translating: more about records

[lambda.git] / week7.mdwn

diff --git a/week7.mdwn b/week7.mdwn
index af8c17e..7544425 100644 (file)
--- a/week7.mdwn
+++ b/week7.mdwn
@@ -4,6 +4,12 @@
 Towards Monads: Safe division
 -----------------------------
 
 Towards Monads: Safe division
 -----------------------------
 

+[This section used to be near the end of the lecture notes for week 6]
+
+We begin by reasoning about what should happen when someone tries to
+divide by zero.  This will lead us to a general programming technique
+called a *monad*, which we'll see in many guises in the weeks to come.
+

 Integer division presupposes that its second argument
 (the divisor) is not zero, upon pain of presupposition failure.
 Here's what my OCaml interpreter says:
 Integer division presupposes that its second argument
 (the divisor) is not zero, upon pain of presupposition failure.
 Here's what my OCaml interpreter says:


@@ -52,12 +58,11 @@ operations.  So we have to jack up the types of the inputs:
 
 <pre>
 let div' (u:int option) (v:int option) =
 
 <pre>
 let div' (u:int option) (v:int option) =

-  match v with
+  match u with

          None -> None
          None -> None

-    | Some 0 -> None
-       | Some y -> (match u with
-                                         None -> None
-                    | Some x -> Some (x / y));;
+       | Some x -> (match v with
+                                 Some 0 -> None
+                               | Some y -> Some (x / y));;

 
 (*
 val div' : int option -> int option -> int option = <fun>
 
 (*
 val div' : int option -> int option -> int option = <fun>


@@ -146,35 +151,6 @@ None objects or real numbers---those details are hidden inside of the
 The definition of `div'` shows exactly what extra needs to be said in
 order to trigger the no-division-by-zero presupposition.
 
 The definition of `div'` shows exactly what extra needs to be said in
 order to trigger the no-division-by-zero presupposition.
 

-For linguists: this is a complete theory of a particularly simply form
-of presupposition projection (every predicate is a hole).
-
-
-
-
-Monads in General
------------------
-
-Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [[Towards Monads]].
-In those notes, we saw a way to separate thinking about error
-conditions (such as trying to divide by zero) from thinking about
-normal arithmetic computations.  We did this by making use of the
-`option` type: in each place where we had something of type `int`, we
-put instead something of type `int option`, which is a sum type
-consisting either of one choice with an `int` payload, or else a `None`
-choice which we interpret as  signaling that something has gone wrong.
-
-The goal was to make normal computing as convenient as possible: when
-we're adding or multiplying, we don't have to worry about generating
-any new errors, so we do want to think about the difference between
-`int`s and `int option`s.  We tried to accomplish this by defining a
-`bind` operator, which enabled us to peel away the `option` husk to get
-at the delicious integer inside.  There was also a homework problem
-which made this even more convenient by mapping any binary operation
-on plain integers into a lifted operation that understands how to deal
-with `int option`s in a sensible way.
-

 [Linguitics note: Dividing by zero is supposed to feel like a kind of
 presupposition failure.  If we wanted to adapt this approach to
 building a simple account of presupposition projection, we would have
 [Linguitics note: Dividing by zero is supposed to feel like a kind of
 presupposition failure.  If we wanted to adapt this approach to
 building a simple account of presupposition projection, we would have


@@ -188,7 +164,30 @@ theory of accommodation, and a theory of the situations in which
 material within the sentence can satisfy presuppositions for other
 material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more
 material within the sentence can satisfy presuppositions for other
 material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more

-sophisticated techniques than the super-simple option monad.]
+sophisticated techniques than the super-simple Option monad.]
+
+
+Monads in General
+-----------------
+
+We've just seen a way to separate thinking about error conditions
+(such as trying to divide by zero) from thinking about normal
+arithmetic computations.  We did this by making use of the `option`
+type: in each place where we had something of type `int`, we put
+instead something of type `int option`, which is a sum type consisting
+either of one choice with an `int` payload, or else a `None` choice
+which we interpret as signaling that something has gone wrong.
+
+The goal was to make normal computing as convenient as possible: when
+we're adding or multiplying, we don't have to worry about generating
+any new errors, so we would rather not think about the difference
+between `int`s and `int option`s.  We tried to accomplish this by
+defining a `bind` operator, which enabled us to peel away the `option`
+husk to get at the delicious integer inside.  There was also a
+homework problem which made this even more convenient by defining a
+`lift` operator that mapped any binary operation on plain integers
+into a lifted operation that understands how to deal with `int
+option`s in a sensible way.

