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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 thining 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
-ints and int options. 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 bindary operation
-on plain integers into a lifted operation that understands how to deal
-with int options in a sensible way.
+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.
+
+
+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
+*)
+
+
+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 div' (u:int option) (v:int option) =
+ match u with
+ None -> None
+ | Some x -> (match v with
+ Some 0 -> None
+ | Some y -> Some (x / y));;
+
+(*
+val div' : int option -> int option -> int option =
+# 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
+*)
+
+
+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:
+
+
+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);;
+
+
+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:
+
+
+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 =
+# add' (Some 12) (Some 4);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+
+
+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.
+
+
+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
+*)
+
+
+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.
[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 options, 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
+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
@@ -38,377 +166,424 @@ 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 examctly 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:
-
-* 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.
- `type 'a option = None | Some of 'a;;`
+Monads in General
+-----------------
-* A way to turn an ordinary value into a monadic value. In Ocaml, we
- did this for any integer n by mapping an arbitrary integer `n` to
- the option `Some n`. To be official, we can define a function
- called unit:
+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.
- `let unit x = Some x;;`
+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:
- `val unit : 'a -> 'a option = `
+* 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 =
+
+ 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`. As we said before, 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 =
+
+ 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 ⋆, 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 =
+
+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 =
+
+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.
- So `unit` is a way to put something inside of a box.
+Although the `unit` and `bind` operation are defined differently for different
+monadic systems, there are some general rules they always have to follow.
-* A bind operation (note the type):
- `let bind m f = match m with None -> None | Some n -> f n;;`
+The Monad Laws
+--------------
- `val bind : 'a option -> ('a -> 'b option) -> 'b option = `
+Just like good robots, monads must obey three laws designed to prevent
+them from hurting the people that use them or themselves.
- `bind` takes two arguments (a monadic object and a function from
- ordinary objects to monadic objects), and returns a monadic
- object.
+* **Left identity: unit is a left identity for the bind operation.**
+ 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`:
- 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
+ # let unit x = Some x;;
+ val unit : 'a -> 'a option =
+ # let ( >>= ) u f = match u with None -> None | Some x -> f x;;
+ val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option =
- `let bind m f = (let x = unwrap m in f x);;`
+ 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`.
- 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`.
+ # unit 2;;
+ - : int option = Some 2
+ # unit 2 >>= unit;;
+ - : int option = Some 2
-So the "Option monad" consists of the polymorphic option type, the
-unit function, and the bind function.
+ Now, for a less trivial instance of a function from `int`s to `int option`s:
-A note on notation: some people use 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, `*`.
+ # let divide x y = if 0 = y then None else Some (x/y);;
+ val divide : int -> int -> int option =
+ # 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 Monad laws
---------------
-Just like good robots, monads must obey three laws designed to prevent
-them from hurting the people that use them or themselves.
+* **Associativity: bind obeys a kind of associativity**. Like this:
-* 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
+ (u >>= f) >>= g == u >>= (fun x -> f x >>= g)
-
-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option =
-# let unit x = Some x;;
-val unit : 'a -> 'a option =
-# unit 2 * unit;;
-- : int option = Some 2
-
+ 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`.
- 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`.
+ 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):
-* Associativity: bind obeys a kind of associativity, like this:
+ # Some 3 >>= unit >>= unit;;
+ - : int option = Some 3
+ # Some 3 >>= (fun x -> unit x >>= unit);;
+ - : int option = Some 3
- (m * f) * g == m * (fun x -> f x * g)
+ # Some 3 >>= divide 6 >>= divide 2;;
+ - : int option = Some 1
+ # Some 3 >>= (fun x -> divide 6 x >>= divide 2);;
+ - : int option = Some 1
- If you don't understand why the lambda form is necessary, you need
- to look again at the type of bind. This is important.
+ # Some 3 >>= divide 2 >>= divide 6;;
+ - : int option = None
+ # Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
+ - : int option = None
- For an illustration of associativity in the option monad:
+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`.
