-* In def 2.5, they say the denotation of an e-type constant <code>α</code> wrt a discourse possibility `(r, g, w)` is whatever object the world `w` associates with <code>α</code>. Since we don't have worlds, this will just be an object.
+* In def 2.5, they say the denotation of an e-type constant <code>α</code> wrt a discourse possibility `(r, h, w)` is whatever object the world `w` associates with <code>α</code>. Since we don't have worlds, this will just be an object.
* They say the denotation of a predicate is whatever extension the world `w` associates with the predicate. Since we don't have worlds, this will just be an extension.
-* 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']]`.
+* They say the denotation of a variable is the object which the store `h` 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, h, w)` is `h[r['x']]`. In our OCaml implementation, that will be `List.nth h (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, h, w)`
+
+we'll just talk about \[[expression]] and let that be a monadic object, implemented in part by a function that takes `(r, h)` 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, h)` is the state that gets updated. Those are implemented as functions from `(r, h)` to `(a, r', h')`, where `a` is a value of type `'a`, and `r', h'` 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) =
- 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 *)
+ let newpeg_and_bind (bound_variable : char) (d : entity) =
+ fun ((r, h) : assignment * store) ->
+ let newindex = List.length h
+ (* first we store d at index newindex in h, 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]
+ in let h' = List.append h [d]
(* next we assign 'x' to location newindex *)
- in let r' = fun variable' ->
- if variable' = variable then newindex else r variable'
- (* the reason for returning a triple with () in first position will emerge *)
- in ((), r',g')
+ in let r' = fun v ->
+ if v = bound_variable then newindex else r v
+ in (r',h')
+
+* 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, h) -> let obj = List.nth h (r 'x') in Q obj`. When `...Q obj` evaluates to `true`, that `(r, h)` 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, h) -> if (let obj = List.nth h (r 'x') in Q obj) then unit_set (r, h) else empty_set)
+
+ We can call the `(fun (r, h) -> ...)` 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:
+
+ <pre><code>s >>= \[[∃x]] >>= \[[Px]]
+ </code></pre>
+
+ 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:
+
+ <pre><code>s updated with \[[∃x]] ≡
+ 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 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, h) -> ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
+
+ <pre><code>s >>= \[[∃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 \[[∃x]]) \[[Px]]
+ </code></pre>
+
+* 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 <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
+
+ Here's how we can represent that:
-* 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]] (or \[[Qx]]) first.
+ <pre><code>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
+ </code></pre>