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 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.
10 (* filter out which bool dpms in a set are true when receiving (r, h) as input *)
11 let truths set (r, h) =
13 let (truth_value, _, _) = one_dpm (r, h)
15 in List.filter test set;;
17 let negate_op (phi : clause) : clause =
19 let new_dpm = fun (r, h) ->
20 (* if one_dpm isn't already false at (r, h),
21 we want to check its behavior when updated with phi
22 set_bind (set_unit one_dpm) phi === phi one_dpm; do you remember why? *)
23 let (truth_value, r', h') = one_dpm (r, h)
24 in let truth_value' = truth_value && (truths (phi one_dpm) (r, h) = [])
25 (* new_dpm must return a (bool, r, h) *)
26 in (truth_value', r', h')
30 **Thanks to Simon Charlow** for catching a subtle error in previous versions of this function. Fixed 1 Dec.
32 * Representing \[[and φ ψ]] is simple:
34 let and_op (phi : clause) (psi : clause) : clause =
35 fun one_dpm -> set_bind (phi one_dpm) psi;;
36 (* now u >>= and_op phi psi === u >>= phi >>= psi; do you remember why? *)
39 * Here are `or` and `if`:
41 (These probably still manifest the bug Simon spotted.)
43 let or_op (phi : clause) (psi : clause) =
44 fun one_dpm -> set_unit (
47 truths (phi one_dpm) (r, h) <> [] ||
48 truths (set_bind (negate_op phi one_dpm) psi) (r, h) <> []
49 ) in (truth_value', r, h))
51 let if_op (phi : clause) (psi : clause) : clause =
52 fun one_dpm -> set_unit (
54 let truth_value' = List.for_all (fun one_dpm ->
55 let (truth_value, _, _) = one_dpm (r, h)
56 in truth_value = false || truths (psi one_dpm) (r, h) <> []
58 in (truth_value', r, h));;
61 * Now let's test everything we've developed:
63 type entity = Bob | Carol | Ted | Alice;;
64 let domain = [Bob; Carol; Ted; Alice];;
65 type assignment = char -> int;;
66 type store = entity list;;
67 type 'a dpm = assignment * store -> 'a * assignment * store;;
68 let dpm_unit (x : 'a) : 'a dpm = fun (r, h) -> (x, r, h);;
69 let dpm_bind (u: 'a dpm) (f : 'a -> 'b dpm) : 'b dpm =
71 let (a, r', h') = u (r, h)
75 type 'a set = 'a list;;
76 let set_empty : 'a set = [];;
77 let set_unit (x : 'a) : 'a set = [x];;
78 let set_bind (u : 'a set) (f : 'a -> 'b set) : 'b set =
79 List.concat (List.map f u);;
81 type clause = bool dpm -> bool dpm set;;
85 (* this generalizes the getx function from hint 4 *)
86 let get (var : char) : entity dpm =
88 let obj = List.nth h (r var)
91 (* this generalizes the proposal for \[[Q]] from hint 4 *)
92 let lift_predicate (f : entity -> bool) : entity dpm -> clause =
94 let eliminator = fun truth_value ->
95 if truth_value = false
97 else dpm_bind entity_dpm (fun e -> dpm_unit (f e))
98 in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator);;
100 (* doing the same thing for binary predicates *)
101 let lift_predicate2 (f : entity -> entity -> bool) : entity dpm -> entity dpm -> clause =
102 fun entity1_dpm entity2_dpm ->
103 let eliminator = fun truth_value ->
104 if truth_value = false
106 else dpm_bind entity1_dpm (fun e1 -> dpm_bind entity2_dpm (fun e2 -> dpm_unit (f e1 e2)))
107 in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator);;
109 let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
112 let new_index = List.length h
113 in let h' = List.append h [d]
114 in let r' = fun var ->
115 if var = var_to_bind then new_index else r var
116 in (truth_value, r', h')
119 let exists var : clause =
120 let extend one_dpm (d : entity) =
121 dpm_bind one_dpm (new_peg_and_assign var d)
122 in fun one_dpm -> List.map (fun d -> extend one_dpm d) domain
124 (* include negate_op, and_op, or_op, and if_op as above *)
128 (* some handy utilities *)
129 let (>>=) = set_bind;;
132 let initial_set = [fun (r,h) -> (true,r,h)];;
133 let initial_r = fun var -> failwith ("no value for " ^ (Char.escaped var));;
135 (* do any of the dpms in the set return (true, _, _) when given (initial_r, []) as input? *)
136 List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (initial_r, []) in truth_value) dpm_set <> [];;
138 (* let's define some predicates *)
139 let male e = (e = Bob || e = Ted);;
140 let wife_of e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
141 let kisses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Alice));;
142 let misses e1 e2 = ((e1,e2) = (Bob, Carol) || (e1,e2) = (Ted, Carol));;
144 (* "a man x has a wife y" *)
145 let antecedent = fun one_dpm -> exists 'x' one_dpm >>= lift_predicate male getx >>= exists 'y' >>= lift_predicate2 wife_of getx gety;;
147 (* "if a man x has a wife y, x kisses y" *)
148 run (initial_set >>= if_op antecedent (lift_predicate2 kisses getx gety));;
149 (* Bob has wife Carol, and kisses her; and Ted has wife Alice and kisses her; so this is true! *)
151 (* "if a man x has a wife y, x misses y" *)
152 run (initial_set >>= if_op antecedent (lift_predicate2 misses getx gety));;
153 (* Bob has wife Carol, and misses her; but Ted misses only Carol, not his wife Alice; so this is false! *)