X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=b78e01e7659c821a504c269b6d2e3b5d2b928c91;hp=c81ff1253ceca9ee8cb724163cee5e139bcfe234;hb=4d3716c93c54b77c70549da836c90d9683fadb41;hpb=1713e01a3a0982e0f8fc68ed93035cea6ca8f46e diff --git a/week7.mdwn b/week7.mdwn index c81ff125..b78e01e7 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -314,7 +314,7 @@ invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Pres Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur. The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.--> -* [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from but hosted the link above is to a local copy. +* [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. @@ -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]]##