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 [[!toc]]
 
-Monads
-------
-
-Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
-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 monad: 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 just an integer, or else some special value which
-we could interpret as signaling that something had 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
-to do several things.  First, we would have to make use of the
-polymorphism of the `option` type.  In the arithmetic example, we only
-made use of `int option`s, but when we're composing natural language
-expression meanings, we'll need to use types like `N int`, `Det Int`,
-`VP int`, and so on.  But that works automatically, because we can use
-any type for the `'a` in `'a option`.  Ultimately, we'd want to have a
-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
-sophisticated techniques than the super-simple option monad.]
-
-So what exactly is a monad?  As usual, we're not going to be pedantic
-about it, but for our purposes, we can consider a monad to be a system
-that provides at least the following three elements:
+Towards Monads: Safe division
+-----------------------------
 
-* A way to build a complex type from some basic type.  In the division
-  example, the polymorphism of the `'a option` type provides a way of
-  building an option out of any other type of object.  People often
-  use a container metaphor: if `x` has type `int option`, then `x` is
-  a box that (may) contain an integer.
+[This section used to be near the end of the lecture notes for week 6]
 
-    `type 'a option = None | Some of 'a;;`
+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.
 
-* A way to turn an ordinary value into a monadic value.  In Ocaml, we
-  did this for any integer `n` by mapping it to
-  the option `Some n`.  To be official, we can define a function
-  called unit:
+Integer division presupposes that its second argument
+(the divisor) is not zero, upon pain of presupposition failure.
+Here's what my OCaml interpreter says:
 
-    `let unit x = Some x;;`
+    # 12/0;;
+    Exception: Division_by_zero.
 
-    `val unit : 'a -> 'a option = <fun>`
+So we want to explicitly allow for the possibility that
+division will return something other than a number.
+We'll use OCaml's `option` type, which works like this:
 
-    So `unit` is a way to put something inside of a box.
+    # type 'a option = None | Some of 'a;;
+    # None;;
+    - : 'a option = None
+    # Some 3;;
+    - : int option = Some 3
 
-* A bind operation (note the type):
+So if a division is normal, we return some number, but if the divisor is
+zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
 
-     `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
-     object.
+<pre>
+let div' (x:int) (y:int) =
+  match y with
+	  0 -> None
+    | _ -> Some (x / y);;
+
+(*
+val div' : int -> int -> int option = fun
+# div' 12 2;;
+- : int option = Some 6
+# div' 12 0;;
+- : int option = None
+# div' (div' 12 2) 3;;
+Characters 4-14:
+  div' (div' 12 2) 3;;
+        ^^^^^^^^^^
+Error: This expression has type int option
+       but an expression was expected of type int
+*)
+</pre>
 
-     Intuitively, the interpretation of what `bind` does is like this:
-     the first argument computes a monadic object m, which will
-     evaluate to a box containing some ordinary value, call it `x`.
-     Then the second argument uses `x` to compute a new monadic
-     value.  Conceptually, then, we have
+This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
+just the behavior we were hoping for.  But we want to be able to use
+the output of the safe-division function as input for further division
+operations.  So we have to jack up the types of the inputs:
 
-    `let bind m f = (let x = unwrap m in f x);;`
+<pre>
+let div' (u:int option) (v:int option) =
+  match v with
+	  None -> None
+    | Some 0 -> None
+	| Some y -> (match u with
+					  None -> None
+                    | Some x -> Some (x / y));;
+
+(*
+val div' : int option -> int option -> int option = <fun>
+# div' (Some 12) (Some 2);;
+- : int option = Some 6
+# div' (Some 12) (Some 0);;
+- : int option = None
+# div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
+</pre>
 
-    The guts of the definition of the `bind` operation amount to
-    specifying how to unwrap the monadic object `m`.  In the bind
-    opertor for the option monad, we unwraped the option monad by
-    matching the monadic object `m` with `Some n`--whenever `m`
-    happend to be a box containing an integer `n`, this allowed us to
-    get our hands on that `n` and feed it to `f`.
+Beautiful, just what we need: now we can try to divide by anything we
+want, without fear that we're going to trigger any system errors.
 
