X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=d44f824b05968d6de85e0cbc5b46ef97c77205b5;hp=882a62327762366dc361db0e9435c22c31d5cd3f;hb=8cf1fe240800a66d644f907fad8d618b014efd7d;hpb=f212354152a53c6a3ab018c7874570c600f463b9 diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 882a6232..d44f824b 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -1,9 +1,22 @@ + + +* 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.
@@ -15,17 +28,16 @@
Deferring the "property P" part, this corresponds to:

u updated with \[[∃x]] ≡
-		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
+		let extend one_dpm (d : entity) =
+			dpm_bind one_dpm (new_peg_and_assign 'x' d)
+		in set_bind 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) =
-			(* we want to return a function that we can bind to a bool dpm *)
-			fun (truth_value : bool) ->
-				fun ((r, h) : assignment * store) ->
+		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 *)
@@ -35,60 +47,65 @@
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')
+					in (truth_value, r', h');;

-	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.
+	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.)

-	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:
+	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]]

or, being explicit about which "bind" operation we're representing here with `>>=`, that is:

-	bind_set (bind_set u \[[∃x]]) \[[Px]]
+	set_bind (set_bind u \[[∃x]]) \[[Px]]

-*	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](
+*	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 [week7](

type assignment = char -> entity;;
type 'a reader = assignment -> 'a;;

-		let unit_reader (x : 'a) = fun r -> x;;
+		let reader_unit (value : 'a) : 'a reader = fun r -> value;;

fun r ->
let a = u r
in let u' = f a
in u' r;;

-		let getx = fun r -> r 'x';;
+	Here the type of a sentential clause is:
+
+		type clause = bool reader;;
+
+	Here are meanings for singular terms and predicates:

-		let lift (predicate : entity -> bool) =
+		let getx : entity reader = fun r -> r 'x';;
+
+
+		let lift (predicate : entity -> bool) : lifted_unary =
fun r ->

-	`lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be:
+	The meaning of \[[Qx]] would then be:

\[[Q]] ≡ lift q
\[[x]] ≡ getx
\[[Qx]] ≡ \[[Q]] \[[x]] ≡
fun r ->
let obj = getx r

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 *)
+		let shift (var_to_bind : char) (clause : clause) : lifted_unary =
-				fun (r : assignment) ->
+				fun 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
@@ -96,12 +113,13 @@

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 r -> exists (fun (obj : entity) -> lifted_predicate (unit_reader obj) r)
+		fun (lifted_predicate : lifted_unary) ->
+			fun r -> exists (fun (obj : entity) ->

That would be the meaning of \[[∃]], which we'd use like this:

-	\[[∃]] \[[Q]]
+	\[[∃]] ( \[[Q]] )

or this:
@@ -118,11 +136,12 @@
in clause r'
in let lifted_exists =
fun lifted_predicate ->
-				fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
+				fun r -> exists (fun obj -> lifted_predicate (reader_unit obj) r)

-	which we can simplify as:
+	which we can simplify to:

+

-			let shifted'' r new_value =
+			let shifted r new_value =
let r' = fun var -> if var = 'x' then new_value else r var
-			in fun r -> exists (shifted'' 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:
@@ -171,9 +191,11 @@
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're able to interpret claims like:
+	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).

-	>	If ∃y (farmer y and ∃x y owns x) then (y beats x).
+	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.
``````