X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=7544425c4bbcca2c3a1b72cf0cf088199a18586f;hp=3336a49685b38bf087eb3a9f67915f3bf8a32ea0;hb=d9e25980d9b3e62e89ab731ed5fbc33126df57e6;hpb=f10b70d2b7589e44135dd0e47c87d3965ee08918 diff --git a/week7.mdwn b/week7.mdwn index 3336a496..7544425c 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -1,414 +1,603 @@ [[!toc]] -Monads ------- - -Start by (re)reading the discussion of monads in the lecture notes for -week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2). -In those notes, we saw a way to separate thining about error -conditions (such as trying to divide by zero) from thinking about -normal arithmetic computations. We did this by making use of the -Option monad: in each place where we had something of type `int`, we -put instead something of type `int option`, which is a sum type -consisting either of just an integer, or else some special value which -we could interpret as signaling that something had gone wrong. -The goal was to make normal computing as convenient as possible: when -we're adding or multiplying, we don't have to worry about generating -any new errors, so we do want to think about the difference between -ints and int options. We tried to accomplish this by defining a -`bind` operator, which enabled us to peel away the option husk to get -at the delicious integer inside. There was also a homework problem -which made this even more convenient by mapping any bindary operation -on plain integers into a lifted operation that understands how to deal -with int options in a sensible way. +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. + +Integer division presupposes that its second argument +(the divisor) is not zero, upon pain of presupposition failure. +Here's what my OCaml interpreter says: + + # 12/0;; + Exception: Division_by_zero. + +So 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: + + # type 'a option = None | Some of 'a;; + # None;; + - : 'a option = None + # Some 3;; + - : 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. + +
```+let div' (x:int) (y:int) =
+  match y with
+	  0 -> None
+    | _ -> Some (x / y);;
+
+(*
+val div' : int -> int -> int option = fun
+# div' 12 2;;
+- : int option = Some 6
+# div' 12 0;;
+- : int option = None
+# div' (div' 12 2) 3;;
+Characters 4-14:
+  div' (div' 12 2) 3;;
+        ^^^^^^^^^^
+Error: This expression has type int option
+       but an expression was expected of type int
+*)
+```
+ +This starts off well: dividing 12 by 2, no problem; dividing 12 by 0, +just the behavior we were hoping for. But we want to be able to use +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) =
+  match u with
+	  None -> None
+	| Some x -> (match v with
+				  Some 0 -> None
+				| Some y -> Some (x / y));;
+
+(*
+val div' : int option -> int option -> int option =
+# div' (Some 12) (Some 2);;
+- : int option = Some 6
+# div' (Some 12) (Some 0);;
+- : int option = None
+# div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
+```
+ +Beautiful, just what we need: now we can try to divide by anything we +want, without fear that we're going to trigger any system errors. + +I prefer to line up the `match` alternatives by using OCaml's +built-in tuple type: + +
```+let div' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (_, Some 0) -> None
+	| (Some x, Some y) -> Some (x / y);;
+```
+ +So far so good. But what if we want to combine division with +other arithmetic operations? We need to make those other operations +aware of the possibility that one of their arguments has triggered a +presupposition failure: + +
```+let add' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (Some x, Some y) -> Some (x + y);;
+
+(*
+val add' : int option -> int option -> int option =
+# add' (Some 12) (Some 4);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+```
+ +This works, but is somewhat disappointing: the `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. + +
```+let bind' (u: int option) (f: int -> (int option)) =
+  match u with
+	  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 div' (u: int option) (v: int option) =
+  bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;
+
+(*
+#  div' (div' (Some 12) (Some 2)) (Some 3);;
+- : int option = Some 2
+#  div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+# add' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
+```
```-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option =
-# let unit x = Some x;;
-val unit : 'a -> 'a option =
-# unit 2 * unit;;
-- : int option = Some 2
-```
- -The parentheses is the magic for telling Ocaml that the -function to be defined (in this case, the name of the function -is `*`, pronounced "bind") is an infix operator, so we write -`m * f` or `( * ) m f` instead of `* m f`. + Wadler and others try to make this look nicer by phrasing it like this, + where U, V, and W are schematic for any expressions with the relevant monadic type: -* Associativity: bind obeys a kind of associativity, like this: + (U >>= fun x -> V) >>= fun y -> W == U >>= fun x -> (V >>= fun y -> W) - `(m * f) * g == m * (fun x -> f x * g)` + Some examples of associativity in the Option monad (bear in + mind that in the Ocaml implementation of integer division, 2/3 + evaluates to zero, throwing away the remainder): - If you don't understand why the lambda form is necessary (the "fun - x" part), you need to look again at the type of bind. + # Some 3 >>= unit >>= unit;; + - : int option = Some 3 + # Some 3 >>= (fun x -> unit x >>= unit);; + - : int option = Some 3 - For an illustration of associativity in the option monad: + # Some 3 >>= divide 6 >>= divide 2;; + - : int option = Some 1 + # Some 3 >>= (fun x -> divide 6 x >>= divide 2);; + - : int option = Some 1 -
```-Some 3 * unit * unit;;
-- : int option = Some 3
-Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
-```