-
-Some 3 * unit * unit;;
-- : int option = Some 3
-Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
-
+* **Right identity: unit is a right identity for bind.** That is,
+ `u >>= unit == u` for all monad objects `u`. For instance,
- Of course, associativity must hold for arbitrary functions of
- type `'a -> M 'a`, where `M` is the monad type. It's easy to
- convince yourself that the bind operation for the option monad
- obeys associativity by dividing the inputs into cases: if `m`
- matches `None`, both computations will result in `None`; if
- `m` matches `Some n`, and `f n` evalutes to `None`, then both
- computations will again result in `None`; and if the value of
- `f n` matches `Some r`, then both computations will evaluate
- to `g r`.
+ # Some 3 >>= unit;;
+ - : int option = Some 3
+ # None >>= unit;;
+ - : 'a option = None
-* Right identity: unit is a right identity for bind. That is,
- `m * unit == m` for all monad objects `m`. For instance,
-
-# Some 3 * unit;;
-- : int option = Some 3
-
+More details about monads
+-------------------------
-Now, if you studied algebra, you'll remember that a *monoid* is an
+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 ).
+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 .
+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. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible.
-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.
+* [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.
+
-The intensionality monad
-------------------------
-
-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.
-
-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";;`.
-
-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. 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:
+* [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.
+
-
-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
-
+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.
-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 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.
+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.
-
-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
+Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-
+You may sometimes see:
-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 >> 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 just means:
-Let's test compliance with the left identity law:
+ u >>= fun _ -> v
-
-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-
+that is:
-We'll assume that this and the other laws always hold.
+ bind u (fun _ -> v)
-We now build up some extensional meanings:
+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.
- let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+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:
-This function says that everyone always left, except for Cam in world
-2 (i.e., `left 2 'c' == false`).
+ # let even x = (x mod 2 = 0);;
+ val g : int -> bool =
-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:
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
-
-let extapp fn arg w = fn w (arg w);;
+ # let lift g = fun u -> bind u (fun x -> Some (g x));;
+ val lift : ('a -> 'b) -> 'a option -> 'b option =
-extapp left ann 1;;
-# - : bool = true
+`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:
-extapp left cam 2;;
-# - : bool = false
-
+ # 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 =
-`extapp` stands for "extensional function application".
-So Ann left in world 1, but Cam didn't leave in world 2.
+`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.
-A transitive predicate:
+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`!
- let saw w x y = (w < 2) && (y < x);;
- extapp (extapp saw bill) ann 1;; (* true *)
- extapp (extapp saw bill) ann 2;; (* false *)
+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 world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone
-in world two.
+In general, any lift/map operation can be relied on to satisfy these laws:
-Good. Now for intensions:
+ * lift id = id
+ * lift (compose f g) = compose (lift f) (lift g)
- let intapp fn arg w = fn w arg;;
+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:
-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.
+ List.map (fun x -> f (g x)) lst
+ List.map f (List.map g lst)
-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:
+Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
-
-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+ ap [f] [x; y] = [f x; f y]
+ ap (Some f) (Some x) = Some (f x)
-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 *)
-
+and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
-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.)
+ 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
-Likewise for extensional transitive predicates like "saw":
+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:
-
-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 *)
-
+ [[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"):
+to:
-
-let think (w:s) (p:s->t) (x:e) =
- match (x, p 2) with ('a', false) -> false | _ -> p w;;
-
+ [1; 1; 2; 1; 3; 1; 2; 4]
-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.
+That is the `join` operation.
-
-intapp (lift (intapp think
- (intapp (lift left)
- (unit 'b'))))
- (unit 'a')
-1;; (* true *)
-
+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:
-So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
+ 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
-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.
-
-intapp (lift (intapp think
- (intapp (lift left)
- (unit 'c'))))
- (unit 'a')
-1;; (* false *)
-
-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.
+Monad outlook
+-------------
-*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.
+We're going to be using monads for a number of different things in the
+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.
-Finally, note that within an intensional grammar, extensional funtion
-application is essentially just bind:
+But first, we'll look at several linguistic applications for monads, based
+on what's called the *Reader monad*.
+
+##[[Reader monad for Variable Binding]]##
+
+##[[Reader monad for Intensionality]]##
-
-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-