X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=591fd31d9421859776cb4220a92ecb8a3076f16e;hp=46ca442dfeba847526e8f6a5683ce51609aee528;hb=26574827b88c32f00baaf941c4eac1aaebac839d;hpb=458cadab1427b0fc0f7bc8689f1dddb18b2201e7;ds=sidebyside diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 46ca442d..591fd31d 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -1,7 +1,7 @@ -* 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: +* 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 our starting set `u` with \[[∃xPx]] as: -
``````s >>= \[[∃x]] >>= \[[Px]]
+	u >>=set \[[∃x]] >>=set \[[Px]]

What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
@@ -12,38 +12,45 @@

>	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:
+	Deferring the "property P" part, this corresponds to:

-	s updated with \[[∃x]] ≡
-		s >>= (fun (r, h) -> List.map (fun d -> newpeg_and_bind 'x' d) domain)
+	u updated with \[[∃x]] ≡
+		let extend_one = fun one_dpm ->
+			fun truth_value ->
+				if truth_value = false
+				then empty_set
+				else List.map (fun d -> new_peg_and_assign 'x' d) domain
+		in bind_set u extend_one

+
+	where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
+
+		let new_peg_and_assign (var_to_bind : char) (d : entity) =
+			fun ((r, h) : assignment * store) ->
+				(* first we calculate an unused index *)
+				let newindex = List.length h
+				(* next we store d at h[newindex], which is at the very end of h *)
+				(* the following line achieves that in a simple but inefficient way *)
+				in let h' = List.append h [d]
+				(* next we assign 'x' to location newindex *)
+				in let r' = fun v ->
+					if v = var_to_bind then newindex else r v
+				(* the reason for returning true as an initial element should now be apparent *)
+				in (true, r',h')

-	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.
+	What's going on here? For each `bool dpm` in `u` that wraps a `true`, we collect `dpm`s that are the result of extending their input `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities, and binding the variable `x` to the index of that peg. For `bool dpm`s in `u` that wrap `false`, we just discard them. We could if we wanted instead return `unit_set (unit_dpm false)`.

-	A later step can then filter out all the possibilities in which the entity `d` we did that with doesn't have property P.
+	A later step can then filter out all the `dpm`s according to 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:
+	So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:

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

or, being explicit about which "bind" operation we're representing here with `>>=`, that is:

-	bind_set (bind_set s \[[∃x]]) \[[Px]]
+	bind_set (bind_set u \[[∃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, h) ->
-			let u = unit_set (r, h)
-			in let descendents = u >>= \[[φ]]
-			in if descendents = empty_set then u else empty_set
-
-
-
+*	Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).
``````