+ # Some 3 >>= divide 2 >>= divide 6;; + - : int option = None + # Some 3 >>= (fun x -> divide 2 x >>= divide 6);; + - : int option = None - Of course, associativity must hold for arbitrary functions of - type `'a -> M 'a`, where `M` is the monad type. It's easy to - convince yourself that the bind operation for the option monad - obeys associativity by dividing the inputs into cases: if `m` + Of course, associativity must hold for *arbitrary* functions of + type `'a -> 'b m`, where `m` is the monad type. It's easy to + convince yourself that the `bind` operation for the Option monad + obeys associativity by dividing the inputs into cases: if `u` matches `None`, both computations will result in `None`; if - `m` matches `Some n`, and `f n` evalutes to `None`, then both + `u` matches `Some x`, and `f x` evalutes to `None`, then both computations will again result in `None`; and if the value of - `f n` matches `Some r`, then both computations will evaluate - to `g r`. + `f x` matches `Some y`, then both computations will evaluate + to `g y`. -* Right identity: unit is a right identity for bind. That is, - `m * unit == m` for all monad objects `m`. For instance, +* **Right identity: unit is a right identity for bind.** That is, + `u >>= unit == u` for all monad objects `u`. For instance, -
```-# Some 3 * unit;;
-- : int option = Some 3
-```
```-Extensional types                 Intensional types       Examples
--------------------------------------------------------------------
-
-S         s->t                    s->t                    John left
-DP        s->e                    s->e                    John
-VP        s->e->t                 s->(s->e)->t            left
-Vt        s->e->e->t              s->(s->e)->(s->e)->t    saw
-Vs        s->t->e->t              s->(s->t)->(s->e)->t    thought
-```
+* [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps): +invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992. + -This system is modeled on the way Montague arranged his grammar. -(There are significant simplifications: for instance, determiner -phrases are thought of as corresponding to individuals rather than to -generalized quantifiers.) If you're curious about the initial `s`'s -in the extensional types, they're there because the behavior of these -expressions depends on which world they're evaluated at. If you are -in a situation in which you can hold the evaluation world constant, -you can further simplify the extensional types. (Usually, the -dependence of the extension of an expression on the evaluation world -is hidden in a superscript, or built into the lexical interpretation -function.) - -The main difference between the intensional types and the extensional -types is that in the intensional types, the arguments are functions -from worlds to extensions: intransitive verb phrases like "left" now -take intensional concepts as arguments (type s->e) rather than plain -individuals (type e), and attitude verbs like "think" now take -propositions (type s->t) rather than truth values (type t). - -The intenstional types are more complicated than the intensional -types. Wouldn't it be nice to keep the complicated types to just -those attitude verbs that need to worry about intensions, and keep the -rest of the grammar as extensional as possible? This desire is -parallel to our earlier desire to limit the concern about division by -zero to the division function, and let the other functions ignore -division-by-zero problems as much as possible. - -So here's what we do: - -In Ocaml, we'll use integers to model possible worlds: - - type s = int;; - type e = char;; - type t = bool;; - -Characters (characters in the computational sense, i.e., letters like -`'a'` and `'b'`, not Kaplanian characters) will model individuals, and -Ocaml booleans will serve for truth values. +* [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf): +in M. Broy, editor, *Marktoberdorf Summer School on Program Design +Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems +sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer, +editors, *Advanced Functional Programming*, Springer Verlag, +LNCS 925, 1995. Some errata fixed August 2001. + -
```-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;

-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-```
+There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you. -In our monad, the intension of an extensional type `'a` is `s -> 'a`, -a function from worlds to extensions. Our unit will be the constant -function (an instance of the K combinator) that returns the same -individual at each world. +In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category +Theory](/advanced_topics/monads_in_category_theory) notes do so, for example. -Then `ann = unit 'a'` is a rigid designator: a constant function from -worlds to individuals that returns `'a'` no matter which world is used -as an argument. +Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference. -Let's test compliance with the left identity law: +You may sometimes see: -
```-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-```
+ u >> v -We'll assume that this and the other laws always hold. +That just means: -We now build up some extensional meanings: + u >>= fun _ -> v - let left w x = match (w,x) with (2,'c') -> false | _ -> true;; +that is: -This function says that everyone always left, except for Cam in world -2 (i.e., `left 2 'c' == false`). + bind u (fun _ -> v) -Then the way to evaluate an extensional sentence is to determine the -extension of the verb phrase, and then apply that extension to the -extension of the subject: +You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used. -
```-let extapp fn arg w = fn w (arg w);;
+The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example:

-extapp left ann 1;;
-# - : bool = true
+	# let even x = (x mod 2 = 0);;
+	val g : int -> bool =

-extapp left cam 2;;
-# - : bool = false
-```
