assignment7 tweaks

[lambda.git] / hints / assignment_7_hint_5.mdwn

diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn
index 139597f..46ca442 100644 (file)
--- a/hints/assignment_7_hint_5.mdwn
+++ b/hints/assignment_7_hint_5.mdwn
@@ -1,2 +1,49 @@
 
+*      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 >>= \[[&exist;x]] >>= \[[Px]]
+       </code></pre>
+
+       What does \[[&exist;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 &exist;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 \[[&exist;x]] &equiv;
+               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 \[[&exist;x]], then `s` updated with \[[&exist;x]] updated with \[[Px]] is just:
+
+       <pre><code>s >>= \[[&exist;x]] >>= \[[Px]]
+       </code></pre>
+
+       or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
+
+       <pre><code>bind_set (bind_set s \[[&exist;x]]) \[[Px]]
+       </code></pre>
+
+*      In def 3.1 on p. 14, GS&V define `s` updated with \[[not &phi;]] as:
+
+       >       { i &elem; s | i does not subsist in s[&phi;] }
+
+       where `i` *subsists* in <code>s[&phi;]</code> if there are any `i'` that *extend* `i` in <code>s[&phi;]</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 >>= \[[&phi;]]
+                       in if descendents = empty_set then u else empty_set
+               </code></pre>
+