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
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
----------------
## 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)?
+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
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?
+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
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.
+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 several problems. The first is that we have two
+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
+`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.
+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 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.
+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
+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`.
+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