X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=a0ac32a3e8066a71968a56684027591a2eff228c;hp=07cbd648c5eba86e4919784bce57c24969fe1925;hb=4bbcf7dfee362fee1c3a23650912a2e259f2f51a;hpb=679852d7a5f67d6da3443b959acb1a41f3558c85 diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 07cbd648..a0ac32a3 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -8,40 +8,38 @@ > 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]]`. + We can defer that to a later step, where we do `... >>= \[[Px]]`. GS&V continue: - > 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. + > 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'` as 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
+		let extend_one = fun (one_dpm : bool dpm) ->
+			List.map (fun d -> bind_dpm one_dpm (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') + (* we want to return a function that we can bind to a bool dpm *) + fun (truth_value : bool) -> + fun ((r, h) : assignment * store) -> + (* first we calculate an unused index *) + let new_index = List.length h + (* next we store d at h[new_index], 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 new_index *) + in let r' = fun var -> + if var = var_to_bind then new_index else r var + (* we pass through the same truth_value that we started with *) + in (truth_value, r', h') - What's going on in this representation of `u` updated with \[[∃x]]? 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)`. + What's going on in this representation of `u` updated with \[[∃x]]? For each `bool dpm` in `u`, we collect `dpm`s that are the result of passing through their `bool`, but 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. - 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. + A later step can then filter out all the `dpm`s where 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: @@ -53,4 +51,131 @@ entity `d` we did that with doesn't have property P. 
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). +* Let's compare this to what \[[∃xPx]] would look like on a non-dynamic semantics, for example, where we use a simple reader monad to implement variable binding. Reminding ourselves, we'd be working in a framework like this. (Here we implement environments or assignments as functions from variables to entities, instead of as lists of pairs of variables and entities. An assignment `r` here is what `fun c -> List.assoc c r` would have been in [week6]( +/reader_monad_for_variable_binding).) + + type assignment = char -> entity;; + type 'a reader = assignment -> 'a;; + + let unit_reader (x : 'a) = fun r -> x;; + + let bind_reader (u : 'a reader) (f : 'a -> 'b reader) = + fun r -> + let a = u r + in let u' = f a + in u' r;; + + let getx = fun r -> r 'x';; + + let lift (predicate : entity -> bool) = + fun entity_reader -> + fun r -> + let obj = entity_reader r + in unit_reader (predicate obj) + + `lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be: + + 
\[[Q]] ≡ lift q
+	\[[x]] ≡ getx
+	\[[Qx]] ≡ \[[Q]] \[[x]] ≡
+		fun r ->
+			let obj = getx r
+			in unit_reader (q obj)
+
+ + Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]: + + let shift (var_to_bind : char) (clause : bool reader) = + (* we return a lifted predicate, that is a entity reader -> bool reader *) + fun entity_reader -> + fun (r : assignment) -> + let new_value = entity_reader r + (* remember here we're implementing assignments as functions rather than as lists of pairs *) + in let r' = fun var -> if var = var_to_bind then new_value else r var + in clause r' + + Now, how would we implement quantifiers in this setting? I'll assume we have a function `exists` of type `(entity -> bool) -> bool`. That is, it accepts a predicate as argument and returns `true` if any element in the domain satisfies that predicate. We could implement the reader-monad version of that like this: + + fun (lifted_predicate : entity reader -> bool reader) -> + fun r -> exists (fun (obj : entity) -> + lifted_predicate (unit_reader obj) r) + + That would be the meaning of \[[∃]], which we'd use like this: + + 
\[[∃]] \[[Q]]
+
+ + or this: + + 
\[[∃]] ( \[[lambda x]] \[[Qx]] )
+
+ + If we wanted to compose \[[∃]] with \[[lambda x]], we'd get: + + let shift var_to_bind clause = + fun entity_reader r -> + let new_value = entity_reader r + in let r' = fun var -> if var = var_to_bind then new_value else r var + in clause r' + in let lifted_exists = + fun lifted_predicate -> + fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r) + in fun bool_reader -> lifted_exists (shift 'x' bool_reader) + + which we can simplify as: + + let shifted clause = + fun entity_reader r -> + let new_value = entity_reader r + in let r' = fun var -> if var = 'x' then new_value else r var + in clause r' + in let lifted_exists = + fun lifted_predicate -> + fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r) + in fun bool_reader -> lifted_exists (shifted bool_reader) + + fun bool_reader -> + let shifted' = + fun entity_reader r -> + let new_value = entity_reader r + in let r' = fun var -> if var = 'x' then new_value else r var + in bool_reader r' + in fun r -> exists (fun obj -> shifted' (unit_reader obj) r) + + fun bool_reader -> + let shifted'' r obj = + let new_value = (unit_reader obj) r + in let r' = fun var -> if var = 'x' then new_value else r var + in bool_reader r' + in fun r -> exists (fun obj -> shifted'' r obj) + + fun bool_reader -> + let shifted'' r obj = + let new_value = obj + in let r' = fun var -> if var = 'x' then new_value else r var + in bool_reader r' + in fun r -> exists (shifted'' r) + + fun bool_reader -> + let shifted'' r new_value = + let r' = fun var -> if var = 'x' then new_value else r var + in bool_reader r' + in fun r -> exists (shifted'' r) + + This gives us a value for \[[∃x]], which we use like this: + + 
\[[∃x]] ( \[[Qx]] )
+
+ + Contrast the way we use \[[∃x]] in GS&V's system. Here we don't have a function that takes \[[Qx]] as an argument. Instead we have a operation that gets bound in a discourse chain: + + 
u >>= \[[∃x]] >>= \[[Qx]]
+
+ + The crucial difference in GS&V's system is that the distinctive effect of the \[[∃x]]---to allocate new pegs in the store and associate variable `x` with the objects stored there---doesn't last only while interpreting clauses supplied as arguments to \[[∃x]]. Instead, it persists through the discourse, possibly affecting the interpretation of claims outside the logical scope of the quantifier. This is how we'll able to interpret claims like: + + > If ∃x (man x and ∃y y is wife of x) then (x kisses y). + + +* Can you figure out how to handle \[[not φ]] and the other connectives? If not, here are some [more hints](/hints/assignment_7_hint_6). But try to get as far as you can on your own. +