-* 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 \[[Px]].
+* 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 a function from entities to `bool`s, `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.
+* 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 ascribed 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 apply a filter that only lets through those `bool dpm`s whose outputs are `(true, r, h)`, where the entity that `r` and `h` associate with variable `x` has the property P. So what we want is:
+* 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.
- let test = (fun (truth_value, r, h) ->
- truth_value && (let obj = List.nth h (r 'x') in Q obj))
- in List.filter test u
+ Then what we want is something like this:
- Persuade yourself that in general:
+ 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))
- List.filter (test : 'a -> bool) (u : 'a set) : 'a set
+ 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.
- is the same as:
-
- bind_set u (fun a -> if test a then unit_set a else empty_set)
-
- Hence substituting in our above formula, we can derive:
-
- let test = (fun (truth_value, r, h) ->
- truth_value && (let obj = List.nth h (r 'x') in Q obj))
- in bind_set u (fun a -> if test a then unit_set a else empty_set)
-
- or simplifying:
-
- bind_set u (fun (truth_value, r, h) ->
- if truth_value && (let obj = List.nth h (r 'x') in Q obj)
- then unit_set (true, r, h) else empty_set)
-
- We can call the `(fun (truth_value, r, h) -> ...)` part \[[Qx]] and then updating `u` with \[[Qx]] will be:
+ We can call the `(fun one_dpm -> ...)` part \[[Qx]] and then updating `u` with \[[Qx]] will be:
bind_set u \[[Qx]]
u >>= \[[Qx]]
-* Now our second question: how do we decompose the behavior here ascribed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
+* 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`. Except that we don't have just one `dpm` to compose it with, but a set of them. However, we'll leave that to our predicates to deal with. We'll just make \[[x]] be an operation on a single `dpm`. Let's call the `dpm` we start with `v`. Then what we want to do 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)
- in bind_dpm v (fun _ -> getx)
+ in (obj, r, h);;
- What's going on here? Our starting `dpm` is a kind of monadic box. We don't care what value is inside that monadic box, which is why we're binding it to a function of the form `fun _ -> ...`. What we return is a new monadic box, which takes `(r, h)` as input and returns the entity they associate with variable `x` (together with unaltered versions of `r` and `h`).
+* Now what do we do with predicates? As before, we suppose we have a function `q` that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting `q` into a function that takes `entity dpm`s as arguments and returns `bool dpm`s:
-* Now what do we do with predicates? We suppose we're given a function Q that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting Q into a function that takes `entity dpm`s as arguments and returns `bool dpm`s:
+ fun entity_dpm -> bind_dpm entity_dpm (fun e -> unit_dpm (q e))
- fun entity_dpm -> bind_dpm entity_dpm (fun e -> unit_dpm (Q e))
+ Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `unit_set`:
- Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `(bool dpm) set`. This is easily done with `unit_set`:
+ fun entity_dpm -> unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (q e)))
- 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 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 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`.
If we let that be \[[Q]], then \[[Q]] \[[x]] would be:
let obj = List.nth h (r 'x')
in (obj, r, h)
in let entity_dpm = getx
- in unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (Q e)))
+ 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 eliminator)
+ <!--
or, simplifying:
let getx = fun (r, h) ->
let obj = List.nth h (r 'x')
in (obj, r, h)
- in unit_set (bind_dpm getx (fun e -> unit_dpm (Q e)))
+ 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 eliminator)
+ -->
- which 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 fun (r, h) ->
- let (a, r', h') = getx (r, h)
- in let u' = (fun e -> unit_dpm (Q e)) a
- in unit_set (u' (r', h'))
-
- which is:
-
- fun (r, h) ->
- let obj = List.nth h (r 'x')
- in let (a, r', h') = (obj, r, h)
- in let u' = (fun e -> unit_dpm (Q e)) a
- in unit_set (u' (r', h'))
-
- which is:
-
- fun (r, h) ->
- let obj = List.nth h (r 'x')
- in let u' = unit_dpm (Q obj)
- in unit_set (u' (r, h))
-
- which is:
-
- fun (r, h) ->
- let obj = List.nth h (r 'x')
- in unit_set (unit_dpm (Q obj) (r, h))
-
- which is:
-
- fun (r, h) ->
- let obj = List.nth h (r 'x')
- in unit_set ((Q obj, r, h))
-
-
-
-
-
-
-
-
-, so really \[[x]] will need to be a monadic operation on *a set of* `entity dpm`s. But one thing at a time. First, let's figure out what operation should be performed on each starting `dpm`. Let's call the `dpm` we start with `v`. Then what we want to do is:
+ 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 eliminator)
+
+ which can be further simplified to:
+
+ <!--
+ let eliminator = fun truth_value ->
+ if truth_value
+ then (fun (r, h) ->
+ let obj = List.nth h (r 'x')
+ let (a, r', h') = (obj, 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 eliminator)
+
+ 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 eliminator)
+ -->
+
+ 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 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 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 eliminator)
+
+ Can you persuade yourself that these are equivalent?)
