X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=b78e01e7659c821a504c269b6d2e3b5d2b928c91;hp=351a6d953fac41901c96823c8bf31829b51e6733;hb=3cc38c05eaf8329c74c11a9df20a2ca8e30e563d;hpb=2a5b500ba1a4da0ebae07837c82810ab418a8812 diff --git a/week7.mdwn b/week7.mdwn index 351a6d95..b78e01e7 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -105,7 +105,7 @@ that provides at least the following three elements: The guts of the definition of the `bind` operation amount to specifying how to unbox the monadic value `u`. In the `bind` - opertor for the option monad, we unboxed the monadic value by + operator for the option monad, we unboxed the monadic value by matching it with the pattern `Some x`---whenever `u` happened to be a box containing an integer `x`, this allowed us to get our hands on that `x` and feed it to `f`. @@ -125,14 +125,8 @@ that provides at least the following three elements: useful way. So the "option/maybe monad" consists of the polymorphic `option` type, the -`unit`/return function, and the `bind` function. With the option monad, we can -think of the "safe division" operation: +`unit`/return function, and the `bind` function. - # let safe_divide num den = if 0 = den then None else Some (num/den);; - val safe_divide : int -> int -> int option = - -as basically a function from two integers to an integer, except with -this little bit of option plumbing on the side. A note on notation: Haskell uses the infix operator `>>=` to stand for `bind`. Chris really hates that symbol. Following Wadler, he prefers to @@ -160,7 +154,7 @@ Similarly: is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x -> bind v (fun y -> f x y))`. Those who did last week's homework may recognize -this. +this last expression. (Note that the above "do" notation comes from Haskell. We're mentioning it here because you're likely to see it when reading about monads. It won't work in @@ -219,16 +213,23 @@ them from hurting the people that use them or themselves. type, we have `(unit x) * f == f x`. For instance, `unit` is itself a function of type `'a -> 'a m`, so we can use it for `f`: - # let ( * ) u f = match u with None -> None | Some x -> f x;; - val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = # let unit x = Some x;; val unit : 'a -> 'a option = + # let ( * ) u f = match u with None -> None | Some x -> f x;; + val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = + + 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 + `u * f` or `( * ) u f` instead of `* u f`. Now: # unit 2;; - : int option = Some 2 # unit 2 * unit;; - : int option = Some 2 + # let divide x y = if 0 = y then None else Some (x/y);; + val divide : int -> int -> int option = # divide 6 2;; - : int option = Some 3 # unit 2 * divide 6;; @@ -239,10 +240,6 @@ them from hurting the people that use them or themselves. # unit 0 * divide 6;; - : int option = None -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 -`u * f` or `( * ) u f` instead of `* u f`. * **Associativity: bind obeys a kind of associativity**. Like this: @@ -268,7 +265,7 @@ is `*`, pronounced "bind") is an infix operator, so we write # Some 3 * (fun x -> divide 2 x * divide 6);; - : int option = None -Of course, associativity must hold for arbitrary functions of +Of course, associativity must hold for *arbitrary* functions of type `'a -> 'a 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` @@ -305,24 +302,26 @@ See also . Here are some papers that introduced monads into functional programming: -* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1). +* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. * [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. - The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers. + * [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 paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required. + + +* [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from but the link above is to a local copy. There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh. 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_notes/monads_in_category_theory) notes do so, for example. -Here are some of the specifics. You don't have to master these; they're collected here for your reference. +Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference. You may sometimes see: @@ -354,7 +353,7 @@ also define a lift operation for binary functions: # 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 = -`lift (+)` 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. +`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. 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`! @@ -379,7 +378,7 @@ Another general monad operation is called `ap` in Haskell---short for "apply." ( and so on. Here are the laws that any `ap` operation can be relied on to satisfy: ap (unit id) u = u - ap (ap (ap (return compose) u) v) w = ap u (ap v w) + 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 @@ -396,11 +395,12 @@ That is the `join` operation. 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: + 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 - lift f u = u >>= compose unit f - lift f u = ap (unit f) u join m2 = m2 >>= id u >>= f = join (lift f u) u >> v = u >>= (fun _ -> v) @@ -420,236 +420,7 @@ Continuation monad. In the meantime, we'll look at several linguistic applications for monads, based on what's called the *reader monad*. +##[[Reader monad]]## -The reader monad ----------------- - -Introduce - -Heim and Kratzer's "Predicate Abstraction Rule" - - - -The intensionality monad ------------------------- -... -intensional function application. In Shan (2001) [Monads for natural -language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that -making expressions sensitive to the world of evaluation is -conceptually the same thing as making use of a *reader monad* (which -we'll see again soon). This technique was beautifully re-invented -by Ben-Avi and Winter (2007) in their paper [A modular -approach to -intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf), -though without explicitly using monads. - -All of the code in the discussion below can be found here: [[intensionality-monad.ml]]. -To run it, download the file, start OCaml, and say - - # #use "intensionality-monad.ml";; - -Note the extra `#` attached to the directive `use`. - -Here's the idea: since people can have different attitudes towards -different propositions that happen to have the same truth value, we -can't have sentences denoting simple truth values. If we did, then if John -believed that the earth was round, it would force him to believe -Fermat's last theorem holds, since both propositions are equally true. -The traditional solution is to allow sentences to denote a function -from worlds to truth values, what Montague called an intension. -So if `s` is the type of possible worlds, we have the following -situation: - - -
-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
-
- -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, like -addition or multiplication, 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. - - type 'a intension = s -> 'a;; - let unit x (w:s) = x;; - - let ann = unit 'a';; - let bill = unit 'b';; - let cam = unit 'c';; - -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. - -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. - -Let's test compliance with the left identity law: - - # let bind u f (w:s) = f (u w) w;; - val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = - # bind (unit 'a') unit 1;; - - : char = 'a' - -We'll assume that this and the other laws always hold. - -We now build up some extensional meanings: - - let left w x = match (w,x) with (2,'c') -> false | _ -> true;; - -This function says that everyone always left, except for Cam in world -2 (i.e., `left 2 'c' == false`). - -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: - - let extapp fn arg w = fn w (arg w);; - - extapp left ann 1;; - # - : bool = true - - extapp left cam 2;; - # - : bool = false - -`extapp` stands for "extensional function application". -So Ann left in world 1, but Cam didn't leave in world 2. - -A transitive predicate: - - let saw w x y = (w < 2) && (y < x);; - extapp (extapp saw bill) ann 1;; (* true *) - extapp (extapp saw bill) ann 2;; (* false *) - -In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone -in world two. - -Good. Now for intensions: - - let intapp fn arg w = fn w arg;; - -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. - -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: - - let lift pred w arg = bind arg (fun x w -> pred w x) w;; - - 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 *) - -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.) - -Likewise for extensional transitive predicates like "saw": - - 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 *) - -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"): - - let think (w:s) (p:s->t) (x:e) = - match (x, p 2) with ('a', false) -> false | _ -> p w;; - -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. - - intapp (lift (intapp think - (intapp (lift left) - (unit 'b')))) - (unit 'a') - 1;; (* true *) - -So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave). - -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. - - intapp (lift (intapp think - (intapp (lift left) - (unit 'c')))) - (unit 'a') - 1;; (* false *) - -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. - -*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 - +##[[Intensionality monad]]##