X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_4.mdwn;h=046a45b9c05008b7d137f0c4122e17c211115c30;hp=6b33b7a544199181d47a92a48b2ea3b06dbc1716;hb=6097d678d4529d775014d378a5d42f7739be8db4;hpb=c6f572fd09dbe53dff30a91cd07b318b3e5edfd6 diff --git a/hints/assignment_7_hint_4.mdwn b/hints/assignment_7_hint_4.mdwn index 6b33b7a5..046a45b9 100644 --- a/hints/assignment_7_hint_4.mdwn +++ b/hints/assignment_7_hint_4.mdwn @@ -17,13 +17,13 @@ then let obj = List.nth h (r 'x') in q obj else false in (truth_value', r, h) - in bind_set u (fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)) + in set_bind u (fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)) - The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `bind_set` operation. This works by taking each `dpm` in the set and returning a `unit_set` of a filtered `dpm`. The definition of `bind_set` takes care of collecting together all of the `unit_set`s that result for each different set element we started with. + The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `set_bind` operation. This works by taking each `dpm` in the set and returning a `set_unit` of a filtered `dpm`. The definition of `set_bind` takes care of collecting together all of the `set_unit`s that result for each different set element we started with. We can call the `(fun one_dpm -> ...)` part \[[Qx]] and then updating `u` with \[[Qx]] will be: - bind_set u \[[Qx]] + set_bind u \[[Qx]] or as it's written using Haskell's infix notation for bind: @@ -39,11 +39,11 @@ * Now what do we do with predicates? As before, we suppose we have a function `q` that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting `q` into a function that takes `entity dpm`s as arguments and returns `bool dpm`s: - fun entity_dpm -> bind_dpm entity_dpm (fun e -> unit_dpm (q e)) + fun entity_dpm -> dpm_bind entity_dpm (fun e -> dpm_unit (q e)) - Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `unit_set`: + Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `set_unit`: - fun entity_dpm -> unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (q e))) + fun entity_dpm -> set_unit (dpm_bind entity_dpm (fun e -> dpm_unit (q e))) Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to monadically bind \[[Qx]] to them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so. @@ -53,9 +53,9 @@ let eliminator : bool -> bool dpm = fun truth_value -> if truth_value = false - then unit_dpm false - else bind_dpm entity_dpm (fun e -> unit_dpm (q 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 (q e)) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) Applied to an `entity_dpm`, that yields a function that we can bind to a `bool dpm set` and that will transform the doubly-wrapped `bool` into a new `bool dpm set`. @@ -67,9 +67,9 @@ in let entity_dpm = getx in let eliminator = fun truth_value -> if truth_value = false - then unit_dpm false - else bind_dpm entity_dpm (fun e -> unit_dpm (q 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 (q e)) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) - If we simplify and unpack the definition of `bind_dpm`, that's equivalent to: + If we simplify and unpack the definition of `dpm_bind`, that's equivalent to: let getx = fun (r, h) -> let obj = List.nth h (r 'x') @@ -93,10 +93,10 @@ if truth_value then (fun (r, h) -> let (a, r', h') = getx (r, h) - in let u' = (fun e -> unit_dpm (q e)) a + in let u' = (fun e -> dpm_unit (q e)) a in u' (r', h') - ) else unit_dpm false - in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator) + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) which can be further simplified to: @@ -106,19 +106,19 @@ then (fun (r, h) -> let obj = List.nth h (r 'x') let (a, r', h') = (obj, r, h) - in let u' = (fun e -> unit_dpm (q e)) a + in let u' = (fun e -> dpm_unit (q e)) a in u' (r', h') - ) else unit_dpm false - in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator) + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) let eliminator = fun truth_value -> if truth_value then (fun (r, h) -> let obj = List.nth h (r 'x') - in let u' = unit_dpm (q obj) + in let u' = dpm_unit (q obj) in u' (r, h) - ) else unit_dpm false - in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator) + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) --> let eliminator = fun truth_value -> @@ -126,8 +126,8 @@ then (fun (r, h) -> let obj = List.nth h (r 'x') in (q obj, r, h) - ) else unit_dpm false - in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator) + ) else dpm_unit false + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) This is a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. @@ -140,13 +140,13 @@ then let obj = List.nth h (r 'x') in q obj else false in (truth_value', r, h)) - in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator) + in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator) Can you persuade yourself that these are equivalent?) * Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We monadically bind this operaration to whatever `bool dpm set` we already have on hand: - bind_set u \[[Qx]] + set_bind u \[[Qx]] or: