ass10 hint tweaks

[lambda.git] / hints / assignment_7_hint_5.mdwn

diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn
index 40df3b8..d44f824 100644 (file)
--- a/hints/assignment_7_hint_5.mdwn
+++ b/hints/assignment_7_hint_5.mdwn
@@ -1,3 +1,15 @@
+<!--
+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 \[[&exist;x]]? As we said, GS&V really tell us how to interpret \[[&exist;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 \[[&exist;xPx]] as:
 
        <pre><code>u >>= \[[&exist;x]] >>= \[[Px]]
 *      How shall we handle \[[&exist;x]]? As we said, GS&V really tell us how to interpret \[[&exist;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 \[[&exist;xPx]] as:
 
        <pre><code>u >>= \[[&exist;x]] >>= \[[Px]]


@@ -17,8 +29,8 @@
 
        <pre><code>u updated with \[[&exist;x]] &equiv;
                let extend one_dpm (d : entity) =
 
        <pre><code>u updated with \[[&exist;x]] &equiv;
                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):
        </code></pre>
 
        where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):


@@ -46,18 +58,18 @@
 
        or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
 
 
        or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
 

-       <pre><code>bind_set (bind_set u \[[&exist;x]]) \[[Px]]
+       <pre><code>set_bind (set_bind u \[[&exist;x]]) \[[Px]]

        </code></pre>
 
        </code></pre>
 

-*      Let's compare this to what \[[&exist;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 \[[&exist;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;;
 
 /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 r ->
                                let a = u r
                                in let u' = f a


@@ -77,7 +89,7 @@
                        fun entity_reader ->
                                fun r ->
                                        let obj = entity_reader r
                        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:
 
 
        The meaning of \[[Qx]] would then be:
 


@@ -86,7 +98,7 @@
        \[[Qx]] &equiv; \[[Q]] \[[x]] &equiv;
                fun r ->
                        let obj = getx r
        \[[Qx]] &equiv; \[[Q]] \[[x]] &equiv;
                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)]]:
        </code></pre>
 
        Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]:


@@ -103,7 +115,7 @@
 
                fun (lifted_predicate : lifted_unary) ->
                        fun r -> exists (fun (obj : entity) ->
 
                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 \[[&exist;]], which we'd use like this:
 
                        
        That would be the meaning of \[[&exist;]], which we'd use like this:
 


@@ -124,7 +136,7 @@
                                in clause r'
                in let lifted_exists =
                        fun lifted_predicate ->
                                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 fun bool_reader -> lifted_exists (shift 'x' bool_reader)
 
        which we can simplify to:


@@ -137,7 +149,7 @@
                                in clause r'
                in let lifted_exists =
                        fun lifted_predicate ->
                                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 ->
                in fun bool_reader -> lifted_exists (shifted bool_reader)
 
                fun bool_reader ->


@@ -146,11 +158,11 @@
                                        let new_value = entity_reader r
                                        in let r' = fun var -> if var = 'x' then new_value else r var
                                        in bool_reader r'
                                        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 =
 
                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)
                                        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)