
-Jim Pryor
-jim@jimpryor.net