 
 So what exactly is a monad?  We can consider a monad to be a system
 that provides at least the following three elements:
 
 So what exactly is a monad?  We can consider a monad to be a system
 that provides at least the following three elements:


@@ -210,7 +209,12 @@ that provides at least the following three elements:
        discussing earlier (whose value is written `()`). It's also only
        very loosely connected to the "return" keyword in many other
        programming languages like C. But these are the names that the literature
        discussing earlier (whose value is written `()`). It's also only
        very loosely connected to the "return" keyword in many other
        programming languages like C. But these are the names that the literature

-       uses.
+       uses.  [The rationale for "unit" comes from the monad laws
+       (see below), where the unit function serves as an identity,
+       just like the unit number (i.e., 1) serves as the identity
+       object for multiplication.  The rationale for "return" comes
+       from a misguided desire to resonate with C programmers and
+       other imperative types.]

 
        The unit/return operation is a way of lifting an ordinary object into
        the monadic box you've defined, in the simplest way possible. You can think
 
        The unit/return operation is a way of lifting an ordinary object into
        the monadic box you've defined, in the simplest way possible. You can think


@@ -223,7 +227,7 @@ that provides at least the following three elements:
 
        So `unit` is a way to put something inside of a monadic box. It's crucial
        to the usefulness of monads that there will be monadic boxes that
 
        So `unit` is a way to put something inside of a monadic box. It's crucial
        to the usefulness of monads that there will be monadic boxes that

-       aren't the result of that operation. In the option/maybe monad, for
+       aren't the result of that operation. In the Option/Maybe monad, for

        instance, there's also the empty box `None`. In another (whimsical)
        example, you might have, in addition to boxes merely containing integers,
        special boxes that contain integers and also sing a song when they're opened. 
        instance, there's also the empty box `None`. In another (whimsical)
        example, you might have, in addition to boxes merely containing integers,
        special boxes that contain integers and also sing a song when they're opened. 


@@ -232,9 +236,9 @@ that provides at least the following three elements:
        most straightforward way to lift an ordinary value into a monadic value
        of the monadic type in question.
 
        most straightforward way to lift an ordinary value into a monadic value
        of the monadic type in question.
 

-*      Thirdly, an operation that's often called `bind`. This is another
+*      Thirdly, an operation that's often called `bind`. As we said before, this is another

        unfortunate name: this operation is only very loosely connected to
        unfortunate name: this operation is only very loosely connected to

-       what linguists usually mean by "binding." In our option/maybe monad, the
+       what linguists usually mean by "binding." In our Option/Maybe monad, the

        bind operation is:
 
                let bind u f = match u with None -> None | Some x -> f x;;
        bind operation is:
 
                let bind u f = match u with None -> None | Some x -> f x;;


@@ -256,7 +260,7 @@ that provides at least the following three elements:
 
        The guts of the definition of the `bind` operation amount to
        specifying how to unbox the monadic value `u`.  In the `bind`
 
        The guts of the definition of the `bind` operation amount to
        specifying how to unbox the monadic value `u`.  In the `bind`

-       operator for the option monad, we unboxed the monadic value by
+       operator for the Option monad, we unboxed the monadic value by

        matching it with the pattern `Some x`---whenever `u`
        happened to be a box containing an integer `x`, this allowed us to
        get our hands on that `x` and feed it to `f`.
        matching it with the pattern `Some x`---whenever `u`
        happened to be a box containing an integer `x`, this allowed us to
        get our hands on that `x` and feed it to `f`.


@@ -270,26 +274,22 @@ that provides at least the following three elements:
        be defined so as to make sure that the result of `f x` was also
        a singing box. If `f` also wanted to insert a song, you'd have to decide
        whether both songs would be carried through, or only one of them.
        be defined so as to make sure that the result of `f x` was also
        a singing box. If `f` also wanted to insert a song, you'd have to decide
        whether both songs would be carried through, or only one of them.

+        (Are you beginning to realize how wierd and wonderful monads
+       can be?)

 
        There is no single `bind` function that dictates how this must go.
        For each new monadic type, this has to be worked out in an
        useful way.
 
 
        There is no single `bind` function that dictates how this must go.
        For each new monadic type, this has to be worked out in an
        useful way.
 

-So the "option/maybe monad" consists of the polymorphic `option` type, the
+So the "Option/Maybe monad" consists of the polymorphic `option` type, the

 `unit`/return function, and the `bind` function.
 
 
 `unit`/return function, and the `bind` function.
 
