update

[lambda.git] / topics / _week7_monads.mdwn

diff --git a/topics/_week7_monads.mdwn b/topics/_week7_monads.mdwn
index 32f7ac0..7d189e4 100644 (file)
--- a/topics/_week7_monads.mdwn
+++ b/topics/_week7_monads.mdwn
@@ -74,13 +74,14 @@ if `α List` is our box type, we can write the second arrow as
 
 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 (/&epsilon;maid&epsilon;nt@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 (/&epsilon;m@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
 
@@ -108,8 +109,8 @@ if there is a `map` function defined for that box type with the type given above
        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:
 
@@ -124,26 +125,26 @@ Identity box type is a completly invisible box.  With the following
 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))
 
@@ -155,26 +156,21 @@ consider the box type `α List`, with the following operations:
     mid: α -> [α]
     mid a = [a]
  
-    mcompose-crossy: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
-    mcompose-crossy f g a = [c | b <- g a, c <- f b]
+    mcompose: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
+    mcompose f g a = concat (map f (g a))
+                   = foldr (\b -> \gs -> (f b) ++ gs) [] (g a) 
+                   = [c | b <- g a, c <- f b]
+
+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.
 
-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.
 For example, 
 
     let f b = [b, b+1] in
     let g a = [a*a, a+a] in
-    mcompose-crossy f g 7 = [49, 50, 14, 15]
+    mcompose f g 7 = [49, 50, 14, 15]
 
 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,
-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
-
-