X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_6.mdwn;h=fd8f55d4ff6a160aea1b75693e22634993abe579;hp=a371b21c71280a29614f516895747a76fc4f0bba;hb=399197e0af7c4b80f326be60ff2c0caae74b3687;hpb=60fde0202775a36c5b20c370374649d2a90c6af8
diff --git a/hints/assignment_7_hint_6.mdwn b/hints/assignment_7_hint_6.mdwn
index a371b21c..fd8f55d4 100644
--- a/hints/assignment_7_hint_6.mdwn
+++ b/hints/assignment_7_hint_6.mdwn
@@ -5,41 +5,53 @@
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 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.
+ 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) = List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (r, h) in truth_value) set;;
+ 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) ->
- (* we want to check the behavior of one_dpm when updated with the operation phi *)
- (* bind_set (unit_set one_dpm) phi === phi one_dpm; do you remember why? *)
- let truth_value' = truths (phi one_dpm) (r, h) = []
- (* we return a (bool, r, h) so as to constitute a dpm *)
- in (truth_value', r, h)
- in unit_set new_dpm
+ (* 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 -> bind_set (phi one_dpm) psi;;
+ 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) =
- (* NOT: negate_op (and_op (negate_op phi) (negate_op psi)) *)
- fun one_dpm -> unit_set (
+ fun one_dpm -> set_unit (
fun (r, h) ->
- in 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 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 =
- (* NOT: negate_op (and_op phi (negate_op psi)) *)
- fun one_dpm -> unit_set (
+ fun one_dpm -> set_unit (
fun (r, h) ->
- in let truth_value' = List.for_all (fun one_dpm ->
+ 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)
@@ -53,21 +65,23 @@
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 = bool dpm -> bool dpm set;;
+* More:
+
(* this generalizes the getx function from hint 4 *)
let get (var : char) : entity dpm =
fun (r, h) ->
@@ -77,20 +91,20 @@
(* 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);;
(* 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 ->
@@ -102,13 +116,17 @@
in (truth_value, r', h')
(* from hint 5 *)
- let exists var : clause = fun one_dpm ->
- List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign var d)) domain
+ 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 (>>=) = bind_set;;
+ let (>>=) = set_bind;;
let getx = get 'x';;
let gety = get 'y';;
let initial_set = [fun (r,h) -> (true,r,h)];;