 

-A note on notation: Haskell uses the infix operator `>>=` to stand
-for `bind`. Chris really hates that symbol.  Following Wadler, he prefers to
-use an infix five-pointed star &#8902;, or on a keyboard, `*`. Jim on the other hand
-thinks `>>=` is what the literature uses and students won't be able to
-avoid it. Moreover, although &#8902; is OK (though not a convention that's been picked up), overloading the multiplication symbol invites its own confusion
-and Jim feels very uneasy about that. If not `>>=` then we should use
-some other unfamiliar infix symbol (but `>>=` already is such...)
+A note on notation: Haskell uses the infix operator `>>=` to stand for
+`bind`: wherever you see `u >>= f`, that means `bind u f`.
+Wadler uses &#8902;, but that hasn't been widely adopted (unfortunately).

 
 

-In any case, the course leaders will work this out somehow. In the meantime,
-as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
-if you ever see this notation:
+Also, if you ever see this notation:

 
        do
                x <- u
 
        do
                x <- u


@@ -303,17 +303,22 @@ Similarly:
                y <- v
                f x y
 
                y <- v
                f x y
 

-is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x
--> bind v (fun y -> f x y))`. Those who did last week's homework may recognize
-this last expression.
+is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u
+(fun x -> bind v (fun y -> f x y))`. Those who did last week's
+homework may recognize this last expression.  You can think of the
+notation like this: take the singing box `u` and evaluate it (which
+includes listening to the song).  Take the int contained in the
+singing box (the end result of evaluting `u`) and bind the variable
+`x` to that int.  So `x <- u` means "Sing me up an int, which I'll call
+`x`".

 
 (Note that the above "do" notation comes from Haskell. We're mentioning it here
 
 (Note that the above "do" notation comes from Haskell. We're mentioning it here

-because you're likely to see it when reading about monads. It won't work in
+because you're likely to see it when reading about monads. (See our page on [[Translating between OCaml Scheme and Haskell]].) It won't work in

 OCaml. In fact, the `<-` symbol already means something different in OCaml,
 having to do with mutable record fields. We'll be discussing mutation someday
 soon.)
 
 OCaml. In fact, the `<-` symbol already means something different in OCaml,
 having to do with mutable record fields. We'll be discussing mutation someday
 soon.)
 

-As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the list monad. Here the monadic type is:
+As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:

 
        # type 'a list
 
 
        # type 'a list
 


@@ -340,7 +345,7 @@ of `'b list`s into a single `'b list`:
        # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
        - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
 
        # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
        - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
 

-So now we've seen two monads: the option/maybe monad, and the list monad. For any
+So now we've seen two monads: the Option/Maybe monad, and the List monad. For any

 monadic system, there has to be a specification of the complex monad type,
 which will be parameterized on some simpler type `'a`, and the `unit`/return
 operation, and the `bind` operation. These will be different for different
 monadic system, there has to be a specification of the complex monad type,
 which will be parameterized on some simpler type `'a`, and the `unit`/return
 operation, and the `bind` operation. These will be different for different


@@ -360,78 +365,88 @@ Just like good robots, monads must obey three laws designed to prevent
 them from hurting the people that use them or themselves.
 
 *      **Left identity: unit is a left identity for the bind operation.**
 them from hurting the people that use them or themselves.
 
 *      **Left identity: unit is a left identity for the bind operation.**

-       That is, for all `f:'a -> 'a m`, where `'a m` is a monadic
-       type, we have `(unit x) * f == f x`.  For instance, `unit` is itself
+       That is, for all `f:'a -> 'b m`, where `'b m` is a monadic
+       type, we have `(unit x) >>= f == f x`.  For instance, `unit` is itself

        a function of type `'a -> 'a m`, so we can use it for `f`:
 
                # let unit x = Some x;;
                val unit : 'a -> 'a option = <fun>
        a function of type `'a -> 'a m`, so we can use it for `f`:
 
                # let unit x = Some x;;
                val unit : 'a -> 'a option = <fun>

-               # let ( * ) u f = match u with None -> None | Some x -> f x;;
-               val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+               # let ( >>= ) u f = match u with None -> None | Some x -> f x;;
+               val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>

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

 
                # unit 2;;
                - : int option = Some 2
 
                # unit 2;;
                - : int option = Some 2

-               # unit 2 * unit;;
+               # unit 2 >>= unit;;

                - : int option = Some 2
 
                - : int option = Some 2
 

+       Now, for a less trivial instance of a function from `int`s to `int option`s:
+

