X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=sidebyside;f=assignment5.mdwn;h=3989928996c6042a1ba4b155236310aad4a6d5a5;hb=b27052e5da57c36b2cf9dc2f18d7dcf102a841f6;hp=61096c4e7a2b34a344e02d6755fe1e53f8bd30b1;hpb=f91578902b8306ab3e52fb172637405cb8e3fb42;p=lambda.git
diff --git a/assignment5.mdwn b/assignment5.mdwn
index 61096c4e..39899289 100644
--- a/assignment5.mdwn
+++ b/assignment5.mdwn
@@ -13,9 +13,9 @@ Types and OCaml
- : 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;;
@@ -35,38 +35,38 @@ Types and OCaml
2. Throughout this problem, assume that we have
- let rec omega x = omega x;;
+ let rec blackhole x = blackhole x;;
All of the following are well-typed.
Which ones terminate? What are the generalizations?
- omega;;
+ blackhole;;
- omega ();;
+ blackhole ();;
- fun () -> omega ();;
+ fun () -> blackhole ();;
- (fun () -> omega ()) ();;
+ (fun () -> blackhole ()) ();;
- if true then omega else omega;;
+ if true then blackhole else blackhole;;
- if false then omega else omega;;
+ if false then blackhole else blackhole;;
- if true then omega else omega ();;
+ if true then blackhole else blackhole ();;
- if false then omega else omega ();;
+ if false then blackhole else blackhole ();;
- if true then omega () else omega;;
+ if true then blackhole () else blackhole;;
- if false then omega () else omega;;
+ if false then blackhole () else blackhole;;
- if true then omega () else omega ();;
+ if true then blackhole () else blackhole ();;
- if false then omega () else omega ();;
+ if false then blackhole () else blackhole ();;
- let _ = omega in 2;;
+ let _ = blackhole in 2;;
- let _ = omega () in 2;;
+ let _ = blackhole () in 2;;
3. This problem is to begin thinking about controlling order of evaluation.
The following expression is an attempt to make explicit the
@@ -104,15 +104,15 @@ and that "bool" is any boolean. Then we can try the following:
However,
- let rec omega x = omega x in
- if true then omega else omega ();;
+ let rec blackhole x = blackhole x in
+ if true then blackhole else blackhole ();;
terminates, but
- let rec omega x = omega x in
+ let rec blackhole x = blackhole x in
let b = true in
- let y = omega in
- let n = omega () in
+ let y = blackhole in
+ let n = blackhole () in
match b with true -> y | false -> n;;
does not terminate. Incidentally, `match bool with true -> yes |
@@ -123,95 +123,130 @@ and that "bool" is any boolean. Then we can try the following:
[[Hint assignment 5 problem 3]]
-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`:
+Booleans, Church numerals, and v3 lists in OCaml
+------------------------------------------------
- let bind (x: int option) (f: int -> (int option)) =
- match x with None -> None | Some n -> f n;;
+(These questions adapted from web materials by Umut Acar. See
+.)
+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.)
-Booleans, Church numbers, and Church lists in OCaml
----------------------------------------------------
-
-(These questions adapted from web materials by Umut Acar. See .)
+Recall from class System F, or the polymorphic λ-calculus.
-The idea is to get booleans, Church numbers, "Church" lists, and
-binary trees working in OCaml.
+ types Ï ::= c | 'a | Ï1 â Ï2 | â'a. Ï
+ expressions e ::= x | λx:Ï. e | e1 e2 | Î'a. e | e [Ï]
-Recall from class System F, or the polymorphic λ-calculus.
+The boolean type, and its two values, may be encoded as follows:
- Ï ::= α | Ï1 â Ï2 | âα. Ï
- e ::= x | λx:Ï. e | e1 e2 | Îα. e | e [Ï ]
+ bool := â'a. 'a â 'a â 'a
+ true := Î'a. λt:'a. λf :'a. t
+ false := Î'a. λt:'a. λf :'a. f
-Recall that bool may be encoded as follows:
+It's used like this:
- bool := âα. α â α â α
- true := Îα. λt:α. λf :α. t
- false := Îα. λt:α. λf :α. f
+ b [Ï] e1 e2
-(where Ï indicates the type of e1 and e2)
+where b is a boolean value, and Ï is the shared type of e1 and e2.
-Note that each of the following terms, when applied to the
-appropriate arguments, return a result 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;
(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 := âα. α â (α â α) â α
- zero := Îα. λz:α. λs:α â α. z
- succ := λn:nat. Îα. λz:α. λs:α â α. s (n [α] z s)
+ nat := â'a. 'a â ('a â 'a) â 'a
+ zero := Î'a. λz:'a. λs:'a â 'a. z
+ succ := λn:nat. Î'a. λz:'a. λs:'a â 'a. s (n ['a] z s)
-A nat n is deï¬ned 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 â α.
+A nat n is deï¬ned 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 â '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 âa list = Nil | Cons of âa * âa list
+ type 'a list = Nil | Cons of 'a * 'a list
We can encode Ï lists, lists of elements of type Ï as follows:
- Ï list := âα. α â (Ï â α â α) â α
- nilÏ := Îα. λn:α. λc:Ï â α â α. n
- makeListÏ := λh:Ï. λt:Ï list. Îα. λn:α. λc:Ï â α â α. c h (t [α] n c)
+ Ï list := â'a. 'a â (Ï â 'a â 'a) â 'a
+ nil Ï := Î'a. λn:'a. λc:Ï â 'a â 'a. n
+ make_list Ï := λh:Ï. λt:Ï list. Î'a. λn:'a. λc:Ï â 'a â 'a. c h (t ['a] n c)
+
+More generally, the polymorphic list type is:
+
+ list := â'b. â'a. 'a â ('b â 'a â 'a) â 'a
As with nats, recursion is built into the datatype.
We can write functions like map:
map : (Ï â Ï ) â Ï list â Ï list
- = λf :Ï â Ï. λl:Ï list. l [Ï list] nilÏ (λx:Ï. λy:Ï list. consÏ (f x) y
-**Excercise** convert this function to OCaml. Also write an `append` function.
-Test with simple lists.
+
+
+**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)
+
+
+
+
+
+Baby monads
+-----------
+
+Read the material on dividing by zero/towards monads from the end of lecture
+notes for week 6 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`.
+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' (u: int option) (f: int -> (int option)) =
+ match u with None -> None | Some x -> f x;;
-Write a function `inOrder` : Ï tree â Ï list that computes the in-order traversal of a binary tree. You
-may assume the above encoding of lists; deï¬ne any auxiliary functions you need.