-So the "option monad" consists of the polymorphic option type, the
-unit function, and the bind function.  With the option monad, we can
-think of the "safe division" operation
+I prefer to line up the `match` alternatives by using OCaml's
+built-in tuple type:
 
 <pre>
-# let divide num den = if den = 0 then None else Some (num/den);;
-val divide : int -> int -> int option = <fun>
+let div' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (_, Some 0) -> None
+	| (Some x, Some y) -> Some (x / y);;
 </pre>
 
-as basically a function from two integers to an integer, except with
-this little bit of option frill, or option plumbing, on the side.
+So far so good.  But what if we want to combine division with
+other arithmetic operations?  We need to make those other operations
+aware of the possibility that one of their arguments has triggered a
+presupposition failure:
 
-A note on notation: Haskell uses the infix operator `>>=` to stand
-for `bind`.  I really hate that symbol.  Following Wadler, I prefer to
-infix five-pointed star, or on a keyboard, `*`.
+<pre>
+let add' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (Some x, Some y) -> Some (x + y);;
+
+(*
+val add' : int option -> int option -> int option = <fun>
+# add' (Some 12) (Some 4);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
 
+This works, but is somewhat disappointing: the `add'` operation
+doesn't trigger any presupposition of its own, so it is a shame that
+it needs to be adjusted because someone else might make trouble.
 
-The Monad laws
---------------
-
-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.
-     That is, for all `f:'a -> 'a M`, where `'a M` is a monadic
-     object, we have `(unit x) * f == f x`.  For instance, `unit` is a
-     function of type `'a -> 'a option`, so we have
+But we can automate the adjustment.  The standard way in OCaml,
+Haskell, etc., is to define a `bind` operator (the name `bind` is not
+well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
 
 <pre>
-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
-# let unit x = Some x;;
-val unit : 'a -> 'a option = <fun>
+let bind' (u: int option) (f: int -> (int option)) =
+  match u with
+	  None -> None
+    | Some x -> f x;;
 
-# unit 2;;
-- : int option = Some 2
-# unit 2 * unit;;
-- : int option = Some 2
+let add' (u: int option) (v: int option)  =
+  bind' u (fun x -> bind' v (fun y -> Some (x + y)));;
 
-# divide 6 2;;
-- : int option = Some 3
-# unit 2 * divide 6;;
-- : int option = Some 3
+let div' (u: int option) (v: int option) =
+  bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;
 
-# divide 6 0;;
+(*
+#  div' (div' (Some 12) (Some 2)) (Some 3);;
+- : int option = Some 2
+#  div' (div' (Some 12) (Some 0)) (Some 3);;
 - : int option = None
-# unit 0 * divide 6;;
+# add' (div' (Some 12) (Some 0)) (Some 3);;
 - : int option = None
+*)
 </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`.
+Compare the new definitions of `add'` and `div'` closely: the definition
+for `add'` shows what it looks like to equip an ordinary operation to
+survive in dangerous presupposition-filled world.  Note that the new
+definition of `add'` does not need to test whether its arguments are
+None objects or real numbers---those details are hidden inside of the
+`bind'` function.
+
+The definition of `div'` shows exactly what extra needs to be said in
+order to trigger the no-division-by-zero presupposition.
 
-*    Associativity: bind obeys a kind of associativity, like this:
+[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
+to do several things.  First, we would have to make use of the
+polymorphism of the `option` type.  In the arithmetic example, we only
+made use of `int option`s, but when we're composing natural language
+expression meanings, we'll need to use types like `N option`, `Det option`,
+`VP option`, and so on.  But that works automatically, because we can use
+any type for the `'a` in `'a option`.  Ultimately, we'd want to have a
+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
+sophisticated techniques than the super-simple option monad.]
 
-    `(m * f) * g == m * (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.
+Monads in General
+-----------------
 
-    Some examples of associativity in the option monad:
+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.
 
