+**The hints for problem 2 were being actively developed until early Saturday morning. They're pretty stable now. Remember you have a grace period until Sunday Nov. 28 to complete this homework.**
1. Make sure that your operation-counting monad from [[assignment6]] is working. Modify it so that instead of counting operations, it keeps track of the last remainder of any integer division. You can help yourself to the functions:
Since `dpm`s are to be a monad, we have to define a unit and a bind. These are just modeled on the unit and bind for the state monad:
- let unit_dpm (x : 'a) = fun (r, h) -> (x, r, h);;
+ let unit_dpm (value : 'a) : 'a dpm = fun (r, h) -> (value, r, h);;
- let bind_dpm (u : 'a dpm) (f : 'a -> 'b dpm) =
+ let bind_dpm (u : 'a dpm) (f : 'a -> 'b dpm) : 'b dpm =
fun (r, h) ->
let (a, r', h') = u (r, h)
in let u' = f a
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) =
+ let unit_set (value: 'a) : 'a set = [value];;
+ let bind_set (u: 'a set) (f: 'a -> 'b set) : 'b set =
List.concat (List.map f u);;
> \[[expression]] in possibility `(r, h, w)`
- we'll just talk about \[[expression]] and let that be a monadic value, implemented in part by a function that takes `(r, h)` as an argument.
+ we'll just talk about \[[expression]] and let that be a monadic operation, implemented in part by a function that takes `(r, h)` as an argument.
+
+ In particular, the meaning of sentential clauses will be an operation that we monadically bind to an existing `bool dpm set`. Here is its type:
+
+ type clause = bool dpm -> bool dpm set;;
* In def 2.7, GS&V talk about an operation that takes an existing set of discourse possibilities, and *extends* each member in the set by (i) allocating a new location in the store, (ii) putting some entity `d` from the domain in that location, and (iii) assigning variable `x` to that location in the store.
It will be useful to have a shorthand way of referring to this operation:
- let new_peg_and_assign (var_to_bind : char) (d : entity) =
- (* we want to return a function that we can bind to a bool dpm *)
- fun (truth_value : bool) ->
- fun ((r, h) : assignment * store) ->
+ (* we want to return a function that we can bind to a bool dpm *)
+ let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
+ fun truth_value ->
+ fun (r, h) ->
(* first we calculate an unused index *)
let new_index = List.length h
(* next we store d at h[new_index], which is at the very end of h *)
in let r' = fun var ->
if var = var_to_bind then new_index else r var
(* we pass through the same truth_value that we started with *)
- in (truth_value, r', h')
+ in (truth_value, r', h');;
* Is that enough? If not, here are some [more hints](/hints/assignment_7_hint_4). But try to get as far as you can on your own.
Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about sets of `dpm`s; we'll leave that to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. So what we want is:
- let getx = fun (r, h) ->
+ let getx : entity dpm = fun (r, h) ->
let obj = List.nth h (r 'x')
in (obj, r, h);;
* How shall we handle \[[∃x]]? As we said, GS&V really tell us how to interpret \[[∃xPx]], but what they say about this breaks naturally into two pieces, such that we can represent the update of our starting set `u` with \[[∃xPx]] as:
- <pre><code>u >>=<sub>set</sub> \[[∃x]] >>=<sub>set</sub> \[[Px]]
+ <pre><code>u >>= \[[∃x]] >>= \[[Px]]
</code></pre>
What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
Deferring the "property P" part, this corresponds to:
<pre><code>u updated with \[[∃x]] ≡
- let extend_one = fun (one_dpm : bool dpm) ->
+ let extend_one : clause = fun one_dpm ->
List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign 'x' d)) domain
in bind_set u extend_one
</code></pre>
where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
- let new_peg_and_assign (var_to_bind : char) (d : entity) =
- (* we want to return a function that we can bind to a bool dpm *)
- fun (truth_value : bool) ->
- fun ((r, h) : assignment * store) ->
+ let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
+ fun truth_value ->
+ fun (r, h) ->
(* first we calculate an unused index *)
let new_index = List.length h
(* next we store d at h[new_index], which is at the very end of h *)
in let r' = fun var ->
if var = var_to_bind then new_index else r var
(* we pass through the same truth_value that we started with *)
- in (truth_value, r', h')
+ in (truth_value, r', h');;
What's going on in this representation of `u` updated with \[[∃x]]? For each `bool dpm` in `u`, we collect `dpm`s that are the result of passing through their `bool`, but extending their input `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities, and binding the variable `x` to the index of that peg.
