+[[!toc levels=2]]
+
# System F and recursive types
In the simply-typed lambda calculus, we write types like <code>σ
distinguish expression abstraction from type abstraction by also
changing the shape of the lambda.
-This expression is a polymorphic version of the identity function. It
-defines one general identity function that can be adapted for use with
-expressions of any type. In order to get it ready to apply this
-identity function to, say, a variable of type boolean, just do this:
+The expression immediately below is a polymorphic version of the
+identity function. It defines one general identity function that can
+be adapted for use with expressions of any type. In order to get it
+ready to apply this identity function to, say, a variable of type
+boolean, just do this:
<code>(Λ 'a (λ x:'a . x)) [t]</code>
pair-manipulating function, which is the heart of the strategy for
this version of predecessor.
-But of course, the type `Pair` (in this simplified example) is defined
-in terms of Church numbers. If we tried to replace the type for
-Church numbers with a concrete (simple) type, we would have to replace
-each `X` with `(N -> N -> N) -> N`. But then we'd have to replace
-each `N` with `(X -> X) -> X -> X`. And then replace each `X`
-with... ad infinitum. If we had to choose a concrete type built
-entirely from explicit base types, we'd be unable to proceed.
+Could we try to build a system for doing Church arithmetic in which
+the type for numbers always manipulated ordered pairs? The problem is
+that the ordered pairs we need here are pairs of numbers. If we tried
+to replace the type for Church numbers with a concrete (simple) type,
+we would have to replace each `X` with the type for Pairs, `(N -> N ->
+N) -> N`. But then we'd have to replace each of these `N`'s with the
+type for Church numbers, `(X -> X) -> X -> X`. And then we'd have to
+replace each of these `X`'s with... ad infinitum. If we had to choose
+a concrete type built entirely from explicit base types, we'd be
+unable to proceed.
[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
Press, chapter 23.]
Turing complete.
-Types in OCaml
---------------
+## 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 rule 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, it builds a function that
+distributes its argument across the two conjuncts and conjoins the two
+results. So `Ann left and slept` will evaluate to `(\x.and(left
+x)(slept x)) ann`. Following the terminology of 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 three 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 to just a single type. We want to somehow
+guarantee that 'b will always itself be a complex 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 meta-level
+generalizations 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
+
OCaml has type inference: the system can often infer what the type of
an expression must be, based on the type of other known expressions.