-<pre>
-# Some 3 * unit * unit;; 
-- : int option = Some 3
-# Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
+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:
 
-# Some 3 * divide 6 * divide 2;;
-- : int option = Some 1
-# Some 3 * (fun x -> divide 6 x * divide 2);;
-- : int option = Some 1
+*	A complex type that's built around some more basic type. Usually
+	the complex type will be polymorphic, and so can apply to different basic types.
+	In our division example, the polymorphism of the `'a option` type
+	provides a way of building an option out of any other type of object.
+	People often use a container metaphor: if `u` has type `int option`,
+	then `u` is a box that (may) contain an integer.
+
+		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 `x` by mapping it to
+	the option `Some x`.  In the general case, this operation is
+	known as `unit` or `return.` Both of those names are terrible. This
+	operation is only very loosely connected to the `unit` type we were
+	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.  [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
+	of the singleton function as an example: it takes an ordinary object
+	and returns a set containing that object. In the example we've been
+	considering:
+
+		let unit x = Some x;;
+		val unit : 'a -> 'a option = <fun>
+
+	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
+	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. 
+
+	The unit/return operation will always be the simplest, conceptually
+	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
+	unfortunate name: this operation is only very loosely connected to
+	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;;
+		val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+
+	Note the type: `bind` takes two arguments: first, a monadic box
+	(in this case, an `'a option`); and second, a function from
+	ordinary objects to monadic boxes. `bind` then returns a monadic
+	value: in this case, a `'b option` (you can start with, e.g., `int option`s
+	and end with `bool option`s).
+
+	Intuitively, the interpretation of what `bind` does is this:
+	the first argument is a monadic value `u`, which 
+	evaluates to a box that (maybe) contains some ordinary value, call it `x`.
+	Then the second argument uses `x` to compute a new monadic
+	value.  Conceptually, then, we have
+
+		let bind u f = (let x = unbox u in f x);;
+
+	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
+	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`.
+
+	If the monadic box didn't contain any ordinary value,
+	we instead pass through the empty box unaltered.
+
+	In a more complicated case, like our whimsical "singing box" example
+	from before, if the monadic value happened to be a singing box
+	containing an integer `x`, then the `bind` operation would probably
+	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.
+
+So the "option/maybe monad" consists of the polymorphic `option` type, the
+`unit`/return function, and the `bind` function.
+
+
+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).
+
+Also, if you ever see this notation:
+
+	do
+		x <- u
+		f x
+
+That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
+Similarly:
+
+	do
+		x <- u
+		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.  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
+because you're likely to see it when reading about monads. 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.)
+
+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
+
+The `unit`/return operation is:
+
+	# let unit x = [x];;
+	val unit : 'a -> 'a list = <fun>
+
+That is, the simplest way to lift an `'a` into an `'a list` is just to make a
+singleton list of that `'a`. Finally, the `bind` operation is:
+
+	# let bind u f = List.concat (List.map f u);;
+	val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
+	
+What's going on here? Well, consider `List.map f u` first. This goes through all
+the members of the list `u`. There may be just a single member, if `u = unit x`
+for some `x`. Or on the other hand, there may be no members, or many members. In
+any case, we go through them in turn and feed them to `f`. Anything that gets fed
+to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`.
+For example, it might return a list of all that value's divisors. Then we'll
+have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch
+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]
+
+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 systems.
+
+Many monadic systems will also define special-purpose operations that only make
+sense for that system.
+
+Although the `unit` and `bind` operation are defined differently for different
+monadic systems, there are some general rules they always have to follow.
 
