From b32335b5eb7ec1092798f3f77f4316b3709bfb8b Mon Sep 17 00:00:00 2001 From: Chris Date: Wed, 25 Feb 2015 13:31:54 -0500 Subject: [PATCH] generalized conjunction --- topics/_week5_system_F.mdwn | 79 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 79 insertions(+) diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 72d07b31..ff0b341b 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -200,6 +200,85 @@ be strongly normalizing, from which it follows that System F is not Turing complete. +## Polymorphism in natural language + +Is the simply-typed lambda calclus enough for analyzing natural +language, or do we need polymorphic types (or something even more expressive)? + +The classic case study motivating polymorphism in natural language +comes from coordination. (The locus classicus is Partee and Rooth +1983.) + + Ann left and Bill left. + Ann left and slept. + Ann and Bill left. + Ann read and reviewed the book. + +In English (likewise, many other languages), *and* can coordinate +clauses, verb phrases, determiner phrases, transitive verbs, and many +other phrase types. In a garden-variety simply-typed grammar, each +kind of conjunct has a different semantic type, and so we would need +an independent treatment of *and* for each one. Yet there is a strong +intuition that the contribution of *and* remains constant across all +of these uses. Can we capture this using polymorphic types? + + Ann, Bill e + left, slept e -> t + read, reviewed e -> e -> t + +With these basic types, we want to say something like this: + + and:t->t->t = lambda l:t . lambda r:t . l r false + and = lambda 'a . lambda 'b . + lambda l:'a->'b . lambda r:'a->'b . + lambda x:'a . and:'b (l x) (r x) + +The idea is that the basic *and* conjoins expressions of type `t`, and +when *and* conjoins functional types, the result is a function that +distributes its argument across the two conjuncts and conjoins the +result. So `Ann left and slept` will evaluate to `(\x.and(left +x)(slept x)) ann`. Following Partee and Rooth, the strategy of +defining the coordination of expressions with complex types in terms +of the coordination of expressions with less complex types is known as +Generalized Coordination. + +But the definitions just given are not well-formed expressions in +System F. There are several problems. The first is that we have two +definitions of the same word. The intention is for one of the +definitions to be operative when the type of its arguments is type +`t`, but we have no way of conditioning evaluation on the type of an +argument. The second is that for the polymorphic definition, the term +*and* occurs inside of the definition. System F does not have +recursion. The third problem is more subtle. The defintion as given +takes two types as parameters: the type of the first argument expected +by each conjunct, and the type of the result of applying each conjunct +to an argument of that type. We would like to instantiate the +recursive use of *and* in the definition by using the result type. +But fully instantiating the definition as given requires type +application to a pair of types, not just one type. + +So conjunction and disjunction provide a compelling motivation for +polymorphism in natural language, but we don't yet have the ability to +build the polymorphism into a formal system. + +And in fact, discussions of generalized coordination in the +linguistics literature are almost always left as a metageneralization +over a basic simply-typed grammar. For instance, in Hendriks' 1992:74 +dissertation, generalized coordination is implemented as a method for +generating a suitable set of translation rules, which are in turn +expressed in a simply-typed grammar. + +Not incidentally, we're not aware of any programming language that +makes generalized coordination available, despite is naturalness and +ubiquity in natural language. That is, coordination in programming +languages is always at the sentential level. You might be able to evaluate +`delete file1 and delete file2` but never `delete file1 and file2`. + +We'll return to thinking about generalized coordination as we get +deeper into types. There will be an analysis in term of continuations +that will be particularly satisfying. + + #Types in OCaml -- 2.11.0