where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
------- wrong....
+ 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.
- In our framework, we just have to convert the operation <code>>>= \[[ψ]]</code> into another operation <code>>>= \[[ψ]] >>= neg</code>, where `neg` flips the truth-value of all the `bool dpm`s it operates on:
+ (* 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;;
- type clause_op = bool dpm -> bool dpm 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
+ set_bind (set_unit one_dpm) phi === phi one_dpm; do you remember why? *)
+ let (truth_value, r', h') = 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 set_unit new_dpm;;
- let negate_op (phi : clause_op) : clause_op =
- let neg : clause_op = fun one_dpm ->
- unit_set (fun (r, h) ->
- let (truth_value, r', h') = one_dpm (r, h)
- in (not truth_value, r', h'))
- in fun one_dpm -> bind_set (phi one_dpm) neg;;
+ **Thanks to Simon Charlow** for catching a subtle error in previous versions of this function. Fixed 1 Dec.
- let negate_op (phi : clause_op) : clause_op =
- fun one_dpm ->
- if blah
- then unit_set one_dpm
- else empty_set
-------
+* Representing \[[and φ ψ]] is simple:
+ let and_op (phi : clause) (psi : clause) : clause =
+ fun one_dpm -> set_bind (phi one_dpm) psi;;
+ (* now u >>= and_op phi psi === u >>= phi >>= psi; do you remember why? *)
+
-* Representing \[[and φ ψ]] is simple:
+* Here are `or` and `if`:
- let and_op (phi : clause_op) (psi : clause_op) : clause_op =
- fun one_dpm -> bind_set (phi one_dpm) psi;;
+ (These probably still manifest the bug Simon spotted.)
-* We define the other connectives in terms of `not` and `and`:
+ let or_op (phi : clause) (psi : clause) =
+ fun one_dpm -> set_unit (
+ fun (r, h) ->
+ let truth_value' = (
+ truths (phi one_dpm) (r, h) <> [] ||
+ truths (set_bind (negate_op phi one_dpm) psi) (r, h) <> []
+ ) in (truth_value', r, h))
- let or_op (phi : clause_op) (psi : clause_op) =
- negate_op (and_op (negate_op phi) (negate_op psi))
+ let if_op (phi : clause) (psi : clause) : clause =
+ fun one_dpm -> set_unit (
+ 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));;
- let if_op (phi : clause_op) (psi : clause_op) =
- negate_op (and_op phi (negate_op psi));;
* Now let's test everything we've developed:
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 =
+ let dpm_unit (x : 'a) : 'a dpm = fun (r, h) -> (x, r, h);;
+ let dpm_bind (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 =
+ let set_empty : 'a set = [];;
+ let set_unit (x : 'a) : 'a set = [x];;
+ let set_bind (u : 'a set) (f : 'a -> 'b set) : 'b set =
List.concat (List.map f u);;
- type clause_op = bool dpm -> bool dpm set;;
+ 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);;
- let lift_predicate (f : entity -> bool) : entity dpm -> clause_op =
+ (* 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))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
+ then dpm_unit false
+ else dpm_bind entity_dpm (fun e -> dpm_unit (f e))
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator);;
- let lift_predicate2 (f : entity -> entity -> bool) : entity dpm -> entity dpm -> clause_op =
+ (* 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 fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
+ then dpm_unit false
+ else dpm_bind entity1_dpm (fun e1 -> dpm_bind entity2_dpm (fun e2 -> dpm_unit (f e1 e2)))
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator);;
let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
fun truth_value ->
if var = var_to_bind then new_index else r var
in (truth_value, r', h')
- let exists var : clause_op = fun one_dpm ->
- List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign var d)) domain
+ (* from hint 5 *)
+ let exists var : clause =
+ let extend one_dpm (d : entity) =
+ dpm_bind one_dpm (new_peg_and_assign var d)
+ in fun one_dpm -> List.map (fun d -> extend one_dpm d) domain
- (* negate_op, and_op, or_op, and if_op as above *)
+ (* include negate_op, and_op, or_op, and if_op as above *)
- let (>>=) = bind_set;;
- let initial_set = [fun (r,h) -> (true,r,h)];;
+* More:
- let initial_r = fun var -> failwith ("no value for " ^ (Char.escaped var));;
- let run dpm_set =
- let bool_set = List.map (fun one_dpm -> let (value, r, h) = one_dpm (initial_r, []) in value) dpm_set
- in List.exists (fun truth_value -> truth_value) bool_set;;
-
- let male obj = obj = Bob || obj = Ted;;
- let wife_of x y = (x,y) = (Bob, Carol) || (x,y) = (Ted, Alice);;
- let kisses x y = (x,y) = (Bob, Carol) || (x,y) = (Ted, Alice);;
- let misses x y = (x,y) = (Bob, Carol) || (x,y) = (Ted, Carol);;
+ (* some handy utilities *)
+ let (>>=) = set_bind;;
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;;