X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=1b801d1935ad7cf630aac801590b21ae50075991;hp=07cbd648c5eba86e4919784bce57c24969fe1925;hb=1b40bc7e0915e247ecaa6ea9d583b26790c31a74;hpb=679852d7a5f67d6da3443b959acb1a41f3558c85 diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn index 07cbd648..1b801d19 100644 --- a/hints/assignment_7_hint_5.mdwn +++ b/hints/assignment_7_hint_5.mdwn @@ -28,15 +28,15 @@ 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 *) + 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 newindex *) - in let r' = fun v -> - if v = var_to_bind then newindex else r v + (* next we assign 'x' to location new_index *) + in let r' = fun var -> + if var = var_to_bind then new_index else r var (* the reason for returning true as an initial element should now be apparent *) - in (true, r',h') + in (true, 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)`. @@ -53,4 +53,108 @@ 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]] & equiv; 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) entity_reader (v : 'a 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 v 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) : 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 entity_reader v = + fun r -> + let new_value = entity_reader r + in let r' = fun var -> if var = var_to_bind then new_value else r var + in v 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' getx bool_reader) + + which we can simplify to: + + let shifted v = + fun r -> + let new_value = r 'x' + in let r' = fun var -> if var = 'x' then new_value else r var + in v 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) + + and simplifying further: + + fun bool_reader -> + let shifted v = + fun r -> + let new_value = r 'x' + in let r' = fun var -> if var = 'x' then new_value else r var + in v r' + let lifted_predicate = shifted bool_reader + in fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r) + + fun bool_reader -> + let lifted_predicate = fun r -> + let new_value = r 'x' + in let r' = fun var -> if var = 'x' then new_value else r var + in bool_reader r' + in fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r) + + + + + + + +* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6). But try to get as far as you can on your own.