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=139597f9cb07c5d48bed18984ec4747f4b4f3438;hb=26574827b88c32f00baaf941c4eac1aaebac839d;hpb=5ffb50f1092baa2b76c7140b8c388d241240fa38;ds=sidebyside diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 139597f9..591fd31d 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -1,2 +1,56 @@ +* 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: +
``````u >>=set \[[∃x]] >>=set \[[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 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 corresponds to: + +
``````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') + + 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 `dpm`s according to which the +entity `d` we did that with doesn't have property P. + + So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just: + +
``````u >>= \[[∃x]] >>= \[[Px]]
+	``````
+ + or, being explicit about which "bind" operation we're representing here with `>>=`, that is: + +
``````bind_set (bind_set u \[[∃x]]) \[[Px]]
+	``````
+ +* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).