```(+ 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 lambda 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 h' > 2. Reduce arg to a'. > 3. If (h' a') 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 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, till 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` at least 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 option`s; and so `Op` is an operator over `int option`s. At this point, we bring in the monadic machinery. In particular, here is the `⇧` and the `map2` function from the notes on safe division: ⇧ (a: 'a) = Some a;; map2 (g : 'a -> 'b -> 'c) (xx : 'a option) (yy : 'b option) = match xx with | None -> None | Some x -> (match yy with | None -> None | Some y -> Some (g 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 throughout 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 save-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:
```                    [λx]
___________          |
|         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
|    |          |
*  __|___       |
|    |       |
_|__  x  <----|
|  |
/  6
```
```type α = int -> α
⇧x = \n. x
xx >>= k = \n. k (xx n) n
map  f xx = \n. f (xx n)
map2 f xx yy = \n. f (xx n) (yy n)
```
```     __________________
|                |
___|____        ____|_____
|      |        |        |
map2 +  ⇧1    ____|_____  ⇧4
|        |
map2 *  ___|____
|      |
___|____  x
|      |
map2 /  ⇧6
```
```; Jacobson's analysis of "Everyone_i thinks he_i left"
let g = \f u. \x. f (u x) in
let z = \f u. \x. f (u x) x in
let he = \x. x in
let everyone = \P. FORALL x (P x) in

everyone (z thinks (g left he))

~~>  FORALL x (thinks (left x) x)
```
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 (z thinks (g (t bill) (g said (g left he))))

~~> FORALL x (thinks (said (left x) bill) x)
```
```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 those 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. 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: 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 (x:e) (y:e) = y < x;; 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. Therefore 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 x y w = (w < 2) && (y < x);; 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.
```
```type 'a intension = s -> 'a;;
```lift2' saw (unit bill) (unit ann) 1;;  (* true *)