X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_3.mdwn;h=7cc2e74ab92fd0d8d766f9461812782d86f06e88;hp=c678e80b6fa036ae45dfa03913c1c6df2d5a5e3b;hb=30535a559f5df2bcca773bc9b292cbc8e8896174;hpb=00fb05d9ee37681282813893ee69ed7441fe1273

diff --git a/hints/assignment_7_hint_3.mdwn b/hints/assignment_7_hint_3.mdwn
index c678e80b..7cc2e74a 100644
--- a/hints/assignment_7_hint_3.mdwn
+++ b/hints/assignment_7_hint_3.mdwn
@@ -5,22 +5,52 @@
 
 *	They say the denotation of a variable is the object which the store `g` 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, g, w)` is `g[r['x']]`.
 
-We're going to keep all of that, except dropping the worlds. And instead of talking about "\[[expression]] in discourse possibility `(r, g, w)`," we'll just talk about "\[[expression]]" and let that be a monadic object, implemented in part by a function that takes `(r, g)` as an argument.
+We're going to keep all of that, except dropping the worlds. And instead of talking about
 
-More specifically, "\[[expression]]" will be a set of 'a discourse possibility monads, where 'a is the appropriate type for "expression," and the discourse possibility monads are themselves state monads where `(r, g)` is the state that gets updated. Those are implemented as functions from `(r, g)` to `(a, r', g')`, where `a` is a value of type `'a`, and `r', g'` are possibly altered assignment functions and stores.
+>	\[[expression]] in possibility `(r, g, w)`
+
+we'll just talk about \[[expression]] and let that be a monadic object, implemented in part by a function that takes `(r, g)` as an argument.
+
+More specifically, \[[expression]] will be a set of `'a discourse_possibility` monads, where `'a` is the appropriate type for *expression*, and the discourse possibility monads are themselves state monads where `(r, g)` is the state that gets updated. Those are implemented as functions from `(r, g)` to `(a, r', g')`, where `a` is a value of type `'a`, and `r', g'` are possibly altered assignment functions and stores.
 
 *	In def 2.7, GS&V talk about an operation that takes an existing set of discourse possibilities, and extends each member in the set by allocating a new location in the store, and assigning a variable `'x'` to that location, which holds some object `d` from the domain. It will be useful to have a shorthand way of referring to this operation:
 
-		let newpeg_and_bind (variable : char) (d : entity) =
+		let newpeg_and_bind (bound_variable : char) (d : entity) =
 			fun ((r, g) : assignment * store) ->
 				let newindex = List.length g
 				(* first we store d at index newindex in g, which is at the very end *)
 				(* the following line achieves that in a simple but very inefficient way *)
 				in let g' = List.append g [d]
 				(* next we assign 'x' to location newindex *)
-				in let r' = fun variable' ->
-					if variable' = variable then newindex else r variable'
+				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')
 
+*	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.
+
+	They say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the object 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 objects to `bool`s, `s` updated with \[[Qx]] should be `s` filtered by the function `fun (r, g) -> let obj = List.nth g (r 'x') in Q obj`. When `...Q obj` evaluates to `true`, that `(r, g)` pair is retained, else it is discarded.
+
+	Recall that [we said before](/hints/assignment_7_hint_2) that `List.filter (test : 'a -> bool) (u : 'a set) : 'a set` is the same as:
+
+		bind_set u (fun a -> if test a then unit_set a else empty_set)
+
+	Hence, updating `s` with \[[Qx]] should be:
+
+		bind_set s (fun (r, g) -> if (let obj = List.nth g (r 'x') in Q obj) then unit_set (r, g) else empty_set)
+
+	We can call the `(fun (r, g) -> ...)` part \[[Qx]] and then updating `s` with \[[Qx]] will be:
+
+		bind_set s [[Qx]]
+
+	or as it's written using Haskell's infix notation for bind:
+
+		s >>= [[Qx]]
+
+*	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]]
+
+
+