type assignment = char -> entity;;
type 'a reader = assignment -> 'a;;
- let unit_reader (x : 'a) = fun r -> x;;
+ let unit_reader (value : 'a) : 'a reader = fun r -> value;;
- let bind_reader (u : 'a reader) (f : 'a -> 'b reader) =
+ let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
fun r ->
let a = u r
in let u' = f a
in u' r;;
- let getx = fun r -> r 'x';;
+ Here the type of a sentential clause is:
+
+ type clause = bool reader;;
+
+ Here are meanings for singular terms and predicates:
- let lift (predicate : entity -> bool) =
+ let getx : entity reader = fun r -> r 'x';;
+
+ type lifted_unary = entity reader -> bool reader;;
+
+ let lift (predicate : entity -> bool) : lifted_unary =
fun entity_reader ->
fun r ->
let obj = entity_reader r
in unit_reader (predicate obj)
- `lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be:
+ The meaning of \[[Qx]] would then be:
<pre><code>\[[Q]] ≡ lift q
\[[x]] ≡ getx
Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]:
- let shift (var_to_bind : char) (clause : bool reader) =
- (* we return a lifted predicate, that is a entity reader -> bool reader *)
+ let shift (var_to_bind : char) (clause : clause) : lifted_unary =
fun entity_reader ->
- fun (r : assignment) ->
+ fun r ->
let new_value = entity_reader r
(* remember here we're implementing assignments as functions rather than as lists of pairs *)
in let r' = fun var -> if var = var_to_bind then new_value else r var
Now, how would we implement quantifiers in this setting? I'll assume we have a function `exists` of type `(entity -> bool) -> bool`. That is, it accepts a predicate as argument and returns `true` if any element in the domain satisfies that predicate. We could implement the reader-monad version of that like this:
- fun (lifted_predicate : entity reader -> bool reader) ->
+ fun (lifted_predicate : lifted_unary) ->
fun r -> exists (fun (obj : entity) ->
lifted_predicate (unit_reader obj) r)
where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
- Here's how to do that in our framework:
+ 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.
- type clause_op = bool dpm -> bool dpm set;;
-
(* filter out which bool dpms in a set are true when receiving (r, h) as input *)
- let extensions set (r, h) = List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (r, h) in truth_value) set;;
-
- let negate_op (phi : clause_op) : clause_op =
- fun one_dpm -> unit_set (
- fun (r, h) ->
- let truth_value' = extensions (phi one_dpm) (r, h) = []
+ let truths set (r, h) = List.filter (fun one_dpm -> let (truth_value, _, _) = one_dpm (r, h) in truth_value) 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
* Representing \[[and φ ψ]] is simple:
- let and_op (phi : clause_op) (psi : clause_op) : clause_op =
+ let and_op (phi : clause) (psi : clause) : clause =
fun one_dpm -> bind_set (phi one_dpm) psi;;
* Here are `or` and `if`:
- let or_op (phi : clause_op) (psi : clause_op) =
+ let or_op (phi : clause) (psi : clause) =
(* NOT: negate_op (and_op (negate_op phi) (negate_op psi)) *)
fun one_dpm -> unit_set (
fun (r, h) ->
- in let truth_value' = extensions (phi one_dpm) (r, h) <> [] || extensions (bind_set (negate_op phi one_dpm) psi) (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 if_op (phi : clause_op) (psi : clause_op) : clause_op =
+ let if_op (phi : clause) (psi : clause) : clause =
(* NOT: negate_op (and_op phi (negate_op psi)) *)
fun one_dpm -> unit_set (
fun (r, h) ->
in let truth_value' = List.for_all (fun one_dpm ->
let (truth_value, _, _) = one_dpm (r, h)
- in truth_value = false || extensions (psi one_dpm) (r, h) <> []
+ in truth_value = false || truths (psi one_dpm) (r, h) <> []
) (phi one_dpm)
in (truth_value', r, h));;
let bind_set (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;;
+ (* 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) ->
if truth_value = false
else bind_dpm entity_dpm (fun e -> unit_dpm (f e))
in fun one_dpm -> unit_set (bind_dpm 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) ->
if truth_value = false
if var = var_to_bind then new_index else r var
in (truth_value, r', h')
- let exists var : clause_op = fun one_dpm ->
+ (* 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
- (* negate_op, and_op, or_op, and if_op as above *)
+ (* include negate_op, and_op, or_op, and if_op as above *)
+ (* some handy utilities *)
let (>>=) = bind_set;;
+ 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 =
- 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);;
- let getx = get 'x';;
- let gety = get 'y';;
+ (* 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;;
(8 Nov) Lecture notes for [[Week8]].
-(15 Nov) Lecture notes are coming; [[Assignment7]] is here.
+(15 Nov) Lecture notes are coming; [[Assignment7]] is here. Everyone auditing in the class is encouraged to do this assignment, or at least work through the substantial "hints".
[[Upcoming topics]]