X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_4.mdwn;h=046a45b9c05008b7d137f0c4122e17c211115c30;hp=d41e0f0adff4da5916b6388b89a1eda1f762106b;hb=61612ba6746dc4646fb0d5bb1c9e1864bf927848;hpb=458cadab1427b0fc0f7bc8689f1dddb18b2201e7 diff --git a/hints/assignment_7_hint_4.mdwn b/hints/assignment_7_hint_4.mdwn index d41e0f0a..046a45b9 100644 --- a/hints/assignment_7_hint_4.mdwn +++ b/hints/assignment_7_hint_4.mdwn @@ -1,60 +1,63 @@ -* 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 most natural to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. But first we need to get clear on expressions like \[[Px]]. +* At the top of p. 13, GS&V give two examples, one for \[[∃xPx]] and the other for \[[Qx]]. For our purposes, it will be most natural to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. But first we need to get clear on expressions like \[[Qx]] and \[[Px]]. -* GS&V say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the entity 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 entities 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. +* GS&V say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the entity associated with the peg associated with the variable `x` does not have the property Q. In other words, if we let `q` be the function from entities to `bool`s that gives the extension of Q, then `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. - OK, we face two questions then. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here ascribed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"? + OK, so we face two questions. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"? -* Answering the first question: we assume we've got some `bool dpm set` to start with. I won't call this `s` because that's what GS&V use for sets of discourse possibilities, and we don't want to confuse discourse possibilities with `dpm`s. Instead I'll call it `u`. Now what we want to do with `u` is to map each `dpm` it gives us to one that results in `(true, r, h)` only when the entity that `r` and `h` associate with variable `x` has the property Q. I'll assume we have some function Q to start with that maps entities to `bool`s. +* Answering the first question: we assume we've got some `bool dpm set` to start with. I won't call this `s` because that's what GS&V use for sets of discourse possibilities, and we don't want to confuse discourse possibilities with `dpm`s. Instead I'll call it `u`. Now what we want to do with `u` is to map each `dpm` it gives us to one that results in `(true, r, h)` only when the entity that `r` and `h` associate with variable `x` has the property Q. As above, I'll assume Q's extension is given by a function `q` from entities to `bool`s. Then what we want is something like this: - let eliminate_non_Qxs = (fun truth_value -> - fun (r, h) -> - let truth_value' = - if truth_value - then let obj = List.nth h (r 'x') in Q obj - else false - in (truth_value', r, h)) - in bind_set u (fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)) + let eliminator : bool -> bool dpm = + fun truth_value -> + fun (r, h) -> + let truth_value' = + if truth_value + then let obj = List.nth h (r 'x') in q obj + else false + in (truth_value', r, h) + in set_bind u (fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)) - The first three seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` whether its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `bind_set` operation. This works by taking each `dpm` in the set and returning a `unit_set` of a filtered `dpm`. The definition of `bind_set` takes care of collecting together all of the `unit_set`s that result for each different set element we started with. + The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `set_bind` operation. This works by taking each `dpm` in the set and returning a `set_unit` of a filtered `dpm`. The definition of `set_bind` takes care of collecting together all of the `set_unit`s that result for each different set element we started with. We can call the `(fun one_dpm -> ...)` part \[[Qx]] and then updating `u` with \[[Qx]] will be: - bind_set u \[[Qx]] + set_bind u \[[Qx]] or as it's written using Haskell's infix notation for bind: u >>= \[[Qx]] -* Now our second question: how do we decompose the behavior here ascribed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"? +* Now our second question: how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"? - Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about sets of `dpm`s; we'll leave that to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. Then what we want is: + Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about `dpm set`s; we'll leave them to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. So what we want is: - let getx = fun (r, h) -> + let getx : entity dpm = fun (r, h) -> let obj = List.nth h (r 