The goal for these notes is to introduce the Reader Monad, and present two linguistic applications: binding and intensionality. Along the way, we'll continue to think through issues related to order, and a related notion of flow of information.

At this point, we've seen monads in general, and three examples of monads: the Identity monad (invisible boxes), the Option/Maybe monad (option types), and the List monad.

We've also seen an application of the Option/Maybe monad to safe division. The starting point was to allow the division function to return an int option instead of an int. If we divide 6 by 2, we get the answer Some 3. But if we divide 6 by 0, we get the answer None.

The next step was to adjust the other arithmetic functions to teach them what to do if they received None instead of a boxed integer. This meant changing the type of their input from ints to int options. But we didn't need to do this piecemeal; rather, we "lift"ed the ordinary arithmetic operations into the monad using the various tools provided by the monad.

We'll go over this lifting operation in detail in the next section.

Tracing the effect of safe_div on a larger computation

So let's see how this works in terms of a specific computation.

(+ 1 (* (/ 6 2) 4)) in tree format:

 ___________
 |         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
      |    |
      *  __|___
         |    |
        _|__  2
        |  |
        /  6

No matter what order we evaluate it in, this computation should eventually reduce to 13. Given a specific reduction strategy, we can watch the order in which the computation proceeds. Following on the evaluator developed during the previous homework, let's adopt the following reduction strategy:

In order to reduce an expression of the form (head arg), do the following in order:

1. Reduce head to head'
2. Reduce arg to arg'.
3. If (head' arg') is a redex, reduce it.

There are many details left unspecified here, but this will be enough for today. The order in which the computation unfolds will be

1. Reduce head (+ 1) to itself
2. Reduce arg ((* ((/ 6) 2)) 4)
   1. Reduce head (* ((/ 6) 2))
      1. Reduce head * to itself
      2. Reduce arg ((/ 6) 2)
         1. Reduce head (/ 6) to itself
         2. Reduce arg 2 to itself
         3. Reduce ((/ 6) 2) to 3
      3. Reduce (* 3) to itself
   2. Reduce arg 4 to itself
   3. Reduce ((* 3) 4) to 12
3. Reduce ((+ 1) 12) to 13

This reduction pattern follows the structure of the original expression exactly, at each node moving first to the left branch, processing the left branch, then moving to the right branch, and finally processing the results of the two subcomputation. (This is called a depth-first postorder traversal of the tree.)

[diagram with arrows traversing the tree]

It will be helpful to see how the types change as we make adjustments.

type num = int
type contents = Num of num   | Op2 of (num -> num -> num)
type tree = Leaf of contents | Branch of tree * tree

Never mind that these types will allow us to construct silly arithmetric trees such as + * or 2 3. Note that during the reduction sequence, the result of reduction was in every case a well-formed subtree. So the process of reduction could be animated by replacing subtrees with the result of reduction on that subtree, until the entire tree is replaced by a single integer (namely, 13).

Now we replace the number 2 with 0:

 ___________
 |         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
      |    |
      *  __|___
         |    |
        _|__  0
        |  |
        /  6

When we reduce, we get quite a ways into the computation before things break down:

1. Reduce head (+ 1) to itself
2. Reduce arg ((* ((/ 6) 0)) 4)
   1. Reduce head (* ((/ 6) 0))
      1. Reduce head * to itself
      2. Reduce arg ((/ 6) 0)
         1. Reduce head (/ 6) to itself
         2. Reduce arg 0 to itself
         3. Reduce ((/ 6) 0) to ACKKKK

This is where we replace / with safe_div. safe_div returns not an int, but an int option. If the division goes through, the result is Some n, where n is the integer result. But if the division attempts to divide by zero, the result is None.

We could try changing the type of the arithmetic operators from int -> int -> int to int -> int -> int option; but since we now have to anticipate the possibility that any argument might involve division by zero inside of it, it would be better to prepare for the possibility that any subcomputation might return None here. So our operators should have the type int option -> int option -> int option. Let's bring that about by just changing the type num from int to int option, leaving everying else the same:

type num = int option
type contents = Num of num   | Op2 of (num -> num -> num)
type tree = Leaf of contents | Branch of tree * tree

The only difference is that instead of defining our numbers to be simple integers, they are now int options; and so Op is an operator over int options.

