From: jim Date: Thu, 19 Mar 2015 06:25:01 +0000 (-0400) Subject: update, add safe division X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=3d7b137f08796cb3ba458fd9ca58148f2f5ed8e9 update, add safe division --- diff --git a/topics/week7_monads.mdwn b/topics/week7_monads.mdwn index 2cba92ad..d4b1541b 100644 --- a/topics/week7_monads.mdwn +++ b/topics/week7_monads.mdwn @@ -28,7 +28,7 @@ Recall that we've been using lower-case Greek letters use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound type variables. For instance, we might have - P_1 ≡ Int + P_1 ≡ int P_2 ≡ α -> α P_3 ≡ ∀α. α -> α P_4 ≡ ∀α. α -> β @@ -52,9 +52,9 @@ For instance, if `α list` is our box type, and `α` is the type Warning: although our initial motivating examples are naturally thought of as "containers" (lists, trees, and so on, with `α`s as their "elments"), with later examples we discuss it will less intuitive to describe the box types that way. For example, where `R` is some fixed type, `R -> α` is a box type. The *box type* is the type `α list` (or as we might just say, `list`); the *boxed type* is some specific instantiantion of the free type variable `α`. We'll often write boxed types as a box containing the instance of the free -type variable. So if our box type is `α List`, and `α` is instantiated with the specific type `int`, we would write: +type variable. So if our box type is `α list`, and `α` is instantiated with the specific type `int`, we would write: -Int +int for the type of a boxed `int`. (We'll fool with the markup to make this a genuine box later; for now it will just display as underlined.) @@ -69,16 +69,16 @@ P -> Q That is, they are functions from objects of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`. For instance, the following are Kleisli arrows: -Int -> Bool +int -> bool -Int List -> Int List +int list -> int list -In the first, `P` has become `Int` and `Q` has become `Bool`. (The boxed type Q is Bool). +In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type Q is bool). Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where -if `α List` is our box type, we can write the second arrow as +if `α list` is our box type, we can write the second arrow as -Int -> Q +int -> Q We'll need a number of classes of functions to help us maneuver in the presence of box types. We will want to define a different instance of @@ -87,21 +87,21 @@ become clearly shortly.) Here are the types of our crucial functions, together with our pronunciation, and some other names the functions go by. (Usually the type doesn't fix their behavior, which will be discussed below.) -map (/maep/): (P -> Q) -> P -> Q +map (/mæp/): (P -> Q) -> P -> Q -map2 (/m&ash;ptu/): (P -> Q -> R) -> P -> Q -> R +map2 (/mæptu/): (P -> Q -> R) -> P -> Q -> R mid (/εmaidεnt@tI/ aka unit, return, pure): P -> P -mapply (/εm@plai/): P -> Q -> P -> Q +m$ or mapply (/εm@plai/): P -> Q -> P -> Q -mcomp (aka <=<): (Q -> R) -> (P -> Q) -> (P -> R) +<=< or mcomp : (Q -> R) -> (P -> Q) -> (P -> R) -mpmoc (or m-flipcomp, aka >=>): (P -> Q) -> (Q -> R) -> (P -> R) +>=> or mpmoc (m-flipcomp): (P -> Q) -> (Q -> R) -> (P -> R) -mbind (aka >>=): ( Q) -> (Q -> R) -> ( R) +>>= or mbind : (Q) -> (Q -> R) -> (R) -mdnib (or m-flipbind, aka =<<) ( Q) -> (Q -> R) -> ( R) +=<<mdnib (or m-flipbind) (Q) -> (Q -> R) -> (R) join: 2P -> P @@ -112,27 +112,32 @@ be interdefinable. For example, here is how `mcomp` and `mbind` are related: [α] mid a = [a] @@ -179,40 +184,32 @@ consider the box type `α List`, with the following operations: = foldr (\b -> \gs -> (f b) ++ gs) [] (g a) = [c | b <- g a, c <- f b] -These three definitions of `mcomp` are all equivalent. In words, `mcomp f g -a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s; +These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises). + +In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s; each `β` in that list is fed to `f`, which returns a list of `γ`s. The final result is the concatenation of those lists of `γ`s. -For example, +For example: let f b = [b, b+1] in let g a = [a*a, a+a] in mcomp f g 7 ==> [49, 50, 14, 15] -It is easy to see that these definitions obey the monad laws (see exercises). +`g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`. -Contrast the preceding to `mapply`, which operates not on two box-producing *functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus: +Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus: let