-# Some 3 * divide 2 * divide 6;;
-- : int option = None
-# Some 3 * (fun x -> divide 2 x * divide 6);;
-- : int option = None
-</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`
+The Monad Laws
+--------------
+
+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.**
+	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
+	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>
+
+	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 equivalently `( >>= ) u f` instead of `>>= u
+	f`. Now, for a less trivial instance of a function from `int`s
+	to `int option`s:
+
+		# unit 2;;
+		- : int option = Some 2
+		# unit 2 >>= unit;;
+		- : int option = Some 2
+
+		# 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;;
+		- : int option = Some 3
+
+		# divide 6 0;;
+		- : int option = None
+		# unit 0 >>= divide 6;;
+		- : int option = None
+
+
+*	**Associativity: bind obeys a kind of associativity**. Like this:
+
+		(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`.
+
+	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;; 
+		- : int option = Some 3
+		# Some 3 >>= (fun x -> unit x >>= unit);;
+		- : int option = Some 3
+
+		# Some 3 >>= divide 6 >>= divide 2;;
+		- : int option = Some 1
+		# Some 3 >>= (fun x -> divide 6 x >>= divide 2);;
+		- : int option = Some 1
+
+		# Some 3 >>= divide 2 >>= divide 6;;
+		- : int option = None
+		# Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
+		- : 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
-`m` matches `Some n`, and `f n` evalutes to `None`, then both
+`u` matches `Some x`, and `f x` 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`.
+`f x` matches `Some y`, then both computations will evaluate
+to `g y`.
 
-*    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, 
+	`u >>= unit == u` for all monad objects `u`.  For instance,
 
-<pre>
-# Some 3 * unit;;
-- : int option = Some 3
-# None * unit;;
-- : 'a option = None
-</pre>
+		# Some 3 >>= unit;;
+		- : int option = Some 3
+		# None >>= unit;;
+		- : 'a option = None
 
-Now, if you studied algebra, you'll remember that a *monoid* is an
+
+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
-serving as the left and right identity.  That is, temporarily using
-`*` to mean arithmetic multiplication, `1 * n == n == n * 1` for all
-`n`, and `(a * b) * c == a * (b * c)` for all `a`, `b`, and `c`.  As
+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 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>, near the bottom).
+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_topics/monads_in_category_theory).
+See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
 
+Here are some papers that introduced monads into functional programming:
 
-Monad outlook
--------------
+*	[Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991.
 
-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,
-which will enable us to model mutation: variables whose values appear
-to change as the computation progresses.  Later, we will study the
-Continuation monad.
-
-The intensionality monad
-------------------------
+*	[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?**
+<!--	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.-->
 
-In the meantime, we'll see a linguistic application for monads:
-intensional function application.  In Shan (2001) [Monads for natural
-language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that
-making expressions sensitive to the world of evaluation is
-conceptually the same thing as making use of a *reader monad* (which
-we'll see again soon).  This technique was beautifully re-invented
-by Ben-Avi and Winter (2007) in their paper [A modular
-approach to
-intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
-though without explicitly using monads.
+*	[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.-->
 
-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 
+*	[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.
 
-    # #use "intensionality-monad.ml";;
+There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes us laugh.
 
-Note the extra `#` attached to the directive `use`.
+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_topics/monads_in_category_theory) notes do so, for example.
 
-Here's the idea: since people can have different attitudes towards
-different propositions that happen to have the same truth value, we
-can't have sentences denoting simple truth values.  If we did, then if John
-believed that the earth was round, it would force him to believe
-Fermat's last theorem holds, since both propositions are equally true.
-The traditional solution is to allow sentences to denote a function
-from worlds to truth values, what Montague called an intension.  
-So if `s` is the type of possible worlds, we have the following
-situation:
-
-
-<pre>
-Extensional types                 Intensional types       Examples
--------------------------------------------------------------------
-
-S         s->t                    s->t                    John left
-DP        s->e                    s->e                    John
-VP        s->e->t                 s->(s->e)->t            left
-Vt        s->e->e->t              s->(s->e)->(s->e)->t    saw
-Vs        s->t->e->t              s->(s->t)->(s->e)->t    thought
-</pre>
+Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
 