At this point, we bring in the monadic machinery. In particular, here is the ⇧ and the map2 function from the notes on safe division:

⇧ (x : 'a) = Some x

map2 (f : 'a -> 'b -> 'c) (xx : 'a option) (yy : 'b option) =
  match xx with
  | None -> None
  | Some x ->
      (match yy with
      | None -> None
      | Some y -> Some (f x y))

Then we lift the entire computation into the monad by applying ⇧ to the integers, and by applying map2 to the operators. Only, we will replace / with safe_div, defined as follows:

safe_div (xx : 'a option) (yy : 'b option) =
  match xx with
  | None -> None
  | Some x ->
      (match yy with
      | None -> None
      | Some 0 -> None
      | Some y -> Some ((/) x y))

     ___________________
     |                 |
  ___|____         ____|_____
  |      |         |        |
map2 +  ⇧1    _____|_____  ⇧4
              |         |
            map2 *  ____|____
                    |       |
                 ___|____  ⇧0
                 |      |
             safe_div  ⇧6

With these adjustments, the faulty computation now completes smoothly:

1. Reduce head ((map2 +) ⇧1)
2. Reduce arg (((map2 *) ((safe_div ⇧6) ⇧0)) ⇧4)
   1. Reduce head ((map2 *) ((safe_div ⇧6) ⇧0))
      1. Reduce head (map2 *)
      2. Reduce arg ((safe_div ⇧6) ⇧0)
         1. Reduce head (safe_div ⇧6)
         2. Reduce arg ⇧0
         3. Reduce ((safe_div ⇧6) ⇧0) to None
      3. Reduce ((map2 *) None)
   2. Reduce arg ⇧4
   3. Reduce (((map2 *) None) ⇧4) to None
3. Reduce (((map2 +) ⇧1) None) to None

As soon as we try to divide by 0, safe_div returns None. Thanks to the details of map2, the fact that None has been returned by one of the arguments of a map2-ed operator guarantees that the map2-ed operator will pass on the None as its result. So the result of each enclosing computation will be None, up to the root of the tree.

It is unfortunate that we need to continue the computation after encountering our first None. We know immediately at the result of the entire computation will be None, yet we continue to compute subresults and combinations. It would be more efficient to simply jump to the top as soon as None is encoutered. Let's call that strategy Abort. We'll arrive at an Abort operator later in the semester.

So at this point, we can see how the Option/Maybe monad provides plumbing that allows subcomputations to send information from one part of the computation to another. In this case, the safe_div function can send the information that division by zero has been attempted upwards to the rest of the computation. If you think of the plumbing as threaded through the tree in depth-first, postorder traversal, then safe_div drops None into the plumbing half way through the computation, and that None travels through the rest of the plumbing till it comes out of the result faucet at the top of the tree.

Information flowing in the other direction: top to bottom

We can think of this application as facilitating information flow. In the safe_div example, a subcomputation created a message that propagated upwards to the larger computation:

                 message: Division by zero occurred!
                      ^
 ___________          |
 |         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
      |    |          |
      *  __|___  -----|
         |    |
        _|__  0
        |  |
        /  6

(The message was implemented by None.)

We might want to reverse the direction of information flow, making information available at the top of the computation available to the subcomputations:

                    [λn]  
 ___________          |
 |         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
      |    |          |
      *  __|___       |
         |    |       |
        _|__  n  <----|
        |  |
        /  6

We've seen exactly this sort of configuration before: it's exactly what we have when a lambda binds a variable that occurs in a deeply embedded position. Whatever the value of the argument that the lambda form combines with, that is what will be substituted in for free occurrences of that variable within the body of the lambda.

Binding

So our next step is to add a (primitive) version of binding to our computation. We'll allow for just one binding dependency for now, and then generalize later.

Binding is independent of the safe division, so we'll return to a situation in which the Option/Maybe monad hasn't been added. One of the nice properties of programming with monads is that it is possible to add or subtract layers according to the needs of the moment. Since we need simplicity, we'll set the Option/Maybe monad aside for now.

So we'll go back to the point where the num type is simple int, not int options.

type num = int

And we won't be using the map2 or ⇧ from the Option/Maybe monad anymore.

As you might guess, the technique we'll use to arrive at binding will be to use the Reader monad, defined here in terms of mid/⇧ and mbind/>>=:

type α = int -> α
⇧x = \n. x
xx >>= k = \n. k (xx n) n
map  f xx = \n. f (xx n)
ff ¢ xx = \n. (ff n) (xx n)
map2 f xx yy = map f xx ¢ yy = \n. f (xx n) (yy n)

A boxed type in this monad will be a function from an integer to an value in the original type. The mid/⇧ function lifts a value x to a function that expects to receive an integer, but throws away the integer and returns x instead (most values in the computation don't depend on the input integer).

