X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=exercises%2Fassignment7.mdwn;h=e9cf82371162e91fb718094a669a096e5809a337;hp=ea387b93cebea8d9453e76f9b6af6012619c751f;hb=9d1d3d7d480c1c808a7a9fddf39f95037196d506;hpb=46af41468d114dba7d89b800df78fafcabe796dc diff --git a/exercises/assignment7.mdwn b/exercises/assignment7.mdwn index ea387b93..e9cf8237 100644 --- a/exercises/assignment7.mdwn +++ b/exercises/assignment7.mdwn @@ -42,8 +42,13 @@ Instead, we're going to give you almost the complete program, with a few gaps 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 @@ -61,8 +66,8 @@ since it amounts to evaluating terms relative to an assignment 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 @@ -70,18 +75,18 @@ to it here soon. 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. +hole (which may have several occurrences, as in `(Î±,Î±) tree`). We can think of the box type +as a function from a type to a type. -Recall that a monad requires a singleton function `mid : P-> MP`, and a -composition operator like `>=> : (P->Q) -> (Q->R) -> (P->R)`. +Recall that a monad requires a singleton function `mid : P-> P`, and a +composition operator like `>=> : (P->Q) -> (Q->R) -> (P->R)`. 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 `â`. But `>=>` has the virtue that its types flow more naturally from left to right. -Anyway, `mid` and (let's say) `<=<` have to obey the following Monad Laws: +Anyway, `mid` and (let's say) `<=<` have to obey the Monad Laws: mid <=< k = k k <=< mid = k @@ -102,7 +107,7 @@ suitable for `mid` and `<=<`: 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. Using the vocabulary of OCaml, let's say that -"`'a option`" is the type of a boxed `'a`, whatever type `'a` is. +`'a option` is the type of a boxed `'a`, whatever type `'a` is. More specifically, type 'a option = None | Some 'a @@ -114,8 +119,8 @@ More specifically, Show that your composition operator obeys the Monad Laws. 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: +`'a`, let the boxed type be `['a]` or `'a list`, that is, lists of values of type `'a`. The `mid` +is the singleton function `\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))