-This system is modeled on the way Montague arranged his grammar.
-There are significant simplifications: for instance, determiner
-phrases are thought of as corresponding to individuals rather than to
-generalized quantifiers.  If you're curious about the initial `s`'s
-in the extensional types, they're there because the behavior of these
-expressions depends on which world they're evaluated at.  If you are
-in a situation in which you can hold the evaluation world constant,
-you can further simplify the extensional types.  Usually, the
-dependence of the extension of an expression on the evaluation world
-is hidden in a superscript, or built into the lexical interpretation
-function.
-
-The main difference between the intensional types and the extensional
-types is that in the intensional types, the arguments are functions
-from worlds to extensions: intransitive verb phrases like "left" now
-take intensional concepts as arguments (type s->e) rather than plain
-individuals (type e), and attitude verbs like "think" now take
-propositions (type s->t) rather than truth values (type t).
-
-The intenstional types are more complicated than the intensional
-types.  Wouldn't it be nice to keep the complicated types to just
-those attitude verbs that need to worry about intensions, and keep the
-rest of the grammar as extensional as possible?  This desire is
-parallel to our earlier desire to limit the concern about division by
-zero to the division function, and let the other functions, like
-addition or multiplication, ignore division-by-zero problems as much
-as possible.
-
-So here's what we do:
-
-In Ocaml, we'll use integers to model possible worlds:
-
-    type s = int;;
-    type e = char;;
-    type t = bool;;
-
-Characters (characters in the computational sense, i.e., letters like
-`'a'` and `'b'`, not Kaplanian characters) will model individuals, and
-Ocaml booleans will serve for truth values.
+You may sometimes see:
 
-<pre>
-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
+	u >> v
 
-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-</pre>
+That just means:
 
-In our monad, the intension of an extensional type `'a` is `s -> 'a`,
-a function from worlds to extensions.  Our unit will be the constant
-function (an instance of the K combinator) that returns the same
-individual at each world.
+	u >>= fun _ -> v
 
-Then `ann = unit 'a'` is a rigid designator: a constant function from
-worlds to individuals that returns `'a'` no matter which world is used
-as an argument.
+that is:
 
-Let's test compliance with the left identity law:
+	bind u (fun _ -> v)
 
-<pre>
-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = <fun>
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-</pre>
+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.
 
-We'll assume that this and the other laws always hold.
+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:
 
-We now build up some extensional meanings:
+	# let even x = (x mod 2 = 0);;
+	val g : int -> bool = <fun>
 
-    let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
 
-This function says that everyone always left, except for Cam in world
-2 (i.e., `left 2 'c' == false`).
+	# let lift g = fun u -> bind u (fun x -> Some (g x));;
+	val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
 
-Then the way to evaluate an extensional sentence is to determine the
-extension of the verb phrase, and then apply that extension to the
-extension of the subject:
+`lift even` will now be a function from `int option`s to `bool option`s. We can
+also define a lift operation for binary functions:
 
-<pre>
-let extapp fn arg w = fn w (arg w);;
+	# let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));;
+	val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = <fun>
 
-extapp left ann 1;;
-# - : bool = true
+`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.
 
-extapp left cam 2;;
-# - : bool = false
-</pre>
+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`!
 
-`extapp` stands for "extensional function application".
-So Ann left in world 1, but Cam didn't leave in world 2.
+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.
 
-A transitive predicate:
+In general, any lift/map operation can be relied on to satisfy these laws:
 
-    let saw w x y = (w < 2) && (y < x);;
-    extapp (extapp saw bill) ann 1;; (* true *)
-    extapp (extapp saw bill) ann 2;; (* false *)
+	* lift id = id
+	* lift (compose f g) = compose (lift f) (lift g)
 
-In world 1, Ann saw Bill and Cam, and Bill saw Cam.  No one saw anyone
-in world two.
+where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these
+two computations to give the same result:
 
-Good.  Now for intensions:
+	List.map (fun x -> f (g x)) lst
+	List.map f (List.map g lst)
 
-    let intapp fn arg w = fn w arg;;
+Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
 