The mbind/>>= function in this monad takes a monadic value xx, a function k taking non-monadic values into the monad, and returns a function that expects an integer n. It feeds n to xx, which delivers a result in the orginal type, which is fed in turn to k. k returns a monadic value, which upon being fed an integer, also delivers a result in the orginal type.

The map, ¢, and map2 functions corresponding to this mbind are given above. They may look familiar --- we'll comment on this in a moment.

Lifing the computation into the monad, we have the following adjusted types:

type num = int -> int

That is, num is once again replaced with the type of a boxed int. When we were dealing with the Option/Maybe monad, a boxed int had type int option. In this monad, a boxed int has type int -> int.

     __________________
     |                |
  ___|____        ____|_____
  |      |        |        |
map2 +  ⇧1    ____|_____  ⇧4
              |        |
            map2 *  ___|____
                    |      |
                 ___|____  x
                 |      |
               map2 /  ⇧6

It remains only to decide how the variable n will access the input value supplied at the top of the tree. The input value is supposed to be the value that gets used for the variable n. Like every leaf in the tree in argument position, the code we want in order to represent the variable will have the type of a boxed int, namely, int -> int. So we have the following:

var_n = fun (n : int) -> n

That is, variables in this system denote the identity function!

The result of evaluating this tree will be a boxed integer: a function from any integer n to (+ 1 (* (/ 6 n)) 4).

Take a look at the definition of the Reader monad again. The mid takes some object x and returns \n. x. In other words, ⇧x = Kx, for our familiar combinator K, so ⇧ = K. Likewise, the map operation is just function composition, and the mapply/¢ operation is our friend the S combinator. map2 f xx yy for the Reader monad is (f ○ xx) ¢ yy or S (f ○ xx) yy.

We've seen this before as a strategy for translating a binding construct into a set of combinators. To remind you, here is a part of the general scheme for translating a lambda abstract into Combinatory Logic. The translation function [.] translates a lambda term into a term in Combinatory Logic:

[(MN)] = ([M] [N])
[\a.a] = I
[\a.M] = K[M]       (assuming a not free in M)
[\a.(MN)] = S[\a.M][\a.N]

The reason we can make do with this subset of the full translation scheme is that we're making the simplifying assumption that there is at most a single lambda involved. So once again we have the identity function I as the translation of the bound variable, K as the function governing expressions that don't contain an instance of the bound variable, S as an operation that manages ¢, that is, the applicative combination of complex expressions.

Jacobson's Variable Free Semantics as a Reader Monad

We've designed the presentation above to make it as easy as possible to show that Jacobson's Variable Free Semantics (e.g., Jacobson 1999, Towards a Variable-Free Semantics) implements a Reader monad.

More specifically, it will turn out that Jacobson's geach combinator g is exactly our map operator, and her binding combinator z is exactly our mbind (though with the arguments reversed).

Jacobson's system contains two main combinators, g and z. She calls g the Geach rule, and z performs binding. Here is a typical computation. This implementation is based closely on email from Simon Charlow, with beta reduction as performed by the on-line evaluator:

; Jacobson's analysis of "Everyone_i thinks he_i left"
let g = \f xx. \n. f (xx n) in ; or `f ○ xx`
let z = \k xx. \n. k (xx n) n in ; or `S (flip k) xx`, or `Reader.(>>=) xx k`
let he = \n. n in
let everyone = \P. FORALL i (P i) in

everyone (z thinks (g left he))

~~>  FORALL i (thinks (left i) i)

Two things to notice: First, pronouns once again denote identity functions, just as we saw in the reader monad in the previous section.

Second, g plays the role of transmitting a binding dependency for an embedded constituent to a containing constituent.

The basic recipe in Jacobson's system is that you transmit the dependence of a pronoun upwards through the tree using g until just before you are about to combine with the binder, when you finish off with z. Here is an example with a longer chain of g's:

; "Everyone_i thinks that Bill said he_i left"

everyone (z thinks (g (T bill) (g said (g left he))))
; or `everyone (thinks =<< T bill ○ said ○ left ○ I)`

~~> FORALL i (thinks (said (left i) bill) i)

If you compare Jacobson's values for g and z to the functions in the reader monad given above, you'll see that Jacobson's g combinator is exactly our map operator. Furthermore, Jacobson's z combinator is identical to our Reader >>=, except with the order of the arguments reversed (i.e., (z k xx) == (xx >>= k)).

