* In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:
- > { i &elem; s | i does not subsist in s[φ] }
+ > { i ∈ s | i does not subsist in s[φ] }
where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
- Here's how we can represent that:
+ 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.
- <pre><code>bind_set s (fun (r, h) ->
- let u = unit_set (r, h)
- in let descendents = u >>= \[[φ]]
- in if descendents = empty_set then u else empty_set
- </code></pre>
+ (* 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
+ 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;;
+
+
+ **Thanks to Simon Charlow** for catching a subtle error in previous versions of this function. Fixed 1 Dec.
+
+* 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? *)
+
+
+* Here are `or` and `if`:
+
+ (These probably still manifest the bug Simon spotted.)
+
+ 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 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));;
+
+
+* 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 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 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 = 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 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);;
+
+ (* 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 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 ->
+ 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) =
+ dpm_bind 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 (>>=) = 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;;
+
+ (* "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! *)