-The only difference between intensional application and extensional
-application is that we don't feed the evaluation world to the argument.
-(See Montague's rules of (intensional) functional application, T4 -- T10.)
-In other words, instead of taking an extension as an argument,
-Montague's predicates take a full-blown intension.  
+	ap [f] [x; y] = [f x; f y]
+	ap (Some f) (Some x) = Some (f x)
 
-But for so-called extensional predicates like "left" and "saw", 
-the extra power is not used.  We'd like to define intensional versions
-of these predicates that depend only on their extensional essence.
-Just as we used bind to define a version of addition that interacted
-with the option monad, we now use bind to intensionalize an
-extensional verb:
+and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
 
-<pre>
-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+	ap (unit id) u = u
+	ap (ap (ap (unit compose) u) v) w = ap u (ap v w)
+	ap (unit f) (unit x) = unit (f x)
+	ap u (unit x) = ap (unit (fun f -> f x)) u
 
-intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
-intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
-</pre>
+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:
 
-Because `bind` unwraps the intensionality of the argument, when the
-lifted "left" receives an individual concept (e.g., `unit 'a'`) as
-argument, it's the extension of the individual concept (i.e., `'a'`)
-that gets fed to the basic extensional version of "left".  (For those
-of you who know Montague's PTQ, this use of bind captures Montague's
-third meaning postulate.)
+	[[1]; [1;2]; [1;3]; [1;2;4]]
 
-Likewise for extensional transitive predicates like "saw":
+to:
 
-<pre>
-let lift2 pred w arg1 arg2 = 
-  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
-intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
-intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
-</pre>
+	[1; 1; 2; 1; 3; 1; 2; 4]
 
-Crucially, an intensional predicate does not use `bind` to consume its
-arguments.  Attitude verbs like "thought" are intensional with respect
-to their sentential complement, but extensional with respect to their
-subject (as Montague noticed, almost all verbs in English are
-extensional with respect to their subject; a possible exception is "appear"):
+That is the `join` operation.
 
-<pre>
-let think (w:s) (p:s->t) (x:e) = 
-  match (x, p 2) with ('a', false) -> false | _ -> p w;;
-</pre>
+All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions:
 
-Ann disbelieves any proposition that is false in world 2.  Apparently,
-she firmly believes we're in world 2.  Everyone else believes a
-proposition iff that proposition is true in the world of evaluation.
+	lift f u = u >>= compose unit f
+	lift f u = ap (unit f) u
+	lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y)))
+	lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v
+	ap u v = u >>= (fun f -> lift f v)
+	ap u v = lift2 id u v
+	join m2 = m2 >>= id
+	u >>= f = join (lift f u)
+	u >> v = u >>= (fun _ -> v)
+	u >> v = lift2 (fun _ -> id) u v
 
-<pre>
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'b'))))
-       (unit 'a') 
-1;; (* true *)
-</pre>
 
-So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
 
-The `lift` is there because "think Bill left" is extensional wrt its
-subject.  The important bit is that "think" takes the intension of
-"Bill left" as its first argument.
+Monad outlook
+-------------
 
-<pre>
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'c'))))
-       (unit 'a') 
-1;; (* false *)
-</pre>
+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,
+which will enable us to model mutation: variables whose values appear
+to change as the computation progresses.  Later, we will study the
+Continuation monad.
 
-But even in world 1, Ann doesn't believe that Cam left (even though he
-did: `intapp (lift left) cam 1 == true`).  Ann's thoughts are hung up
-on what is happening in world 2, where Cam doesn't leave.
+In the meantime, we'll look at several linguistic applications for monads, based
+on what's called the *reader monad*.
 
-*Small project*: add intersective ("red") and non-intersective
- adjectives ("good") to the fragment.  The intersective adjectives
- will be extensional with respect to the nominal they combine with
- (using bind), and the non-intersective adjectives will take
- intensional arguments.
+##[[Reader monad]]##
 
-Finally, note that within an intensional grammar, extensional funtion
-application is essentially just bind:
+##[[Intensionality monad]]##
 
-<pre>
-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-</pre>