in it that you have to complete. You have to understand enough to
add the last pieces to make the program function.
-We're still writing up the (substantial) exposition of this, and will post a link
-to it here soon.
+You can find the skeleton code [[here|/code/untyped_evaluator.ml]].
+
+We've also prepared a much fuller version, that has user-friendly input
+and printing of results. We'll provide a link to that shortly. It
+will be easiest for you to understand that code if you've
+completed the gaps in the simplified skeleton linked above.
+
## Evaluation in the untyped lambda calculus: environments
function. The difference between the substitute-and-repeat approach
above, and this approach, is one huge step towards monads.
-We're still writing up the exposition of this, too, and will post a link
-to it here soon.
+The skeleton code for this is at the [[same link as before|/code/untyped_evaluator.ml]].
+This part of the exercise is the "V2" part of that code.
## Monads
-Mappables (functors), MapNables (applicatives functors), and Monads
+Mappables (functors), MapNables (applicative functors), and Monads
(composables) are ways of lifting computations from unboxed types into
boxed types. Here, a "boxed type" is a type function with one unsaturated
hole (which may have several occurrences). We can think of the box type
as a function from a type to a type. Call this type function M, and let P, Q, R, and S be schematic variables over types.
Recall that a monad requires a singleton function `mid : P-> MP`, and a
-composition operator `>=> : (P->MQ) -> (Q->MR) -> (P->MR)`. The type for
-the composition operator stated here corrects a typo in the class handout.
-Also, in the handout we called `mid` `𝟭`. But now we've decided that `mid`
-is better. (Think of it as "m" plus "identity", not as the start of "midway".)
+composition operator like `>=> : (P-><u>Q</u>) -> (Q-><u>R</u>) -> (P-><u>R</u>)`.
-We will also move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
+As we said in the notes, we'll move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more
closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
that its types flow more naturally from left to right.
For example, the Identity monad has the identity function `I` for `mid`
and ordinary function composition `○` for `<=<`. It is easy to prove
that the laws hold for any terms `j`, `k`, and `l` whose types are
-suitable for `mid` and `>=>`:
+suitable for `mid` and `<=<`:
mid <=< k == I ○ k == \p. I (k p) ~~> \p. k p ~~> k
k <=< mid == k ○ I == \p. k (I p) ~~> \p. k p ~~> k
(j <=< k) <=< l == (\p. j (k p)) ○ l == \q. (\p. j (k p)) (l q) ~~> \q. j (k (l q))
j <=< (k <=< l) == j ○ (k ○ l) == j ○ (\p. k (l p)) == \q. j ((\p. k (l p)) q) ~~> \q. j (k (l q))
-1. On a number of occasions, we've used the Option type to make our
+1. On a number of occasions, we've used the Option/Maybe type to make our
conceptual world neat and tidy (for instance, think of [[our discussion
of Kaplan's Plexy|topics/week6_plexy]]). As we learned in class, there is a natural monad
-for the Option type. Borrowing the notation of OCaml, let's say that
+for the Option type. Using the vocabulary of OCaml, let's say that
"`'a option`" is the type of a boxed `'a`, whatever type `'a` is.
More specifically,
type 'a option = None | Some 'a
-(If you have trouble keeping straight what is the OCaml terminology for this and what is the Haskell terminology, don't worry, we do too.)
+ (If you have trouble keeping straight what is the OCaml terminology for this and what is the Haskell terminology, don't worry, we do too.)
-Now the obvious singleton for the Option monad is `\p. Some p`. Give
+ Now the obvious singleton for the Option monad is `\p. Some p`. Give
(or reconstruct) either of the composition operators `>=>` or `<=<`.
Show that your composition operator obeys the Monad Laws.
-2. Do the same with lists. That is, given an arbitrary type
+2. Do the same with lists. That is, given an arbitrary type
`'a`, let the boxed type be `['a]` or `'a list`, that is, lists of objects of type `'a`. The singleton
is `\p. [p]`, and the composition operator is:
let (>=>) (j : 'a -> 'b list) (k : 'b -> 'c list) : 'a -> 'c list =
fun a -> List.flatten (List.map k (j a))
-For example:
+ For example:
- let j a = [a; a+1];;
- let k b = [b*b; b+b];;
- (j >=> k) 7 (* ==> [49; 14; 64; 16] *)
+ let j a = [a; a+1];;
+ let k b = [b*b; b+b];;
+ (j >=> k) 7 (* ==> [49; 14; 64; 16] *)