2 * In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:
4 > { i ∈ s | i does not subsist in s[φ] }
6 where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
8 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.
10 (* filter out which bool dpms in a set are true when receiving (r, h) as input *)
11 let truths set (r, h) = List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (r, h) in truth_value) set;;
13 let negate_op (phi : clause) : clause =
15 let new_dpm = fun (r, h) ->
16 (* we want to check the behavior of one_dpm when updated with the operation phi *)
17 (* bind_set (unit_set one_dpm) phi === phi one_dpm; do you remember why? *)
18 let truth_value' = truths (phi one_dpm) (r, h) = []
19 (* we return a (bool, r, h) so as to constitute a dpm *)
20 in (truth_value', r, h)
24 * Representing \[[and φ ψ]] is simple:
26 let and_op (phi : clause) (psi : clause) : clause =
27 fun one_dpm -> bind_set (phi one_dpm) psi;;
29 * Here are `or` and `if`:
31 let or_op (phi : clause) (psi : clause) =
32 (* NOT: negate_op (and_op (negate_op phi) (negate_op psi)) *)
33 fun one_dpm -> unit_set (
35 in let truth_value' = truths (phi one_dpm) (r, h) <> [] || truths (bind_set (negate_op phi one_dpm) psi) (r, h) <> []
36 in (truth_value', r, h))
38 let if_op (phi : clause) (psi : clause) : clause =
39 (* NOT: negate_op (and_op phi (negate_op psi)) *)
40 fun one_dpm -> unit_set (
42 in let truth_value' = List.for_all (fun one_dpm ->
43 let (truth_value, _, _) = one_dpm (r, h)
44 in truth_value = false || truths (psi one_dpm) (r, h) <> []
46 in (truth_value', r, h));;
49 * Now let's test everything we've developed:
51 type entity = Bob | Carol | Ted | Alice;;
52 let domain = [Bob; Carol; Ted; Alice];;
53 type assignment = char -> int;;
54 type store = entity list;;
55 type 'a dpm = assignment * store -> 'a * assignment * store;;
56 let unit_dpm (x : 'a) : 'a dpm = fun (r, h) -> (x, r, h);;
57 let bind_dpm (u: 'a dpm) (f : 'a -> 'b dpm) : 'b dpm =
59 let (a, r', h') = u (r, h)
63 type 'a set = 'a list;;
64 let empty_set : 'a set = [];;
65 let unit_set (x : 'a) : 'a set = [x];;
66 let bind_set (u : 'a set) (f : 'a -> 'b set) : 'b set =
67 List.concat (List.map f u);;
69 type clause = bool dpm -> bool dpm set;;
71 (* this generalizes the getx function from hint 4 *)
72 let get (var : char) : entity dpm =
74 let obj = List.nth h (r var)
77 (* this generalizes the proposal for \[[Q]] from hint 4 *)
78 let lift_predicate (f : entity -> bool) : entity dpm -> clause =
80 let eliminator = fun (truth_value : bool) ->
81 if truth_value = false
83 else bind_dpm entity_dpm (fun e -> unit_dpm (f e))
84 in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
86 (* doing the same thing for binary predicates *)
87 let lift_predicate2 (f : entity -> entity -> bool) : entity dpm -> entity dpm -> clause =
88 fun entity1_dpm entity2_dpm ->
89 let eliminator = fun (truth_value : bool) ->
90 if truth_value = false
92 else bind_dpm entity1_dpm (fun e1 -> bind_dpm entity2_dpm (fun e2 -> unit_dpm (f e1 e2)))
93 in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator);;
95 let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
98 let new_index = List.length h
99 in let h' = List.append h [d]
100 in let r' = fun var ->
101 if var = var_to_bind then new_index else r var
102 in (truth_value, r', h')
105 let exists var : clause = fun one_dpm ->
106 List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign var d)) domain
108 (* include negate_op, and_op, or_op, and if_op as above *)
110 (* some handy utilities *)
111 let (>>=) = bind_set;;
114 let initial_set = [fun (r,h) -> (true,r,h)];;
115 let initial_r = fun var -> failwith ("no value for " ^ (Char.escaped var));;
117 (* do any of the dpms in the set return (true, _, _) when given (initial_r, []) as input? *)
118 List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (initial_r, []) in truth_value) dpm_set <> [];;
120 (* let's define some predicates *)
121 let male e = (e = Bob || e = Ted);;
122 let wife_of e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
123 let kisses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
124 let misses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Carol));;
126 (* "a man x has a wife y" *)
127 let antecedent = fun one_dpm -> exists 'x' one_dpm >>= lift_predicate male getx >>= exists 'y' >>= lift_predicate2 wife_of getx gety;;
129 (* "if a man x has a wife y, x kisses y" *)
130 run (initial_set >>= if_op antecedent (lift_predicate2 kisses getx gety));;
131 (* Bob has wife Carol, and kisses her; and Ted has wife Alice and kisses her; so this is true! *)
133 (* "if a man x has a wife y, x misses y" *)
134 run (initial_set >>= if_op antecedent (lift_predicate2 misses getx gety));;
135 (* Bob has wife Carol, and misses her; but Ted misses only Carol, not his wife Alice; so this is false! *)