(or reconstruct) the composition operator >=> we discussed in class.
Show your composition operator obeys the monad laws.
-2. Do the same with crossy lists. That is, given an arbitrary type
-'a, let the boxed type be a list of objects of type 'a. The singleton
+2. Do the same with lists. That is, given an arbitrary type
+'a, let the boxed type be ['a], i.e., a list of objects of type 'a. The singleton
is `\p.[p]`, and the composition operator is
- >=> (first:P->[Q]) (second:Q->[R]) :(P->[R]) = fun p -> [r | q <- first p, r <- second q]
+ >=> (first:P->[Q]) (second:Q->[R]) :(P->[R]) = List.flatten (List.map f (g a))
-Sanity check:
+For example:
f p = [p, p+1]
s q = [q*q, q+q]
>=> f s 7 = [49, 14, 64, 16]
-3. Do the same for zippy lists. That is, you need to find a
-composition operator such that
-
- f p = [p, p+1]
- s q = [q*q, q+q]
- >=> f s 7 = [49, 16]
-
-and then prove it obeys the monad laws.
We'll need a number of classes of functions to help us maneuver in the
presence of box types. We will want to define a different instance of
-each of these for whichever box type we're dealing with:
+each of these for whichever box type we're dealing with. (This will
+become clearly shortly.)
<code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
<code>map (/maep/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
-<code>map2 (/maeptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
+<code>map2 (/m&ash;ptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
<code>mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
if there are in addition `map2`, `mid`, and `mapply`. (With
`map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.)
-* ***Monad*** ("composable") A MapNable box type is a *Monad* if there
- is in addition an `mcompose` and a `join` such that `mid` is be
+* ***Monad*** ("composables") A MapNable box type is a *Monad* if there
+ is in addition an `mcompose` and a `join` such that `mid` is
a left and right identity for `mcompose`, and `mcompose` is
associative. That is, the following "laws" must hold:
definitions
mid ≡ \p.p
- mcompose ≡ \f\g\x.f(gx)
+ mcompose ≡ \fgx.f(gx)
Id is a monad. Here is a demonstration that the laws hold:
- mcompose mid k == (\f\g\x.f(gx)) (\p.p) k
+ mcompose mid k == (\fgx.f(gx)) (\p.p) k
~~> \x.(\p.p)(kx)
~~> \x.kx
~~> k
- mcompose k mid == (\f\g\x.f(gx)) k (\p.p)
+ mcompose k mid == (\fgx.f(gx)) k (\p.p)
~~> \x.k((\p.p)x)
~~> \x.kx
~~> k
- mcompose (mcompose j k) l == mcompose ((\f\g\x.f(gx)) j k) l
+ mcompose (mcompose j k) l == mcompose ((\fgx.f(gx)) j k) l
~~> mcompose (\x.j(kx)) l
- == (\f\g\x.f(gx)) (\x.j(kx)) l
+ == (\fgx.f(gx)) (\x.j(kx)) l
~~> \x.(\x.j(kx))(lx)
~~> \x.j(k(lx))
- mcompose j (mcompose k l) == mcompose j ((\f\g\x.f(gx)) k l)
+ mcompose j (mcompose k l) == mcompose j ((\fgx.f(gx)) k l)
~~> mcompose j (\x.k(lx))
- == (\f\g\x.f(gx)) j (\x.k(lx))
+ == (\fgx.f(gx)) j (\x.k(lx))
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
mcompose-crossy: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
mcompose-crossy f g a = [c | b <- g a, c <- f b]
+ mcompose-crossy f g a = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
+ mcompose-crossy f g a = concat (map f (g a))
+These three definitions are all equivalent.
In words, `mcompose f g a` feeds the a (which has type α) to g, which
returns a list of βs; each β in that list is fed to f, which returns a
-list of γs. The final result is the concatenation of those lists of γs.
+list of γs.
+
+The final result is the concatenation of those lists of γs.
For example,
let f b = [b, b+1] in
It is easy to see that these definitions obey the monad laws (see exercises).
-There can be multiple monads for any given box type. For isntance,
+There can be multiple monads for any given box type. For instance,
using the same box type and the same mid, we can define
- mcompose-zippy f g a = match (f,g) with
- ([],_) -> []
- (_,[]) -> []
- (f:ftail, g:gtail) -> f(ga) && mcompoze-zippy ftail gtail a
+ mcompose-zippy f g a = foldr (\b -> \gs -> f b ++ gs) (g a) []
+so that
+ mcompose-zippy f g 7 = [49, 14]