'x') in (obj, r, h);; -* Now what do we do with predicates? As before, we suppose we have a function Q that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting Q into a function that takes `entity dpm`s as arguments and returns `bool dpm`s: +* Now what do we do with predicates? As before, we suppose we have a function `q` that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting `q` into a function that takes `entity dpm`s as arguments and returns `bool dpm`s: - fun entity_dpm -> bind_dpm entity_dpm (fun e -> unit_dpm (Q e)) + fun entity_dpm -> dpm_bind entity_dpm (fun e -> dpm_unit (q e)) - Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `unit_set`: + Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `set_unit`: - fun entity_dpm -> unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (Q e))) + fun entity_dpm -> set_unit (dpm_bind entity_dpm (fun e -> dpm_unit (q e))) - Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to compose \[[Qx]] with them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so. + Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to monadically bind \[[Qx]] to them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so. - This is most easily done like this: + This could be handled like this: fun entity_dpm -> - fun truth_value -> - if truth_value = false - then empty_set - else unit_set (bind dpm entity_dpm (fun e -> unit_dpm (Q e))) + let eliminator : bool -> bool dpm = + fun truth_value -> + if truth_value = false + then dpm_unit false + else dpm_bind entity_dpm (fun e -> dpm_unit (q e)) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) - Doing things this way will discard `bool dpm`s that start out wrapping `false`, and will pass through other `bool dpm`s that start out wrapping `true` but which our current filter transforms to a wrapped `false`. You might instead aim for consistency, and always pass through wrapped `false`s, whether they started out that way or are only now being generated; or instead always discard such, and only pass through wrapped `true`s. But what we have here will work fine too. + Applied to an `entity_dpm`, that yields a function that we can bind to a `bool dpm set` and that will transform the doubly-wrapped `bool` into a new `bool dpm set`. If we let that be \[[Q]], then \[[Q]] \[[x]] would be: @@ -62,79 +65,93 @@ let obj = List.nth h (r 'x') in (obj, r, h) in let entity_dpm = getx - in fun truth_value -> + in let eliminator = fun truth_value -> if truth_value = false - then empty_set - else unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (Q e))) + then dpm_unit false + else dpm_bind entity_dpm (fun e -> dpm_unit (q e)) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) + - which is: + If we simplify and unpack the definition of `dpm_bind`, that's equivalent to: let getx = fun (r, h) -> let obj = List.nth h (r 'x') in (obj, r, h) - in fun truth_value -> + in let eliminator = fun truth_value -> if truth_value - then unit_set ( - fun (r, h) -> + then (fun (r, h) -> let (a, r', h') = getx (r, h) - in let u' = (fun e -> unit_dpm (Q e)) a + in let u' = (fun e -> dpm_unit (q e)) a in u' (r', h') - ) else empty_set - - which is: + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) - in fun truth_value -> + which can be further simplified to: + + + + let eliminator = fun truth_value -> + if truth_value + then (fun (r, h) -> + let obj = List.nth h (r 'x') + in (q obj, r, h) + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) + + This is a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. + + (Compare to the \[[Qx]] we had before: - let eliminate_non_Qxs = (fun truth_value -> + let eliminator = (fun truth_value -> fun (r, h) -> let truth_value' = if truth_value - then let obj = List.nth h (r 'x') in Q obj + then let obj = List.nth h (r 'x') in q obj else false in (truth_value', r, h)) - in (fun one_dpm -> unit_set (bind_dpm one_dpm eliminate_non_Qxs)) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) - because that one passed through every `bool dpm` that wrapped a `false`; whereas now we're discarding some of them. But these will work equally well. We can implement either behavior (or, as we said before, the behavior of never passing through a wrapped `false`). + Can you persuade yourself that these are equivalent?) -* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We compose this with a `bool dpm set` we already have on hand: +* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We monadically bind this operaration to whatever `bool dpm set` we already have on hand: - bind_set u \[[Qx]] + set_bind u \[[Qx]] or: - 
u >>=set \[[Qx]]
+	
u >>= \[[Qx]]
 	

 
-*	Can you figure out how to handle \[[∃x]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_5).
+*	Can you figure out how to handle \[[∃x]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_5). But try to get as far as you can on your own.