```\tree ((((+) (1)) (((*) (((/) (6)) (2))) (4))))

___________
|         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
|    |
*  __|___
|    |
_|__  2
|  |
/  6
```
This computation should reduce to 13. But 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 (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)) 3) 1. Reduce head (* ((/ 6) 2)) 1. Reduce head * 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.) It will be helpful to see how the types change as we make adjustments. type num = int type contents = Num of num | Op 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:
```\tree ((((+) (1)) (((*) (((/) (6)) (0))) (4))))

___________
|         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
|    |
*  __|___
|    |
_|__  0
|  |
/  6
```
When we reduce, we get quite a ways into the computation before things go south: 1. Reduce head (+ 1) to itself 2. Reduce arg ((* ((/ 6) 0)) 3) 1. Reduce head (* ((/ 6) 0)) 1. Reduce head * 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`. This means changing the type of the arithmetic operators from `int -> int -> int` to `int -> int -> int option`; and since we now have to anticipate the possibility that any argument might involve division by zero inside of it, here is the net result for our types: type num = int option type contents = Num of num | Op 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: ⇧ (a: 'a) = Just a;; map2 (g : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) = match u with | None -> None | Some x -> (match v 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 `map1` to the operators:
```\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (⇧0))) (⇧4))))

___________________
|                 |
___|____         ____|_____
|      |         |        |
map2 +  ⇧1    _____|_____  ⇧4
|         |
map2 *  ____|____
|       |
___|____  ⇧0
|      |
map2 /  ⇧6
```
```\tree ((((+) (1)) (((*) (((/) (6)) (2))) (4))))

___________
|         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
|    |
*  __|___
|    |
_|__  2
|  |
/  6
```
This computation should reduce to 13. But 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 (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)) 3) 1. Reduce head (* ((/ 6) 2)) 1. Reduce head * 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.) It will be helpful to see how the types change as we make adjustments. type num = int type contents = Num of num | Op 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:
```\tree ((((+) (1)) (((*) (((/) (6)) (0))) (4))))

___________
|         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
|    |
*  __|___
|    |
_|__  0
|  |
/  6
```
```                 message: Division by zero occurred!
^
___________          |
|         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
|    |          |
*  __|___  -----|
|    |
_|__  0
|  |
/  6
```
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
```
```\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (x))) (⇧4))))

__________________
|                |
___|____        ____|_____
|      |        |        |
map2 +  ⇧1    ____|_____  ⇧4
|        |
map2 *  ___|____
|      |
___|____  x
|      |
map2 /  ⇧6
```
It remains only to decide how the variable `x` will access the value input at the top of the tree. Since the input value is supposed to be the value put in place of the variable `x`. 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: x (i:int):int = i That is, variables in this system denote the indentity function! The result of evaluating this tree will be a boxed integer: a function from any integer `x` to `(+ 1 (* (/ 6 x)) 4)`. Take a look at the definition of the reader monad again. The midentity takes some object `a` and returns `\x.a`. In other words, `⇧a = Ka`, so `⇧ = K`. Likewise, `map2` for this monad is the `S` combinator. We've seen this before as a strategy for translating a lambda abstract into a set of combinators. 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 function is that we're making the simplifying assumption that there is at most a single lambda involved. So here you see the I (the translation of the bound variable), the K and the S. ## Jacobson's Variable Free Semantics as a Reader Monad We've designed the discussion so far to make the following claim as easy to show as possible: Jacobson's Variable Free Semantics (e.g., Jacobson 1999, [Towards a Variable-Free Semantics](http://www.springerlink.com/content/j706674r4w217jj5/)) is a reader monad. More specifically, it will turn out that Jacobson's geach combinator *g* is exactly our `lift` operator, and her binding combinator *z* is exactly our `bind` (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:
```; Analysis of "Everyone_i thinks he_i left"
let g = \f g x. f (g x) in
let z = \f g x. f (g 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)
```
Several things to notice: First, pronouns once again denote identity functions. As Jeremy Kuhn has pointed out, this is related to the fact that in the mapping from the lambda calculus into combinatory logic that we discussed earlier in the course, bound variables translated to I, the identity combinator (see additional comments below). We'll return to the idea of pronouns as identity functions in later discussions. Second, *g* plays the role of transmitting a binding dependency for an embedded constituent to a containing constituent. Third, 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 basic recipe in Jacobson's system, then, 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*. (There are examples with longer chains of *g*'s below.) Jacobson's *g* combinator is exactly our `lift` operator: it takes a functor and lifts it into the monad. Furthermore, Jacobson's *z* combinator, which is what she uses to create binding links, is essentially identical to our reader-monad `bind`!
```everyone (z thinks (g left he))

~~> forall w (thinks (left w) w)

everyone (z thinks (g (t bill) (g said (g left he))))

~~> forall w (thinks (said (left w) bill) w)
```
```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 *)