X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=exercises%2Fassignment7.mdwn;h=e9cf82371162e91fb718094a669a096e5809a337;hp=bd614b3807966bd2f0620a5f4b33837cc5b30598;hb=9d1d3d7d480c1c808a7a9fddf39f95037196d506;hpb=b18e37c538c98f47159a96486860a808e2a068dc diff --git a/exercises/assignment7.mdwn b/exercises/assignment7.mdwn index bd614b38..e9cf8237 100644 --- a/exercises/assignment7.mdwn +++ b/exercises/assignment7.mdwn @@ -75,18 +75,18 @@ This part of the exercise is the "V2" part of that code. 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 @@ -107,7 +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 @@ -119,8 +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))