assignment7 tweaks

[lambda.git] / hints / assignment_7_hint_5.mdwn

*       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:

        <pre><code>u >>= \[[&exist;x]] >>= \[[Px]]
        </code></pre>

        What does \[[&exist;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 &exist;xPx. Possibilities in `s` in which no entity has the property P will be eliminated.

        We can defer that to a later step, where we do `... >>= \[[Px]]`. GS&V continue:
 
        >       The referent system of the remaining possibilities will be extended with a new peg, which is associated with `x`. And for each old possibility `i` in `s`, there will be just as many extensions `i[x/d]` in the new state `s'` as there are entities `d` which in the possible world of `i` have the property P.

        Deferring the "property P" part, this corresponds to:

        <pre><code>u updated with \[[&exist;x]] &equiv;
                let extend_one : clause = fun one_dpm ->
                        List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign 'x' d)) domain
                in bind_set u extend_one
        </code></pre>

        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) : 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 *)
                                        (* the following line achieves that in a simple but inefficient way *)
                                        in let h' = List.append h [d]
                                        (* next we assign 'x' to location new_index *)
                                        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');;
        
        What's going on in this representation of `u` updated with \[[&exist;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.

        So if we just call the function `extend_one` defined above \[[&exist;x]], then `u` updated with \[[&exist;x]] updated with \[[Px]] is just:

        <pre><code>u >>= \[[&exist;x]] >>= \[[Px]]
        </code></pre>

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

        <pre><code>bind_set (bind_set u \[[&exist;x]]) \[[Px]]
        </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](
/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 bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
                        fun r ->
                                let a = u r
                                in let u' = f a
                                in u' r;;

        Here the type of a sentential clause is:

                type clause = bool reader;;

        Here are meanings for singular terms and predicates:

                let getx : entity reader = fun r -> r 'x';;

                type lifted_unary = entity reader -> bool reader;;

                let lift (predicate : entity -> bool) : lifted_unary =
                        fun entity_reader ->
                                fun r ->
                                        let obj = entity_reader r
                                        in unit_reader (predicate obj)

        The meaning of \[[Qx]] would then be:

        <pre><code>\[[Q]] &equiv; lift q
        \[[x]] &equiv; getx
        \[[Qx]] &equiv; \[[Q]] \[[x]] &equiv;
                fun r ->
                        let obj = getx r
                        in unit_reader (q obj)
        </code></pre>

        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 : clause) : lifted_unary =
                        fun entity_reader ->
                                fun r ->
                                        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 clause 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 : lifted_unary) ->
                        fun r -> exists (fun (obj : entity) ->
                                lifted_predicate (unit_reader obj) r)
                        
        That would be the meaning of \[[&exist;]], which we'd use like this:

        <pre><code>\[[&exist;]] \[[Q]]
        </code></pre>

        or this:

        <pre><code>\[[&exist;]] ( \[[lambda x]] \[[Qx]] )
        </code></pre>

        If we wanted to compose \[[&exist;]] with \[[lambda x]], we'd get:

                let shift var_to_bind clause =
                        fun entity_reader r ->
                                let new_value = entity_reader r
                                in let r' = fun var -> if var = var_to_bind then new_value else r var
                                in clause 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' bool_reader)

        which we can simplify as:

                let shifted clause =
                        fun entity_reader r ->
                                let new_value = entity_reader r
                                in let r' = fun var -> if var = 'x' then new_value else r var
                                in clause 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)

                fun bool_reader ->
                        let shifted' =
                                fun entity_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)

                fun bool_reader ->
                        let shifted'' r obj =
                                        let new_value = (unit_reader 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)

                fun bool_reader ->
                        let shifted'' r obj =
                                        let new_value = obj
                                        in let r' = fun var -> if var = 'x' then new_value else r var
                                        in bool_reader r'
                        in fun r -> exists (shifted'' r)

                fun bool_reader ->
                        let shifted'' r new_value =
                                        let r' = fun var -> if var = 'x' then new_value else r var
                                        in bool_reader r'
                        in fun r -> exists (shifted'' r)

        This gives us a value for \[[&exist;x]], which we use like this:

        <pre><code>\[[&exist;x]] ( \[[Qx]] )
        </code></pre>

        Contrast the way we use \[[&exist;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:

        <pre><code>u >>= \[[&exist;x]] >>= \[[Qx]]
        </code></pre>

        The crucial difference in GS&V's system is that the distinctive effect of the \[[&exist;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 \[[&exist;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 &exist;x (man x and &exist;y y is wife of x) then (x kisses y).


*       Can you figure out how to handle \[[not &phi;]] 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.