meaning. So we will need to get more sophisticated about types. The
next step in that journey will be to consider System F.
-In the simply-typed lambda calculus, we write types like <code>σ
--> τ</code>. This looks like logical implication. We'll take
-that resemblance seriously when we discuss the Curry-Howard
-correspondence. In the meantime, note that types respect modus
-ponens:
-
-<pre>
-Expression Type Implication
------------------------------------
-fn α -> β α ⊃ β
-arg α α
------- ------ --------
-(fn arg) β β
-</pre>
-
-The implication in the right-hand column is modus ponens, of course.
-
System F was discovered by Girard (the same guy who invented Linear
Logic), but it was independently proposed around the same time by
Reynolds, who called his version the *polymorphic lambda calculus*.
let fst = λx:N. λy:N. x in
let snd = λx:N. λy:N. y in
let pair = λx:N. λy:N. λz:N->N->N. z x y in
- let suc = λn:N. Λα. λs:α->α. λz:α. s (n [α] s z) in
- let shift = λp:Pair. pair (suc (p fst)) (p fst) in
- let pre = λn:N. n [Pair] shift (pair zero zero) snd in
+ let succ = λn:N. Λα. λs:α->α. λz:α. s (n [α] s z) in
+ let shift = λp:Pair. pair (succ (p fst)) (p fst) in
+ let pred = λn:N. n [Pair] shift (pair zero zero) snd in
pre (suc (suc (suc zero)));
Exercise: convince yourself that `zero` has type `N`.
The key to the extra expressive power provided by System F is evident
-in the typing imposed by the definition of `pre`. The variable `n` is
-typed as a Church number, i.e., as `∀α.(α->α)->α->α`. The type
-application `n [Pair]` instantiates `n` in a way that allows it to
-manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`. In
-other words, the instantiation turns a Church number into a
+in the typing imposed by the definition of `pred`. The variable `n`
+is typed as a Church number, i.e., as `N ≡ ∀α.(α->α)->α->α`.
+The type application `n [Pair]` instantiates `n` in a way that allows
+it to manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`.
+In other words, the instantiation turns a Church number into a certain
pair-manipulating function, which is the heart of the strategy for
-this version of predecessor.
-
-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, `(α -> α) -> α -> α`. And then we'd have to
-replace each of these `α`'s with... ad infinitum. If we had to choose
-a concrete type built entirely from explicit base types, we'd be
-unable to proceed.
+this version of computing the predecessor function.
+
+Could we try to accommodate the needs of the predecessor function by
+building 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 `N` 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, which we're imagining is `(Pair -> Pair) ->
+Pair -> Pair`. And then we'd have to replace each of these `Pairs`'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.]
In order to see how this works, we'll apply ω to the identity
function.
-<code>ω id ==</code>
-
- (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)
+<code>ω id ≡ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)</code>
Since the type of the identity function is `∀α.α->α`, it's the
right type to serve as the argument to ω. The definition of
Not only does a fixed-point combinator remain out of reach, we can't
even construct an infinite loop. This means that although we found a
type for ω, there is no general type for Ω ≡ ω
-ω. Furthermore, it turns out that no Turing complete system can
-be strongly normalizing, from which it follows that System F is not
+ω. In fact, it turns out that no Turing complete system can be
+strongly normalizing, from which it follows that System F is not
Turing complete.
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.
+ Type of the argument of "and":
+ Ann left and Bill left. t
+ Ann left and slept. e->t
+ Ann and Bill left. (e->t)-t (i.e, generalize quantifiers)
+ Ann read and reviewed the book. e->e->t
In English (likewise, many other languages), *and* can coordinate
clauses, verb phrases, determiner phrases, transitive verbs, and many
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?
+uses.
+
+Can we capture this using polymorphic types?
Ann, Bill e
left, slept e -> t
With these basic types, we want to say something like this:
and:t->t->t = λl:t. λr:t. l r false
- and = Λα.Λβ.λl:α->β.λr:α->β.λx:α. and [β] (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 β 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.
+ gen_and = Λα.Λβ.λf:(β->t).λl:α->β.λr:α->β.λx:α. f (l x) (r x)
+
+The idea is that the basic *and* (the one defined in the first line)
+conjoins expressions of type `t`. But when *and* conjoins functional
+types (the definition in the second line), it builds a function that
+distributes its argument across the two conjuncts and then applies the
+appropriate lower-order instance of *and*.
+
+ and (Ann left) (Bill left)
+ gen_and [e] [t] and left slept
+ gen_and [e] [e->t] (gen_and [e] [t] and) read reviewed
+
+Following the terminology of Partee and Rooth, this 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, which is why we call the polymorphic part of
+the definition `gen_and`.
+
+In the first line, the basic *and* is ready to conjoin two truth
+values. In the second line, the polymorphic definition of `gen_and`
+makes explicit exactly how the meaning of *and* when it coordinates
+verb phrases depends on the meaning of the basic truth connective.
+Likewise, when *and* coordinates transitive verbs of type `e->e->t`,
+the generalized *and* depends on the `e->t` version constructed for
+dealing with coordinated verb phrases.
+
+On the one hand, this definition accurately expresses the way in which
+the meaning of the conjunction of more complex types relates to the
+meaning of the conjunction of simpler types. On the other hand, it's
+awkward to have to explicitly supply an expression each time that
+builds up the meaning of the *and* that coordinates the expressions of
+the simpler types. We'd like to have that automatically handled by
+the polymorphic definition; but that would require writing code that
+behaved differently depending on the types of its type arguments,
+which goes beyond the expressive power of System F.
And in fact, discussions of generalized coordination in the
linguistics literature are almost always left as a meta-level
implemented as a method for generating a suitable set of translation
rules, which are in turn expressed in a simply-typed grammar.
+There is some work using System F to express generalizations about
+natural language: Ponvert, Elias. 2005. Polymorphism in English Logical
+Grammar. In *Lambda Calculus Type Theory and Natural Language*: 47--60.
+
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