From de64b1646254627a9e0d10fcdea232c74104e7e5 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Thu, 18 Nov 2010 18:15:57 -0500 Subject: [PATCH 1/1] assignment7 tweaks Signed-off-by: Jim Pryor --- hints/assignment_7_hint_3.mdwn | 46 ++++++++++++++++++++++++++++++++++++++---- 1 file changed, 42 insertions(+), 4 deletions(-) diff --git a/hints/assignment_7_hint_3.mdwn b/hints/assignment_7_hint_3.mdwn index 5fbcb411..ed6bc36a 100644 --- a/hints/assignment_7_hint_3.mdwn +++ b/hints/assignment_7_hint_3.mdwn @@ -24,8 +24,7 @@ More specifically, \[[expression]] will be a set of `'a discourse_possibility` m (* next we assign 'x' to location newindex *) in let r' = fun v -> if v = bound_variable then newindex else r v - (* the reason for returning a triple with () in first position will emerge *) - in ((), r',g') + in (r',g') * 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 easiest for us to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. Let's consider expressions like \[[Px]] first. @@ -49,10 +48,49 @@ More specifically, \[[expression]] will be a set of `'a discourse_possibility` m * 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: - 

-	s >>= \[[∃x]] >>= \[[Px]]
+	
s >>= \[[∃x]] >>= \[[Px]]
 	

 
+	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 object 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 objects `d` which in the possible world of `i` have the property P.
+
+	Deferring the "property P" part, this says:
+
+	
s updated with \[[∃x]] ≡
+		s >>= (fun (r, g) -> List.map (fun d -> newpeg_and_bind 'x' d) domain)
+	

+	
+	That is, for each pair `(r, g)` in `s`, we collect the result of extending `(r, g)` by allocating a new peg for object `d`, for each `d` in our whole domain of objects (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 object `d` we did that with doesn't have property P.
+
+	So if we just call the function `(fun (r, g) -> ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
+
+	
s >>= \[[∃x]] >>= \[[Px]]
+	

+
+	or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
+
+	
bind_set (bind_set s \[[∃x]]) \[[Px]]
+	

+
+*	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 s[φ] if there are any `i'` that *extend* `i` in s[φ].
+
+	Here's how we can represent that:
+
+		
bind_set s (fun (r, g) ->
+			let u = unit_set (r, g)
+			in let descendents = u >>= \[[φ]]
+			in if descendents = empty_set then u else empty_set
+		

 
 
-- 
2.11.0