                # let divide x y = if 0 = y then None else Some (x/y);;
                val divide : int -> int -> int option = <fun>
                # divide 6 2;;
                - : int option = Some 3
                # let divide x y = if 0 = y then None else Some (x/y);;
                val divide : int -> int -> int option = <fun>
                # divide 6 2;;
                - : int option = Some 3

-               # unit 2 * divide 6;;
+               # unit 2 >>= divide 6;;

                - : int option = Some 3
 
                # divide 6 0;;
                - : int option = None
                - : int option = Some 3
 
                # divide 6 0;;
                - : int option = None

-               # unit 0 * divide 6;;
+               # unit 0 >>= divide 6;;

                - : int option = None
 
 
 *      **Associativity: bind obeys a kind of associativity**. Like this:
 
                - : int option = None
 
 
 *      **Associativity: bind obeys a kind of associativity**. Like this:
 

-               (u * f) * g == u * (fun x -> f x * g)
+               (u >>= f) >>= g  ==  u >>= (fun x -> f x >>= g)
+
+       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`.
+
+       Wadler and others try to make this look nicer by phrasing it like this,
+       where U, V, and W are schematic for any expressions with the relevant monadic type:

 
 

-       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`.
+               (U >>= fun x -> V) >>= fun y -> W  ==  U >>= fun x -> (V >>= fun y -> W)

 
 

-       Some examples of associativity in the option monad:
+       Some examples of associativity in the Option monad (bear in
+       mind that in the Ocaml implementation of integer division, 2/3
+       evaluates to zero, throwing away the remainder):

 
 

-               # Some 3 * unit * unit;; 
+               # Some 3 >>= unit >>= unit;; 

                - : int option = Some 3
                - : int option = Some 3

-               # Some 3 * (fun x -> unit x * unit);;
+               # Some 3 >>= (fun x -> unit x >>= unit);;

                - : int option = Some 3
 
                - : int option = Some 3
 

-               # Some 3 * divide 6 * divide 2;;
+               # Some 3 >>= divide 6 >>= divide 2;;

                - : int option = Some 1
                - : int option = Some 1

-               # Some 3 * (fun x -> divide 6 x * divide 2);;
+               # Some 3 >>= (fun x -> divide 6 x >>= divide 2);;

                - : int option = Some 1
 
                - : int option = Some 1
 

-               # Some 3 * divide 2 * divide 6;;
+               # Some 3 >>= divide 2 >>= divide 6;;

                - : int option = None
                - : int option = None

-               # Some 3 * (fun x -> divide 2 x * divide 6);;
+               # Some 3 >>= (fun x -> divide 2 x >>= divide 6);;

                - : int option = None
 
                - : int option = None
 

-Of course, associativity must hold for *arbitrary* functions of
-type `'a -> 'a m`, 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 `u`
-matches `None`, both computations will result in `None`; if
-`u` matches `Some x`, and `f x` evalutes to `None`, then both
-computations will again result in `None`; and if the value of
-`f x` matches `Some y`, then both computations will evaluate
-to `g y`.
+       Of course, associativity must hold for *arbitrary* functions of
+       type `'a -> 'b m`, 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 `u`
+       matches `None`, both computations will result in `None`; if
+       `u` matches `Some x`, and `f x` evalutes to `None`, then both
+       computations will again result in `None`; and if the value of
+       `f x` matches `Some y`, then both computations will evaluate
+       to `g y`.

 
 *      **Right identity: unit is a right identity for bind.**  That is, 
 
 *      **Right identity: unit is a right identity for bind.**  That is, 

-       `u * unit == u` for all monad objects `u`.  For instance,
+       `u >>= unit == u` for all monad objects `u`.  For instance,

 
 

-               # Some 3 * unit;;
+               # Some 3 >>= unit;;

                - : int option = Some 3
                - : int option = Some 3

-               # None * unit;;
+               # None >>= unit;;

                - : 'a option = None
 
 
                - : 'a option = None
 
 


@@ -441,41 +456,47 @@ More details about monads
 If you studied algebra, you'll remember that a *monoid* is an
 associative operation with a left and right identity.  For instance,
 the natural numbers along with multiplication form a monoid with 1
 If you studied algebra, you'll remember that a *monoid* is an
 associative operation with a left and right identity.  For instance,
 the natural numbers along with multiplication form a monoid with 1

-serving as the left and right identity.  That is, temporarily using
-`*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
+serving as the left and right identity.  That is, `1 * u == u == u * 1` for all