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad? -`extapp` stands for "extensional function application". -So Ann left in world 1, but Cam didn't leave in world 2. + # let lift g = fun u -> bind u (fun x -> Some (g x));; + val lift : ('a -> 'b) -> 'a option -> 'b option = -A transitive predicate: +`lift even` will now be a function from `int option`s to `bool option`s. We can +also define a lift operation for binary functions: - let saw w x y = (w < 2) && (y < x);; - extapp (extapp saw bill) ann 1;; (* true *) - extapp (extapp saw bill) ann 2;; (* false *) + # let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));; + val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = -In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone -in world two. +`lift2 (+)` will now be a function from `int option`s and `int option`s to `int option`s. This should look familiar to those who did the homework. -Good. Now for intensions: +The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<\$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`! - let intapp fn arg w = fn w arg;; +Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with. -The only difference between intensional application and extensional -application is that we don't feed the evaluation world to the argument. -(See Montague's rules of (intensional) functional application, T4 -- T10.) -In other words, instead of taking an extension as an argument, -Montague's predicates take a full-blown intension. +In general, any lift/map operation can be relied on to satisfy these laws: -But for so-called extensional predicates like "left" and "saw", -the extra power is not used. We'd like to define intensional versions -of these predicates that depend only on their extensional essence. -Just as we used bind to define a version of addition that interacted -with the option monad, we now use bind to intensionalize an -extensional verb: + * lift id = id + * lift (compose f g) = compose (lift f) (lift g) -
```-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these
+two computations to give the same result:

-intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
-intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
-```
+ List.map (fun x -> f (g x)) lst + List.map f (List.map g lst) -Because `bind` unwraps the intensionality of the argument, when the -lifted "left" receives an individual concept (e.g., `unit 'a'`) as -argument, it's the extension of the individual concept (i.e., `'a'`) -that gets fed to the basic extensional version of "left". (For those -of you who know Montague's PTQ, this use of bind captures Montague's -third meaning postulate.) +Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this: -Likewise for extensional transitive predicates like "saw": + ap [f] [x; y] = [f x; f y] + ap (Some f) (Some x) = Some (f x) -
```-let lift2 pred w arg1 arg2 =
-  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
-intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
-intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
-```
+and so on. Here are the laws that any `ap` operation can be relied on to satisfy: -Crucially, an intensional predicate does not use `bind` to consume its -arguments. Attitude verbs like "thought" are intensional with respect -to their sentential complement, but extensional with respect to their -subject (as Montague noticed, almost all verbs in English are -extensional with respect to their subject; a possible exception is "appear"): + ap (unit id) u = u + ap (ap (ap (unit compose) u) v) w = ap u (ap v w) + ap (unit f) (unit x) = unit (f x) + ap u (unit x) = ap (unit (fun f -> f x)) u -
```-let think (w:s) (p:s->t) (x:e) =
-  match (x, p 2) with ('a', false) -> false | _ -> p w;;
-```
+Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where +we went from: -Ann disbelieves any proposition that is false in world 2. Apparently, -she firmly believes we're in world 2. Everyone else believes a -proposition iff that proposition is true in the world of evaluation. + [[1]; [1;2]; [1;3]; [1;2;4]] -
```-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'b'))))
-       (unit 'a')
-1;; (* true *)
-```
+to: -So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave). + [1; 1; 2; 1; 3; 1; 2; 4] -The `lift` is there because "think Bill left" is extensional wrt its -subject. The important bit is that "think" takes the intension of -"Bill left" as its first argument. +That is the `join` operation. -
```-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'c'))))
-       (unit 'a')
-1;; (* false *)
-```
+All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions: -But even in world 1, Ann doesn't believe that Cam left (even though he -did: `intapp (lift left) cam 1 == true`). Ann's thoughts are hung up -on what is happening in world 2, where Cam doesn't leave. + lift f u = u >>= compose unit f + lift f u = ap (unit f) u + lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y))) + lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v + ap u v = u >>= (fun f -> lift f v) + ap u v = lift2 id u v + join m2 = m2 >>= id + u >>= f = join (lift f u) + u >> v = u >>= (fun _ -> v) + u >> v = lift2 (fun _ -> id) u v -*Small project*: add intersective ("red") and non-intersective - adjectives ("good") to the fragment. The intersective adjectives - will be extensional with respect to the nominal they combine with - (using bind), and the non-intersective adjectives will take - intensional arguments. -Finally, note that within an intensional grammar, extensional funtion -application is essentially just bind: -
```-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-```
+Monad outlook +------------- + +We're going to be using monads for a number of different things in the +weeks to come. One major application will be the State monad, +which will enable us to model mutation: variables whose values appear +to change as the computation progresses. Later, we will study the +Continuation monad. + +But first, we'll look at several linguistic applications for monads, based +on what's called the *Reader monad*. + +##[[Reader Monad for Variable Binding]]## + +##[[Reader Monad for Intensionality]]## +