* In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:
> { i ∈ s | i does not subsist in s[φ] }
where `i` *subsists* in s[φ]
if there are any `i'` that *extend* `i` in s[φ]
.
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) =
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) ->
(* 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;;
**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) =
fun one_dpm -> unit_set (
fun (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 =
fun one_dpm -> unit_set (
fun (r, h) ->
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)
in (truth_value', r, h));;
* Now let's test everything we've developed:
type entity = Bob | Carol | Ted | Alice;;
let domain = [Bob; Carol; Ted; Alice];;
type assignment = char -> int;;
type store = entity list;;
type 'a dpm = assignment * store -> 'a * assignment * store;;
let unit_dpm (x : 'a) : 'a dpm = fun (r, h) -> (x, r, h);;
let bind_dpm (u: 'a dpm) (f : 'a -> 'b dpm) : 'b dpm =
fun (r, h) ->
let (a, r', h') = u (r, h)
in let u' = f a
in u' (r', h')
type 'a set = 'a list;;
let empty_set : 'a set = [];;
let unit_set (x : 'a) : 'a set = [x];;
let bind_set (u : 'a set) (f : 'a -> 'b set) : 'b set =
List.concat (List.map f u);;
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) ->
let obj = List.nth h (r var)
in (obj, 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 ->
if truth_value = false
then unit_dpm false
else bind_dpm entity_dpm (fun e -> unit_dpm (f e))
in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
(* 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 ->
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 fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
fun truth_value ->
fun (r, h) ->
let new_index = List.length h
in let h' = List.append h [d]
in let r' = fun var ->
if var = var_to_bind then new_index else r var
in (truth_value, r', h')
(* from hint 5 *)
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';;
let gety = get 'y';;
let initial_set = [fun (r,h) -> (true,r,h)];;
let initial_r = fun var -> failwith ("no value for " ^ (Char.escaped var));;
let run dpm_set =
(* do any of the dpms in the set return (true, _, _) when given (initial_r, []) as input? *)
List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (initial_r, []) in truth_value) dpm_set <> [];;
(* let's define some predicates *)
let male e = (e = Bob || e = Ted);;
let wife_of e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
let kisses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
let misses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Carol));;
(* "a man x has a wife y" *)
let antecedent = fun one_dpm -> exists 'x' one_dpm >>= lift_predicate male getx >>= exists 'y' >>= lift_predicate2 wife_of getx gety;;
(* "if a man x has a wife y, x kisses y" *)
run (initial_set >>= if_op antecedent (lift_predicate2 kisses getx gety));;
(* Bob has wife Carol, and kisses her; and Ted has wife Alice and kisses her; so this is true! *)
(* "if a man x has a wife y, x misses y" *)
run (initial_set >>= if_op antecedent (lift_predicate2 misses getx gety));;
(* Bob has wife Carol, and misses her; but Ted misses only Carol, not his wife Alice; so this is false! *)