[[!toc]]
-Monads
-------
+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:
+
+ # 12/0;;
+ Exception: Division_by_zero.
+
+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:
+
+ # type 'a option = None | Some of 'a;;
+ # None;;
+ - : 'a option = None
+ # Some 3;;
+ - : int option = Some 3
+
+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.
+
+<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>
+
+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:
+
+<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>
+
+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.
+
+I prefer to line up the `match` alternatives by using OCaml's
+built-in tuple type:
+
+<pre>
+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>
+
+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:
+
+<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.
+
+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 bind' (u: int option) (f: int -> (int option)) =
+ match u with
+ None -> None
+ | Some x -> f x;;
+
+let add' (u: int option) (v: int option) =
+ bind' u (fun x -> bind' v (fun y -> Some (x + y)));;
+
+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)));;
+
+(*
+# div' (div' (Some 12) (Some 2)) (Some 3);;
+- : int option = Some 2
+# div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+# add' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
+</pre>
+
+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.
+
+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 monad: in each place where we had something of type `int`, we
+`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 just an integer, or else some special value which
-we could interpret as signaling that something had gone wrong.
+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
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
+So what exactly is a monad? We can consider a monad to be a system
that provides at least the following three elements:
* A complex type that's built around some more basic type. Usually
- it will be polymorphic, and so can apply to different basic types.
+ 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 `x` has type `int option`,
- then `x` is a box that (may) contain an integer.
+ 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 `n` by mapping it to
- the option `Some n`. In the general case, this operation is
+ 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
what linguists usually mean by "binding." In our option/maybe monad, the
bind operation is:
- let bind m f = match m with None -> None | Some n -> f n;;
+ 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
+ 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 options
- and end with bool options).
+ 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 like this:
- the first argument is a monadic value m, which
+ 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 m f = (let x = unbox m in f x);;
+ 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 `m`. In the bind
- opertor for the option monad, we unboxed the option monad by
- matching the monadic value `m` with `Some n`---whenever `m`
- happened to be a box containing an integer `n`, this allowed us to
- get our hands on that `n` and feed it to `f`.
+ 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, then
- we just pass through the empty box unaltered.
+ 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 `n`, then the `bind` operation would probably
- be defined so as to make sure that the result of `f n` was also
- a singing box. If `f` also inserted a song, you'd have to decide
+ 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.
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. With the option monad, we can
-think of the "safe division" operation
-
-<pre>
-# let divide' num den = if den = 0 then None else Some (num/den);;
-val divide' : int -> int -> int option = <fun>
-</pre>
+So the "option/maybe monad" consists of the polymorphic `option` type, the
+`unit`/return function, and the `bind` function.
-as basically a function from two integers to an integer, except with
-this little bit of option plumbing on the side.
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, or on a keyboard, `*`. Jim on the other hand
+use an infix five-pointed star ⋆, 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 ⋆ 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...)
In any case, the course leaders will work this out somehow. In the meantime,
-as you read around, wherever you see `m >>= f`, that means `bind m f`. Also,
+as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
if you ever see this notation:
do
- x <- m
+ x <- u
f x
-That's a Haskell shorthand for `m >>= (\x -> f x)`, that is, `bind m f`.
+That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
Similarly:
do
- x <- m
- y <- n
+ x <- u
+ y <- v
f x y
-is shorthand for `m >>= (\x -> n >>= (\y -> f x y))`, that is, `bind m (fun x
--> bind n (fun y -> f x y))`. Those who did last week's homework may recognize
-this.
+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.
(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
# type 'a list
-The unit/return operation is:
+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:
+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 m f = List.concat (List.map f m);;
+ # 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 m) first. This goes through all
-the members of the list m. There may be just a single member, if `m = unit a`
-for some a. 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.
+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 lists. The surrounding `List.concat ( )` converts that bunch
-of 'b lists into a single 'b list:
+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
+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
+Although the `unit` and `bind` operation are defined differently for different
monadic systems, there are some general rules they always have to follow.
* **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
- value, we have `(unit x) * f == f x`. For instance, `unit` is itself
+ 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`:
-<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 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>
-# unit 2;;
-- : int option = Some 2
-# unit 2 * unit;;
-- : int option = Some 2
+ 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:
-# divide 6 2;;
-- : int option = Some 3
-# unit 2 * divide 6;;
-- : int option = Some 3
+ # unit 2;;
+ - : int option = Some 2
+ # unit 2 * unit;;
+ - : int option = Some 2
-# divide 6 0;;
-- : int option = None
-# unit 0 * divide 6;;
-- : int option = None
-</pre>
+ # 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
-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:
- (m * f) * g == m * (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.
+ x" part), you need to look again at the type of `bind`.