(The T combinator in the derivations above is given by T x <~~> \f. f x; it handles situations in which English word order reverses the usual function/argument order. T is what Curry and Steedman call this combinator. Jacobson calls it "lift", but it shouldn't be confused with the mid and map operations that lift values into the Reader monad we're focusing on here. It does lift values into a different monad, that we'll consider in a few weeks.)

In other words, Jacobson's variable-free semantics is essentially a Reader monad.

One of the peculiar aspects of Jacobson's system is that binding is accomplished not by applying z to the element that will (in some pre-theoretic sense) bind the pronoun, here, everyone, but rather by applying z instead to the predicate that will take everyone as an argument, here, thinks.

The Reader monad for intensionality

Now we'll look at using the Reader monad to do intensional function application.

In Shan (2001) Monads for natural language semantics, Ken shows that making expressions sensitive to the world of evaluation is conceptually the same thing as making use of the Reader monad. This technique was beautifully re-invented by Ben-Avi and Winter (2007) in their paper A modular approach to intensionality, 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.

First, the familiar linguistic problem:

Bill left.  
Cam left.
Ann believes that Bill left.
Ann believes that Cam left.

We want an analysis on which the first three sentences can be true at the same time that the last sentence is false. If sentences denoted simple truth values or booleans, we have a problem: if the sentences Bill left and Cam left are both true, they denote the same object, and Ann's beliefs can't distinguish between them.

The traditional solution to the problem sketched above 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:

Extensional types              Intensional types       Examples
-------------------------------------------------------------------

S         t                    s->t                    John left
DP        e                    s->e                    John
VP        e->t                 (s->e)->s->t            left
Vt        e->e->t              (s->e)->(s->e)->s->t    saw
Vs        t->e->t              (s->t)->(s->e)->s->t    thought

This system is modeled on the way Montague arranged his grammar. There are significant simplifications compared to Montague: for instance, determiner phrases are thought of here as corresponding to individuals rather than to generalized quantifiers.

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 so-called "individual 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). In addition, the result of each predicate is an intension. This expresses the fact that the set of people who left in one world may be different than the set of people who left in a different world.

Normally, the dependence of the extension of a predicate to the world of evaluation is hidden inside of an evaluation coordinate, or built into the the lexical meaning function, but we've made it explicit here in the way that the intensionality monad makes most natural.

The intensional types are more complicated than the extensional types. Wouldn't it be nice to make the complicated types available for expressions like 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. chars (characters in the computational sense, i.e., letters like 'a' and 'b', not Kaplanian characters) will model individuals, and OCaml bools will serve for truth values:

type s = int
type e = char
type t = bool

let ann = 'a'
let bill = 'b'
let cam = 'c'

let left1 (x : e) = true 
let saw1 (y : e) (x : e) = x < y 

left1 ann (* true *)
saw1 bill ann (* true *)
saw1 ann bill (* false *)

So here's our extensional system: everyone left, including Ann; and Ann saw Bill (saw1 bill ann), but Bill didn't see Ann. (Note that the word order we're using is VOS, verb-object-subject.)

Now we add intensions. Because different people leave in different worlds, the meaning of leave must depend on the world in which it is being evaluated:

let left (x : e) (w : s) = match (x, w) with ('c', 2) -> false | _ -> true
left ann 1 (* true: Ann left in world 1 *)
left cam 2 (* false: Cam didn't leave in world 2 *) 

This new definition says that everyone always left, except that in world 2, Cam didn't leave.

Note that although this general left is sensitive to world of evaluation, it does not have the fully intensionalized type given in the chart above, which was (s->e)->s->t. This is because left does not exploit the additional resolving power provided by making the subject an individual concept. In semantics jargon, we say that leave is extensional with respect to its first argument.

We will adopt the general strategy of defining predicates in a way that they take arguments of the lowest type that will allow us to make all the distinctions the predicate requires. When it comes time to combine this predicate with monadic arguments, we'll have to make use of various lifting predicates.

