X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=40df3b81971f5b8c31139c6fcfd7bc942fbdce24;hp=07cbd648c5eba86e4919784bce57c24969fe1925;hb=ddcb99cc42fe32a7a16ba2fe78b9228c354807d0;hpb=679852d7a5f67d6da3443b959acb1a41f3558c85 diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 07cbd648..40df3b81 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -1,49 +1,45 @@ +* How shall we handle \[[∃x]]? As we said, GS&V really tell us how to interpret \[[∃xPx]], but for our purposes, what they say about this can be broken naturally into two pieces, such that we represent the update of our starting set `u` 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: - - 
u >>=set \[[∃x]] >>=set \[[Px]]
+	
u >>= \[[∃x]] >>= \[[Px]]
 	

 
+	(Extra credit: how does the discussion on pp. 25-29 of GS&V bear on the possibility of this simplification?)
+
 	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]]`.
+	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
-		in bind_set u extend_one
+		let extend one_dpm (d : entity) =
+			bind_dpm one_dpm (new_peg_and_assign 'x' d)
+		in bind_set u (fun one_dpm -> List.map (fun d -> extend one_dpm d) domain)
 	

 
 	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')
+		let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
+			fun truth_value ->
+				fun (r, h) ->
+					(* 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)`.
-
-	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.
+	What's going on in this proposed representation of \[[∃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 where the entity `d` we did that with doesn't have property P. (Again, consult GS&V pp. 25-9 for extra credit.)
 
-	So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
+	If we call the function `(fun one_dom -> List.map ...)` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
 
 	
u >>= \[[∃x]] >>= \[[Px]]
 	

@@ -53,4 +49,142 @@ 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 (value : 'a) : 'a reader = fun r -> value;;
+
+		let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
+			fun r ->
+				let a = u r
+				in let u' = f a
+				in u' r;;
+
+	Here the type of a sentential clause is:
+
+		type clause = bool reader;;
+
+	Here are meanings for singular terms and predicates:
+
+		let getx : entity reader = fun r -> r 'x';;
+
+		type lifted_unary = entity reader -> bool reader;;
+
+		let lift (predicate : entity -> bool) : lifted_unary =
+			fun entity_reader ->
+				fun r ->
+					let obj = entity_reader r
+					in unit_reader (predicate obj)
+
+	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 : clause) : lifted_unary =
+			fun entity_reader ->
+				fun r ->
+					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 : lifted_unary) ->
+			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 to:
+
+	
+
+		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 some 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).
+
+	See the discussion on pp. 24-5 of GS&V.
+
+
+*	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.
+