Some examples of associativity in the option monad:
-<pre>
-# Some 3 * unit * unit;;
-- : int option = Some 3
-# Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
+ # 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 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
-</pre>
+ # 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
+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 `m`
+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,
+ `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
More details about monads
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
+`*` to mean arithmetic multiplication, `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 [[Advanced Notes/Monads in Category Theory]]. See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
+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>.
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).
+* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991.
* [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.
- 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 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.-->
* [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.
+<!-- 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.
+ 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.-->
-There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+* [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.
-In the presentation we gave above, which is the usual one in functional programming, 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, and define for example bind in terms of them. This is discussed some in the [[Advanced Notes/Monads in Category Theory]].
+There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+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.
-Here are some of the specifics. 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.
You may sometimes see:
- m >> n
+ u >> v
That just means:
- m >>= fun _ -> n
+ u >>= fun _ -> v
that is:
- bind m (fun _ -> n)
+ bind u (fun _ -> v)
-You could also do `bind m (fun x -> n)`; 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>
-`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 m -> bind m (fun x -> Some (g x));;
+ # let lift g = fun u -> bind u (fun x -> Some (g x));;
val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
`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:
- # let lift2 g = fun m n -> bind m (fun x -> bind n (fun y -> Some (g x y)));;
+ # 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>
-`lift (+)` 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 `<$>`.) And indeed when we're working with list monad, `lift f` is exactly `List.map f`!
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) 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 pre-defined `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.
In general, any lift/map operation can be relied on to satisfy these laws:
* lift id = id
* lift (compose f g) = compose (lift f) (lift g)
-where id is `fun x -> x` and `compose a b` is `fun x -> a (b 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
+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:
List.map (fun x -> f (g x)) lst
-
List.map f (List.map g lst)
Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
ap [f] [x; y] = [f x; f y]
ap (Some f) (Some x) = Some (f x)
-and so on. Here are the laws that any ap operation can be relied on to satisfy:
+and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
- ap (unit id) v = v
- ap (ap (ap (return compose) u) v) w = ap u (ap v 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 y) = ap (unit (fun f -> f y)) u
+ 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
we went from:
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:
- lift f m = ap (unit f) m
- lift2 f m n = ap (lift f m) n = ap (ap (unit f) m) n
- ap m n = lift2 id m n
- lift f m = m >>= compose unit f
- lift f m = ap (unit f) m
+ 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
- m >>= f = join (lift f m)
- m >> n = m >>= (fun _ -> n)
- m >> n = lift2 (fun _ -> id) m n
+ u >>= f = join (lift f u)
+ u >> v = u >>= (fun _ -> v)
+ u >> v = lift2 (fun _ -> id) u v
In the meantime, we'll look at several linguistic applications for monads, based
on what's called the *reader monad*.
+##[[Reader monad]]##
-The reader monad
-----------------
-
-Introduce
-
-Heim and Kratzer's "Predicate Abstraction Rule"
+##[[Intensionality monad]]##
-
-
-The intensionality monad
-------------------------
-...
-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.
-
-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";;
-
-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
-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>
-
-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.
-
-<pre>
-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
-
-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-</pre>
-
-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.
-
-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.
-
-Let's test compliance with the left identity law:
-
-<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>
-
-We'll assume that this and the other laws always hold.
-
-We now build up some extensional meanings:
-
- let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
-
-This function says that everyone always left, except for Cam in world
-2 (i.e., `left 2 'c' == false`).
-
-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:
-
-<pre>
-let extapp fn arg w = fn w (arg w);;
-
-extapp left ann 1;;
-# - : bool = true
-
-extapp left cam 2;;
-# - : bool = false
-</pre>
-
-`extapp` stands for "extensional function application".
-So Ann left in world 1, but Cam didn't leave in world 2.
-
-A transitive predicate:
-
- let saw w x y = (w < 2) && (y < x);;
- extapp (extapp saw bill) ann 1;; (* true *)
- extapp (extapp saw bill) ann 2;; (* false *)
-
-In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone
-in world two.
-
-Good. Now for intensions:
-
- let intapp fn arg w = fn w arg;;
-
-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.
-
-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:
-
-<pre>
-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
-
-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>
-
-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.)
-
-Likewise for extensional transitive predicates like "saw":
-
-<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>
-
-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"):
-
-<pre>
-let think (w:s) (p:s->t) (x:e) =
- match (x, p 2) with ('a', false) -> false | _ -> p w;;
-</pre>
-
-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.
-
-<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.
-
-<pre>
-intapp (lift (intapp think
- (intapp (lift left)
- (unit 'c'))))
- (unit 'a')
-1;; (* false *)
-</pre>
-
-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.
-
-*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.
-
-Finally, note that within an intensional grammar, extensional funtion
-application is essentially just bind:
-
-<pre>
-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-</pre>