+
+* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We monadically bind this operaration to whatever `bool dpm set` we already have on hand:
+ bind_set u \[[Qx]]
- So that's how \[[x]] should operate on a single `dpm`. How should it operate on a set of them? Well, we just have to take each member of the set and return a `unit_set` of the operation we perform on each of them. The `bind_set` operation takes care of joining all those `unit_set`s together. So, where `u` is the set of `dpm`s we start with, we have:
-
- let handle_each = fun v ->
- (* as above *)
- let getx = fun (r, h) ->
- let obj = List.nth h (r 'x')
- in (obj, r, h)
- in let result = bind_dpm v (fun _ -> getx)
- (* we return a unit_set of each result *)
- in unit_set result
- in bind_set u handle_each
-
- This is a computation that takes a bunch of `_ dpm`s and returns `dpm`s that return their input discourse possibilities unaltered, together with the entity those discouse possibilities associate with variable 'x'. We can take \[[x]] to be the `handle_each` function defined above.
-
-
-* They say the denotation of a variable is the entity which the store `h` assigns to the index that the assignment function `r` assigns to the variable. In other words, if the variable is `'x'`, its denotation wrt `(r, h, w)` is `h[r['x']]`. In our OCaml implementation, that will be `List.nth h (r 'x')`.
-
-
-
-* Now how shall we handle \[[∃x]]. As we said, GS&V really tell us how to interpret \[[∃xPx]], but what they say about this breaks naturally into two pieces, such that we can represent the update of `s` with \[[∃xPx]] as:
-
- <pre><code>s >>= \[[∃x]] >>= \[[Px]]
- </code></pre>
-
- What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
-
- > Suppose an information state `s` is updated with the sentence ∃xPx. Possibilities in `s` in which no entity has the property P will be eliminated.
-
- We can defer that to a later step, where we do `... >>= \[[Px]]`.
-
- > The referent system of the remaining possibilities will be extended with a new peg, which is associated with `x`. And for each old possibility `i` in `s`, there will be just as many extensions `i[x/d]` in the new state `s'` and there are entities `d` which in the possible world of `i` have the property P.
-
- Deferring the "property P" part, this says:
-
- <pre><code>s updated with \[[∃x]] ≡
- s >>= (fun (r, h) -> List.map (fun d -> newpeg_and_bind 'x' d) domain)
- </code></pre>
-
- That is, for each pair `(r, h)` in `s`, we collect the result of extending `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities (here designated `domain`), and binding the variable `x` to the index of that peg.
-
- A later step can then filter out all the possibilities in which the entity `d` we did that with doesn't have property P.
-
- So if we just call the function `(fun (r, h) -> ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
-
- <pre><code>s >>= \[[∃x]] >>= \[[Px]]
- </code></pre>
-
- or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
+ or:
- <pre><code>bind_set (bind_set s \[[∃x]]) \[[Px]]
+ <pre><code>u >>= \[[Qx]]
</code></pre>
-* In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:
-
- > { i &elem; s | i does not subsist in s[φ] }
-
- where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
-
- Here's how we can represent that:
-
- <pre><code>bind_set s (fun (r, h) ->
- let u = unit_set (r, h)
- in let descendents = u >>= \[[φ]]
- in if descendents = empty_set then u else empty_set
- </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.