where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
- Here's how to do that in our framework. Instead of a possibility subsisting in an updated set of possibilities, we ask what is returned by extensions of a `dpm` when they're given a particular (r, h) as input.
+ Here's how to do that in our framework. Instead of asking whether a possibility subsists in an updated set of possibilities, we ask what is returned by extensions of a `dpm` when they're given a particular (r, h) as input.
(* filter out which bool dpms in a set are true when receiving (r, h) as input *)
- let truths set (r, h) = List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (r, h) in truth_value) set;;
+ let truths set (r, h) =
+ let test one_dpm =
+ let (truth_value, _, _) = one_dpm (r, h)
+ in truth_value
+ in List.filter test set;;
let negate_op (phi : clause) : clause =
fun one_dpm ->
let new_dpm = fun (r, h) ->
- (* we want to check the behavior of one_dpm when updated with the operation phi *)
- (* bind_set (unit_set one_dpm) phi === phi one_dpm; do you remember why? *)
- let truth_value' = truths (phi one_dpm) (r, h) = []
- (* we return a (bool, r, h) so as to constitute a dpm *)
+ (* if one_dpm isn't already false at (r, h),
+ we want to check its behavior when updated with phi
+ bind_set (unit_set one_dpm) phi === phi one_dpm; do you remember why? *)
+ let (truth_value, _, _) = one_dpm (r, h)
+ in let truth_value' = truth_value && (truths (phi one_dpm) (r, h) = [])
+ (* new_dpm must return a (bool, r, h) *)
in (truth_value', r, h)
- in unit_set new_dpm
+ in unit_set new_dpm;;
+ **Note: Simon pointed out a subtle error in this code, which we will look into fixing. At the moment, the subtle error is still there.**
+
* Representing \[[and φ ψ]] is simple:
let and_op (phi : clause) (psi : clause) : clause =
fun one_dpm -> bind_set (phi one_dpm) psi;;
+ (* now u >>= and_op phi psi === u >>= phi >>= psi; do you remember why? *)
+
* Here are `or` and `if`:
let or_op (phi : clause) (psi : clause) =
- (* NOT: negate_op (and_op (negate_op phi) (negate_op psi)) *)
fun one_dpm -> unit_set (
fun (r, h) ->
- in let truth_value' = truths (phi one_dpm) (r, h) <> [] || truths (bind_set (negate_op phi one_dpm) psi) (r, h) <> []
- in (truth_value', r, h))
+ let truth_value' = (
+ truths (phi one_dpm) (r, h) <> [] ||
+ truths (bind_set (negate_op phi one_dpm) psi) (r, h) <> []
+ ) in (truth_value', r, h))
let if_op (phi : clause) (psi : clause) : clause =
- (* NOT: negate_op (and_op phi (negate_op psi)) *)
fun one_dpm -> unit_set (
fun (r, h) ->
- in let truth_value' = List.for_all (fun one_dpm ->
+ let truth_value' = List.for_all (fun one_dpm ->
let (truth_value, _, _) = one_dpm (r, h)
in truth_value = false || truths (psi one_dpm) (r, h) <> []
) (phi one_dpm)
type clause = bool dpm -> bool dpm set;;
+* More:
+
(* this generalizes the getx function from hint 4 *)
let get (var : char) : entity dpm =
fun (r, h) ->
(* this generalizes the proposal for \[[Q]] from hint 4 *)
let lift_predicate (f : entity -> bool) : entity dpm -> clause =
fun entity_dpm ->
- let eliminator = fun (truth_value : bool) ->
+ let eliminator = fun truth_value ->
if truth_value = false
then unit_dpm false
else bind_dpm entity_dpm (fun e -> unit_dpm (f e))
(* doing the same thing for binary predicates *)
let lift_predicate2 (f : entity -> entity -> bool) : entity dpm -> entity dpm -> clause =
fun entity1_dpm entity2_dpm ->
- let eliminator = fun (truth_value : bool) ->
+ let eliminator = fun truth_value ->
if truth_value = false
then unit_dpm false
else bind_dpm entity1_dpm (fun e1 -> bind_dpm entity2_dpm (fun e2 -> unit_dpm (f e1 e2)))
in (truth_value, r', h')
(* from hint 5 *)
- let exists var : clause = fun one_dpm ->
- List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign var d)) domain
+ let exists var : clause =
+ let extend one_dpm (d : entity) =
+ bind_dpm one_dpm (new_peg_and_assign var d)
+ in fun one_dpm -> List.map (fun d -> extend one_dpm d) domain
(* include negate_op, and_op, or_op, and if_op as above *)
+* More:
+
(* some handy utilities *)
let (>>=) = bind_set;;
let getx = get 'x';;