edits

[lambda.git] / week7.mdwn

diff --git a/week7.mdwn b/week7.mdwn
index 62ef89f..ba845b2 100644 (file)
--- a/week7.mdwn
+++ b/week7.mdwn
@@ -48,22 +48,24 @@ that provides at least the following three elements:
   use a container metaphor: if `x` has type `int option`, then `x` is
   a box that (may) contain an integer.
 
   use a container metaphor: if `x` has type `int option`, then `x` is
   a box that (may) contain an integer.
 

-    type 'a option = None | Some of 'a;;
+    `type 'a option = None | Some of 'a;;`

 
 * A way to turn an ordinary value into a monadic value.  In Ocaml, we
   did this for any integer n by mapping an arbitrary integer `n` to
   the option `Some n`.  To be official, we can define a function
   called unit:
 
 
 * A way to turn an ordinary value into a monadic value.  In Ocaml, we
   did this for any integer n by mapping an arbitrary integer `n` to
   the option `Some n`.  To be official, we can define a function
   called unit:
 

-    let unit x = Some x;;
-    val unit : 'a -> 'a option = <fun>
+    `let unit x = Some x;;`
+
+    `val unit : 'a -> 'a option = <fun>`

 
     So `unit` is a way to put something inside of a box.
 
 * A bind operation (note the type):
 
 
     So `unit` is a way to put something inside of a box.
 
 * A bind operation (note the type):
 

-     let bind m f = match m with None -> None | Some n -> f n;;
-     val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+     `let bind m f = match m with None -> None | Some n -> f n;;`
+
+     `val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>`

 
      `bind` takes two arguments (a monadic object and a function from
      ordinary objects to monadic objects), and returns a monadic
 
      `bind` takes two arguments (a monadic object and a function from
      ordinary objects to monadic objects), and returns a monadic


@@ -75,7 +77,7 @@ that provides at least the following three elements:
      Then the second argument uses `x` to compute a new monadic
      value.  Conceptually, then, we have
 
      Then the second argument uses `x` to compute a new monadic
      value.  Conceptually, then, we have
 

-    let bind m f = (let x = unwrap m in f x);;
+    `let bind m f = (let x = unwrap m in f x);;`

 
     The guts of the definition of the `bind` operation amount to
     specifying how to unwrap the monadic object `m`.  In the bind
 
     The guts of the definition of the `bind` operation amount to
     specifying how to unwrap the monadic object `m`.  In the bind


@@ -112,17 +114,17 @@ val unit : 'a -> 'a option = <fun>
 - : int option = Some 2
 </pre>
 
 - : int option = Some 2
 </pre>
 

-       The parentheses is the magic for telling Ocaml that the
-       function to be defined (in this case, the name of the function
-       is `*`, pronounced "bind") is an infix operator, so we write
-       `m * f` or `( * ) m f` instead of `* m f`.
+The parentheses is the magic for telling Ocaml that the
+function to be defined (in this case, the name of the function
+is `*`, pronounced "bind") is an infix operator, so we write
+`m * f` or `( * ) m f` instead of `* m f`.

 
 *    Associativity: bind obeys a kind of associativity, like this:
 
 
 *    Associativity: bind obeys a kind of associativity, like this:
 

-    (m * f) * g == m * (fun x -> f x * g)
+    `(m * f) * g == m * (fun x -> f x * g)`

 
 

-    If you don't understand why the lambda form is necessary, you need
-    to look again at the type of bind.  This is important.  
+    If you don't understand why the lambda form is necessary (the "fun
+    x" part), you need to look again at the type of bind.

 
     For an illustration of associativity in the option monad:
 
 
     For an illustration of associativity in the option monad:
 


@@ -133,15 +135,15 @@ Some 3 * (fun x -> unit x * unit);;
 - : int option = Some 3
 </pre>
 
 - : int option = Some 3
 </pre>
 

-       Of course, associativity must hold for arbitrary functions of
-       type `'a -> M 'a`, where `M` is the monad type.  It's easy to
-       convince yourself that the bind operation for the option monad
-       obeys associativity by dividing the inputs into cases: if `m`
-       matches `None`, both computations will result in `None`; if
-       `m` matches `Some n`, and `f n` evalutes to `None`, then both
-       computations will again result in `None`; and if the value of
-       `f n` matches `Some r`, then both computations will evaluate
-       to `g r`.
+Of course, associativity must hold for arbitrary functions of
+type `'a -> M 'a`, where `M` is the monad type.  It's easy to
+convince yourself that the bind operation for the option monad
+obeys associativity by dividing the inputs into cases: if `m`
+matches `None`, both computations will result in `None`; if
+`m` matches `Some n`, and `f n` evalutes to `None`, then both
+computations will again result in `None`; and if the value of
+`f n` matches `Some r`, then both computations will evaluate
+to `g r`.

 
 *    Right identity: unit is a right identity for bind.  That is, 
      `m * unit == m` for all monad objects `m`.  For instance,
 
 *    Right identity: unit is a right identity for bind.  That is, 
      `m * unit == m` for all monad objects `m`.  For instance,


@@ -162,7 +164,7 @@ arguments of a monoid operation) the two arguments of the bind are of
 different types.  But if we generalize bind so that both arguments are
 of type `'a -> M 'a`, then we get plain identity laws and
 associativity laws, and the monad laws are exactly like the monoid
 different types.  But if we generalize bind so that both arguments are
 of type `'a -> M 'a`, then we get plain identity laws and
 associativity laws, and the monad laws are exactly like the monoid

-laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>).
+laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>, near the bottom).

 
 
 Monad outlook
 
 
 Monad outlook


@@ -189,8 +191,11 @@ intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
 though without explicitly using monads.
 
 All of the code in the discussion below can be found here: [[intensionality-monad.ml]].
 though without explicitly using monads.
 
 All of the code in the discussion below can be found here: [[intensionality-monad.ml]].

-To run it, download the file, start Ocaml, and say `# #use
-"intensionality-monad.ml";;`. 
+To run it, download the file, start Ocaml, and say 
+
+    # #use "intensionality-monad.ml";;
+
+Note the extra `#` attached to the directive `use`.

 
 Here's the idea: since people can have different attitudes towards
 different propositions that happen to have the same truth value, we
 
 Here's the idea: since people can have different attitudes towards
 different propositions that happen to have the same truth value, we