gs = [(\a->a*a),(\a->a+a)] in - let xs = [7,5] in - mapply gs xs ==> [49,25,14,10] - - - - + let xs = [7, 5] in + mapply gs xs ==> [49, 25, 14, 10] -[[!toc]] +As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to identify the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the Monadic functions of the Option/Maybe box type. -Towards Monads: Safe division ------------------------------ -[This section used to be near the end of the lecture notes for week 6] - -We begin by reasoning about what should happen when someone tries to -divide by zero. This will lead us to a general programming technique -called a *monad*, which we'll see in many guises in the weeks to come. +Safe division +------------- Integer division presupposes that its second argument (the divisor) is not zero, upon pain of presupposition failure. @@ -221,9 +218,9 @@ Here's what my OCaml interpreter says: # 12/0;; Exception: Division_by_zero. -So we want to explicitly allow for the possibility that +Say we want to explicitly allow for the possibility that division will return something other than a number. -We'll use OCaml's `option` type, which works like this: +To do that, we'll use OCaml's `option` type, which works like this: # type 'a option = None | Some of 'a;; # None;; @@ -232,24 +229,23 @@ We'll use OCaml's `option` type, which works like this: - : int option = Some 3 So if a division is normal, we return some number, but if the divisor is -zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function. +zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function. 
-let div' (x:int) (y:int) =
+let safe_div (x:int) (y:int) =
   match y with
-	  0 -> None
+    | 0 -> None
     | _ -> Some (x / y);;
 
 (*
-val div' : int -> int -> int option = fun
-# div' 12 2;;
+val safe_div : int -> int -> int option = fun
+# safe_div 12 2;;
 - : int option = Some 6
-# div' 12 0;;
+# safe_div 12 0;;
 - : int option = None
-# div' (div' 12 2) 3;;
-Characters 4-14:
-  div' (div' 12 2) 3;;
-        ^^^^^^^^^^
+# safe_div (safe_div 12 2) 3;;
+# safe_div (safe_div 12 2) 3;;
+            ~~~~~~~~~~~~~
 Error: This expression has type int option
        but an expression was expected of type int
 *)
@@ -261,7 +257,7 @@ the output of the safe-division function as input for further division
 operations. So we have to jack up the types of the inputs:
 
 
-let div' (u:int option) (v:int option) =
+let safe_div2 (u:int option) (v:int option) =
   match u with
 	  None -> None
 	| Some x -> (match v with
@@ -269,12 +265,12 @@ let div' (u:int option) (v:int option) =
 				| Some y -> Some (x / y));;
 
 (*
-val div' : int option -> int option -> int option = 
-# div' (Some 12) (Some 2);;
+val safe_div2 : int option -> int option -> int option = 
+# safe_div2 (Some 12) (Some 2);;
 - : int option = Some 6
-# div' (Some 12) (Some 0);;
+# safe_div2 (Some 12) (Some 0);;
 - : int option = None
-# div' (div' (Some 12) (Some 0)) (Some 3);;
+# safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
 - : int option = None
 *)
 

@@ -286,12 +282,12 @@ I prefer to line up the `match` alternatives by using OCaml's
 built-in tuple type:
 
 
-let div' (u:int option) (v:int option) =
+let safe_div2 (u:int option) (v:int option) =
   match (u, v) with
-	  (None, _) -> None
+    | (None, _) -> None
     | (_, None) -> None
     | (_, Some 0) -> None
-	| (Some x, Some y) -> Some (x / y);;
+    | (Some x, Some y) -> Some (x / y);;
 

 
 So far so good. But what if we want to combine division with
@@ -300,62 +296,87 @@ aware of the possibility that one of their arguments has triggered a
 presupposition failure:
 
 
-let add' (u:int option) (v:int option) =
+let safe_add (u:int option) (v:int option) =
   match (u, v) with
-	  (None, _) -> None
+    | (None, _) -> None
     | (_, None) -> None
     | (Some x, Some y) -> Some (x + y);;
 
 (*
-val add' : int option -> int option -> int option = 
-# add' (Some 12) (Some 4);;
+val safe_add : int option -> int option -> int option = 
+# safe_add (Some 12) (Some 4);;
 - : int option = Some 16
-# add' (div' (Some 12) (Some 0)) (Some 4);;
+# safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
 - : int option = None
 *)
 

 
-This works, but is somewhat disappointing: the `add'` operation
+This works, but is somewhat disappointing: the `safe_add` operation
 doesn't trigger any presupposition of its own, so it is a shame that
 it needs to be adjusted because someone else might make trouble.
