+[[!toc levels=2]]
+
# System F and recursive types
In the simply-typed lambda calculus, we write types like <code>σ
System F:
---------
- types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
- expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
+ types τ ::= c | α | τ1 -> τ2 | ∀'a. τ
+ expressions e ::= x | λx:τ. e | e1 e2 | Λα. e | e [τ]
In the definition of the types, "`c`" is a type constant. Type
constants play the role in System F that base types play in the
simply-typed lambda calculus. So in a lingusitics context, type
-constants might include `e` and `t`. "`'a`" is a type variable. The
+constants might include `e` and `t`. "α" is a type variable. The
tick mark just indicates that the variable ranges over types rather
than over values; in various discussion below and later, type variable
can be distinguished by using letters from the greek alphabet
(α, β, etc.), or by using capital roman letters (X, Y,
etc.). "`τ1 -> τ2`" is the type of a function from expressions of
-type `τ1` to expressions of type `τ2`. And "`∀'a. τ`" is called a
+type `τ1` to expressions of type `τ2`. And "`∀α. τ`" is called a
universal type, since it universally quantifies over the type variable
-`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually
-have at least one free occurrence of `'a` somewhere inside of it.
+`'a`. You can expect that in `∀α. τ`, the type `τ` will usually
+have at least one free occurrence of `α` somewhere inside of it.
In the definition of the expressions, we have variables "`x`" as usual.
Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda
type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
In addition to variables, abstracts, and applications, we have two
-additional ways of forming expressions: "`Λ'a. e`" is called a *type
+additional ways of forming expressions: "`Λα. e`" is called a *type
abstraction*, and "`e [τ]`" is called a *type application*. The idea
is that <code>Λ</code> is a capital <code>λ</code>: just
like the lower-case <code>λ</code>, <code>Λ</code> binds
<code>Λ</code> binds type variables instead of expression
variables. So in the expression
-<code>Λ 'a (λ x:'a . x)</code>
+<code>Λ α (λ x:α . x)</code>
-the <code>Λ</code> binds the type variable `'a` that occurs in
+the <code>Λ</code> binds the type variable `α` that occurs in
the <code>λ</code> abstract. Of course, as long as type
variables are carefully distinguished from expression variables (by
tick marks, Grecification, or capitalization), there is no need to
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>
+<code>(Λ α (λ x:α . x)) [t]</code>
This type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `'a`. Not
+values) specifies the value of the type variable `α`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
-<code>((Λ 'a (λ x:'a . x)) [t]): (b -> b)</code>
+<code>((Λ α (λ x:α . x)) [t]): (b -> b)</code>
Likewise, if we had instantiated the type variable as an entity (base
type `e`), the resulting identity function would have been a function
of type `e -> e`:
-<code>((Λ 'a (λ x:'a . x)) [e]): (e -> e)</code>
+<code>((Λ α (λ x:α . x)) [e]): (e -> e)</code>
-Clearly, for any choice of a type `'a`, the identity function can be
-instantiated as a function from expresions of type `'a` to expressions
-of type `'a`. In general, then, the type of the uninstantiated
+Clearly, for any choice of a type `α`, the identity function can be
+instantiated as a function from expresions of type `α` to expressions
+of type `α`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
-<code>(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)</code>
+<code>(Λ α (λ x:α . x)): (∀ α . α -> α)</code>
Pred in System F
----------------
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.