+<!--
+I would like to propose one pedegogical suggestion (due to Ken), which
+is to separate peg addition from non-determinacy by explicitly adding
+a "Let" construction to GSV's logic, i.e., "Let of var * term *
+clause", whose interpretation adds a peg, assigns var to it, sets the
+value to the value computed by term, and evaluates the clause with the
+new peg in place. This can be added easily, especially since you have
+supplied a procedure that handles the main essence of the
+construction. Once the Let is in place, adding the existential is
+purely dealing with nondeterminism.
+-->
+
* 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:
<pre><code>u >>= \[[∃x]] >>= \[[Px]]
<pre><code>u updated with \[[∃x]] ≡
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)
+ 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)
</code></pre>
where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
- <pre><code>bind_set (bind_set u \[[∃x]]) \[[Px]]
+ <pre><code>set_bind (set_bind u \[[∃x]]) \[[Px]]
</code></pre>
-* 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](
+* 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](
/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 reader_unit (value : 'a) : 'a reader = fun r -> value;;
- let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
+ let reader_bind (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
fun r ->
let a = u r
in let u' = f a
fun entity_reader ->
fun r ->
let obj = entity_reader r
- in unit_reader (predicate obj)
+ in reader_unit (predicate obj)
The meaning of \[[Qx]] would then be:
\[[Qx]] ≡ \[[Q]] \[[x]] ≡
fun r ->
let obj = getx r
- in unit_reader (q obj)
+ in reader_unit (q obj)
</code></pre>
Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]:
fun (lifted_predicate : lifted_unary) ->
fun r -> exists (fun (obj : entity) ->
- lifted_predicate (unit_reader obj) r)
+ lifted_predicate (reader_unit obj) r)
That would be the meaning of \[[∃]], which we'd use like this:
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)
in fun bool_reader -> lifted_exists (shift 'x' bool_reader)
which we can simplify to:
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)
in fun bool_reader -> lifted_exists (shifted bool_reader)
fun bool_reader ->
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)
+ in fun r -> exists (fun obj -> shifted' (reader_unit obj) r)
fun bool_reader ->
let shifted'' r obj =
- let new_value = (unit_reader obj) r
+ let new_value = (reader_unit 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)