 
 But we can automate the adjustment. The standard way in OCaml,
-Haskell, etc., is to define a `bind` operator (the name `bind` is not
-well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
+Haskell, and other functional programming languages, is to use the monadic
+`bind` operator, `>>=`. (The name "bind" is not well chosen from our
+perspective, but this is too deeply entrenched by now.) As mentioned above,
+there needs to be a different `>>=` operator for each Monad or box type you're working with.
+Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
+symbol and it will calculate from the context of the surrounding type constraints what
+monad you must have meant. In OCaml, the `>>=` or `bind` operator is not pre-defined, but we will
+give you a library that has definitions for all the standard monads, as in Haskell.
+For now, though, we will define our `bind` operation by hand:
 
 
-let bind' (u: int option) (f: int -> (int option)) =
+let bind (u: int option) (f: int -> (int option)) =
   match u with
-	  None -> None
+    |  None -> None
     | Some x -> f x;;
 
-let add' (u: int option) (v: int option) =
-  bind' u (fun x -> bind' v (fun y -> Some (x + y)));;
+let safe_add3 (u: int option) (v: int option) =
+  bind u (fun x -> bind v (fun y -> Some (x + y)));;
+
+(* This is really just `map2 (+)`, using the `map2` operation that corresponds to
+   definition of `bind`. *)
 
-let div' (u: int option) (v: int option) =
-  bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;
+let safe_div3 (u: int option) (v: int option) =
+  bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;
+
+(* This goes back to some of the simplicity of the original safe_div, without the complexity
+   introduced by safe_div2. *)
+

+
+The above definitions look even simpler if you focus on the fact that `safe_add3` can be written as simply `map2 (+)`, and that `safe_div3` could be written as `u >>= fun x -> v >>= fun y -> if 0 = y then None else Some (x / y)`. Haskell has an even more user-friendly notation for this, namely:
+
+    safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
+    safe_div3 u v = do {x <- u;
+                        y <- v;
+                        if 0 == y then Nothing else return (x `div` y)}
+
+Let's see our new functions in action:
 
 (*
-#  div' (div' (Some 12) (Some 2)) (Some 3);;
+# safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
 - : int option = Some 2
-#  div' (div' (Some 12) (Some 0)) (Some 3);;
+#  safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
 - : int option = None
-# add' (div' (Some 12) (Some 0)) (Some 3);;
+# safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
 - : int option = None
 *)
-Compare the new definitions of `add'` and `div'` closely: the definition -for `add'` shows what it looks like to equip an ordinary operation to +Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition +for `safe_add3` shows what it looks like to equip an ordinary operation to survive in dangerous presupposition-filled world. Note that the new -definition of `add'` does not need to test whether its arguments are +definition of `safe_add3` does not need to test whether its arguments are None objects or real numbers---those details are hidden inside of the -`bind'` function. +`bind` function. -The definition of `div'` shows exactly what extra needs to be said in -order to trigger the no-division-by-zero presupposition. +The definition of `safe_div3` shows exactly what extra needs to be said in +order to trigger the no-division-by-zero presupposition. Here, too, we don't +need to keep track of what presuppositions may have already failed +for whatever reason on our inputs. -[Linguitics note: Dividing by zero is supposed to feel like a kind of +(Linguitics note: Dividing by zero is supposed to feel like a kind of presupposition failure. If we wanted to adapt this approach to building a simple account of presupposition projection, we would have to do several things. First, we would have to make use of the @@ -368,6 +389,5 @@ theory of accommodation, and a theory of the situations in which material within the sentence can satisfy presuppositions for other material that otherwise would trigger a presupposition violation; but, not surprisingly, these refinements will require some more -sophisticated techniques than the super-simple Option monad.] - +sophisticated techniques than the super-simple Option monad.)