 `u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`.  As
 presented here, a monad is not exactly a monoid, because (unlike the
 arguments of a monoid operation) the two arguments of the bind are of
 different types.  But it's possible to make the connection between
 monads and monoids much closer. This is discussed in [Monads in Category
 `u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`.  As
 presented here, a monad is not exactly a monoid, because (unlike the
 arguments of a monoid operation) the two arguments of the bind are of
 different types.  But it's possible to make the connection between
 monads and monoids much closer. This is discussed in [Monads in Category

-Theory](/advanced_notes/monads_in_category_theory).
-See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
+Theory](/advanced_topics/monads_in_category_theory).
+
+See also:
+
+*      [Haskell wikibook on Monad Laws](http://www.haskell.org/haskellwiki/Monad_Laws).
+*      [Yet Another Haskell Tutorial on Monad Laws](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Definition)
+*      [Haskell wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads)
+*      [Haskell wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads)
+*      [Haskell wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation)
+*      [Yet Another Haskell Tutorial on do-notation](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Do_Notation)
+

 
 Here are some papers that introduced monads into functional programming:
 
 
 Here are some papers that introduced monads into functional programming:
 

-*      [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991.
+*      [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible.
+
+*      [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
+invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
+<!--   This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
+       Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
+       The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.-->

 
 *      [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
 in M. Broy, editor, *Marktoberdorf Summer School on Program Design
 Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
 sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
 editors, *Advanced Functional Programming*, Springer Verlag, 
 
 *      [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
 in M. Broy, editor, *Marktoberdorf Summer School on Program Design
 Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
 sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
 editors, *Advanced Functional Programming*, Springer Verlag, 

-LNCS 925, 1995. Some errata fixed August 2001.  This paper has a great first
-line: **Shall I be pure, or impure?**
+LNCS 925, 1995. Some errata fixed August 2001.

 <!--   The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.-->
 
 <!--   The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.-->
 

-*      [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
-invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
-<!--   This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
-       Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
-       The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.-->
-
-*      [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from <https://www.cs.indiana.edu/cgi-pub/c311/doku.php?id=home> but the link above is to a local copy.

 
 

-There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you.

 
 In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category
 
 In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category

-Theory](/advanced_notes/monads_in_category_theory) notes do so, for example.
+Theory](/advanced_topics/monads_in_category_theory) notes do so, for example.

 
 Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
 
 
 Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
 


@@ -493,12 +514,12 @@ that is:
 
 You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
 
 
 You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
 

-The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example:
+The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example:

 
        # let even x = (x mod 2 = 0);;
        val g : int -> bool = <fun>
 
 
        # let even x = (x mod 2 = 0);;
        val g : int -> bool = <fun>
 

-`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad?

 
        # let lift g = fun u -> bind u (fun x -> Some (g x));;
        val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
 
        # let lift g = fun u -> bind u (fun x -> Some (g x));;
        val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>


@@ -511,7 +532,7 @@ also define a lift operation for binary functions:
 
 `lift2 (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.
 
 
 `lift2 (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.
 

-The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`!

 
 Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
 
 
 Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
 


@@ -538,7 +559,7 @@ and so on. Here are the laws that any `ap` operation can be relied on to satisfy
        ap (unit f) (unit x) = unit (f x)
        ap u (unit x) = ap (unit (fun f -> f x)) u
 
        ap (unit f) (unit x) = unit (f x)
        ap u (unit x) = ap (unit (fun f -> f x)) u
 

-Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the list monad, there was a step where
+Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where

 we went from:
 
        [[1]; [1;2]; [1;3]; [1;2;4]]
 we went from:
 
        [[1]; [1;2]; [1;3]; [1;2;4]]


@@ -568,15 +589,15 @@ Monad outlook
 -------------
 
 We're going to be using monads for a number of different things in the
 -------------
 
 We're going to be using monads for a number of different things in the

-weeks to come.  The first main application will be the State monad,
+weeks to come.  One major application will be the State monad,

 which will enable us to model mutation: variables whose values appear
 to change as the computation progresses.  Later, we will study the
 Continuation monad.
 
 which will enable us to model mutation: variables whose values appear
 to change as the computation progresses.  Later, we will study the
 Continuation monad.
 

-In the meantime, we'll look at several linguistic applications for monads, based
-on what's called the *reader monad*.
+But first, we'll look at several linguistic applications for monads, based
+on what's called the *Reader monad*.

 
 

-##[[Reader monad]]##
+##[[Reader Monad for Variable Binding]]##

 
 

-##[[Intensionality monad]]##
+##[[Reader Monad for Intensionality]]##