-* At the top of p. 13 (this is in between defs 2.8 and 2.9), GS&V give two examples, one for \[[∃xPx]] and the other for \[[Qx]]. In fact it will be most natural to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. But first we need to get clear on expressions like \[[Qx]].
+* At the top of p. 13, GS&V give two examples, one for \[[∃xPx]] and the other for \[[Qx]]. For our purposes, it will be most natural to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. But first we need to get clear on expressions like \[[Qx]] and \[[Px]].
* GS&V say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the entity associated with the peg associated with the variable `x` does not have the property Q. In other words, if we let `q` be the function from entities to `bool`s that gives the extension of Q, then `s` updated with \[[Qx]] should be `s` filtered by the function `fun (r, h) -> let obj = List.nth h (r 'x') in q obj`. When `... q obj` evaluates to `true`, that `(r, h)` pair is retained, else it is discarded.
- OK, we face two questions then. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
+ OK, so we face two questions. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
* Answering the first question: we assume we've got some `bool dpm set` to start with. I won't call this `s` because that's what GS&V use for sets of discourse possibilities, and we don't want to confuse discourse possibilities with `dpm`s. Instead I'll call it `u`. Now what we want to do with `u` is to map each `dpm` it gives us to one that results in `(true, r, h)` only when the entity that `r` and `h` associate with variable `x` has the property Q. As above, I'll assume Q's extension is given by a function `q` from entities to `bool`s.
Then what we want is something like this:
- let eliminate_non_Qxs = (fun truth_value ->
- fun (r, h) ->
- let truth_value' =
- if truth_value
- then let obj = List.nth h (r 'x') in q obj
- else false
- in (truth_value', r, h))
- in bind_set u (fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs))
+ let eliminator : bool -> bool dpm =
+ fun truth_value ->
+ fun (r, h) ->
+ let truth_value' =
+ if truth_value
+ then let obj = List.nth h (r 'x') in q obj
+ else false
+ in (truth_value', r, h)
+ in bind_set u (fun one_dpm -> unit_set (bind_dpm one_dpm eliminator))
The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `bind_set` operation. This works by taking each `dpm` in the set and returning a `unit_set` of a filtered `dpm`. The definition of `bind_set` takes care of collecting together all of the `unit_set`s that result for each different set element we started with.
* Now our second question: how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
- Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about sets of `dpm`s; we'll leave that to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. So what we want is:
+ Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about `dpm set`s; we'll leave them to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. So what we want is:
- let getx = fun (r, h) ->
+ let getx : entity dpm = fun (r, h) ->
let obj = List.nth h (r 'x')
in (obj, r, h);;
fun entity_dpm -> unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (q e)))
- Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to compose \[[Qx]] with them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so.
+ Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to monadically bind \[[Qx]] to them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so.
This could be handled like this:
fun entity_dpm ->
- let eliminate_non_Qxs = fun truth_value ->
- if truth_value = false
- then unit_dpm false
- else bind_dpm entity_dpm (fun e -> unit_dpm (q e))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ let eliminator : bool -> bool dpm =
+ fun truth_value ->
+ if truth_value = false
+ then unit_dpm false
+ else bind_dpm entity_dpm (fun e -> unit_dpm (q e))
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
Applied to an `entity_dpm`, that yields a function that we can bind to a `bool dpm set` and that will transform the doubly-wrapped `bool` into a new `bool dpm set`.
let obj = List.nth h (r 'x')
in (obj, r, h)
in let entity_dpm = getx
- in let eliminate_non_Qxs = fun truth_value ->
+ in let eliminator = fun truth_value ->
if truth_value = false
then unit_dpm false
else bind_dpm entity_dpm (fun e -> unit_dpm (q e))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ <!--
or, simplifying:
let getx = fun (r, h) ->
let obj = List.nth h (r 'x')
in (obj, r, h)
- in let eliminate_non_Qxs = fun truth_value ->
+ in let eliminator = fun truth_value ->
if truth_value
then bind_dpm getx (fun e -> unit_dpm (q e))
else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ -->
- unpacking the definition of `bind_dpm`, that is:
+ If we simplify and unpack the definition of `bind_dpm`, that's equivalent to:
let getx = fun (r, h) ->
let obj = List.nth h (r 'x')
in (obj, r, h)
- in let eliminate_non_Qxs = fun truth_value ->
+ in let eliminator = fun truth_value ->
if truth_value
then (fun (r, h) ->
let (a, r', h') = getx (r, h)
in let u' = (fun e -> unit_dpm (q e)) a
in u' (r', h')
) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
- continuing to simplify:
+ which can be further simplified to:
- let eliminate_non_Qxs = fun truth_value ->
+ <!--
+ let eliminator = fun truth_value ->
if truth_value
then (fun (r, h) ->
let obj = List.nth h (r 'x')
in let u' = (fun e -> unit_dpm (q e)) a
in u' (r', h')
) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
- let eliminate_non_Qxs = fun truth_value ->
+ let eliminator = fun truth_value ->
if truth_value
then (fun (r, h) ->
let obj = List.nth h (r 'x')
in let u' = unit_dpm (q obj)
in u' (r, h)
) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ -->
- let eliminate_non_Qxs = fun truth_value ->
+ let eliminator = fun truth_value ->
if truth_value
then (fun (r, h) ->
let obj = List.nth h (r 'x')
in (q obj, r, h)
) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
This is a function that takes a `bool dpm` as input and returns a `bool dpm set` as output.
(Compare to the \[[Qx]] we had before:
- let eliminate_non_Qxs = (fun truth_value ->
+ let eliminator = (fun truth_value ->
fun (r, h) ->
let truth_value' =
if truth_value
then let obj = List.nth h (r 'x') in q obj
else false
in (truth_value', r, h))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)
+ in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
Can you persuade yourself that these are equivalent?)
or:
- <pre><code>u >>=<sub>set</sub> \[[Qx]]
+ <pre><code>u >>= \[[Qx]]
</code></pre>
* Can you figure out how to handle \[[∃x]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_5). But try to get as far as you can on your own.