Likewise, although see depends on the world of evaluation, it is extensional in both of its syntactic arguments:

let saw (y : e) (x : e) (w : s) = (w < 2) && (x < y)
saw bill ann 1 (* true: Ann saw Bill in world 1 *)
saw bill ann 2 (* false: no one saw anyone in world 2 *)

This (again, partially) intensionalized version of see coincides with the saw1 function we defined above for world 1; in world 2, no one saw anyone.

Just to keep things straight, let's review the facts:

     World 1: Everyone left.
              Ann saw Bill, Ann saw Cam, Bill saw Cam, no one else saw anyone.              
     World 2: Ann left, Bill left, Cam didn't leave.
              No one saw anyone.

Now we are ready for the intensionality monad:

type 'a intension = s -> 'a
let mid x = fun (w : s) -> x
let (>>=) xx k = fun (w : s) -> let x = xx w in let yy = k x in yy w
                 (* or just `fun w -> k (xx w) w` *)

Then the individual concept mid ann is a rigid designator: a constant function from worlds to individuals that returns 'a' no matter which world is used as an argument. This is a typical kind of thing for a monadic mid to do.

Then combining a predicate like left which is extensional in its subject argument with an intensional subject like mid ann is simply mbind in action:

(mid ann >>= left) 1  (* true: Ann left in world 1 *)
(mid cam >>= left) 2  (* false: Cam didn't leave in world 2 *)

As usual, >>= takes a monadic box containing Ann, extracts Ann, and feeds her to the function left. In linguistic terms, we take the individual concept mid ann, apply it to the world of evaluation in order to get hold of an individual ('a'), then feed that individual to the predicate left.

We can arrange for a transitive verb that is extensional in both of its arguments to take intensional arguments:

let map2' f xx yy = xx >>= (fun x -> yy >>= (fun y -> f x y))

This is almost the same map2 predicate we defined in order to allow addition in our division monad example. The difference is that this variant operates on verb meanings that take extensional arguments but returns an intensional result. Thus the original map2 predicate has mid (f x y) where we have just f x y here.

The use of >>= here to combine left with an individual concept, and the use of map2' to combine see with two intensional arguments closely parallels the two of Montague's meaning postulates (in PTQ) that express the relationship between extensional verbs and their uses in intensional contexts.

map2' saw (mid bill) (mid ann) 1  (* true *)
map2' saw (mid bill) (mid ann) 2  (* false *)

Ann did see Bill in world 1, but Ann didn't see Bill in world 2.

Finally, we can define our intensional verb thinks. Think is intensional with respect to its sentential complement (it takes complements of type s -> t), though still extensional with respect to its subject (type e). (As Montague noticed, almost all verbs in English are extensional with respect to their subject; a possible exception is appear.)

let thinks (p : s->t) (x : e) (w : s) = 
  match (x, p 2) with ('a', false) -> false | _ -> p w

In every world, Ann fails to believe any proposition that is false in world 2. Apparently, she stably thinks we may be in world 2. Otherwise, everyone believes a proposition iff that proposition is true in the world of evaluation.

(mid ann >>= thinks (mid bill >>= left)) 1  (* true *)

So in world 1, Ann thinks that Bill left (because in worlds 1 and 2, Bill did leave).

But although Cam left in world 1:

(mid cam >>= left) 1  (* true *)

he didn't leave in world 2, so Ann doesn't in world 1 believe that Cam left:

(mid ann >>= thinks (mid cam >>= left)) 1  (* false *)

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 mbind), and the non-intersective adjectives will take intensional arguments.

Connections with variable binding: the rigid individual concepts generated by mid ann and the like correspond to the numerical constants, that don't interact with the environment in any way, in the variable binding examples we considered earlier on the page. If we had any non-contingent predicates that were wholly insensitive to intensional effects, they would be modeled using map2 and would correspond to the operations like map2 (+) in the earlier examples. As it is, our predicates lift and saw, though only sensitive to the extension of their arguments, nonetheless are sensitive to the world of evaluation for their bool output. So these are somewhat akin, at the predicate level, to expressions like var_n, at the singular term level, in the variable bindings examples. Our predicate thinks shows yet a further kind of interaction with the intensional structures we introduced: namely, its output can depend upon evaluating its complement relative to other possible worlds. We didn't discuss analogues of this in the variable binding case, but they exist: namely, they are expressions like let x = 2 in ... and forall x ..., that have the effect of supplying their arguments with an environment or assignment function that is shifted from the one they are themselves being evaluated with.

notes: cascade, general env