+[[!toc levels=2]]
+
# System F and recursive types
In the simply-typed lambda calculus, we write types like <code>σ
continuations.)
System F enhances the simply-typed lambda calculus with abstraction
-over types. In order to state System F, we'll need to adopt the
-notational convention that "<code>x:α</code>" represents a
-expression whose type is <code>α</code>.
+over types. Normal lambda abstraction abstracts (binds) an expression
+(a term); type abstraction abstracts (binds) a type.
+
+In order to state System F, we'll need to adopt the
+notational convention (which will last throughout the rest of the
+course) that "<code>x:α</code>" represents an expression `x`
+whose type is <code>α</code>.
Then System F can be specified as follows (choosing notation that will
match up with usage in O'Caml, whose type system is based on System F):
- System F:
- types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
- expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
-
-In the definition of the types, "`c`" is a type constant (e.g., `e` or
-`t`). "`'a`" is a type variable (the tick mark just indicates that
-the variable ranges over types rather than values). "`τ1 -> τ2`" is
-the type of a function from expressions of type `τ1` to expressions of
-type `τ2`. And "`∀'a. τ`" is called a universal type, since it
-universally quantifies over the type variable `'a`.
-
-In the definition of the expressions, we have variables "`x`".
+ System F:
+ ---------
+ 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`. "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 "`∀α. τ`" is called a
+universal type, since it universally quantifies over the type variable
+`'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
calculus, except that they have their shrug variable annotated with a
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 a type
-abstraction, and "`e [τ]`" is 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
-variables in its body; unlike <code>λ</code>,
-<code>Λ</code> binds type variables. So in the expression
-
-<code>Λ 'a (λ x:'a . x)</code>
-
-the <code>Λ</code> binds the type variable `'a` that occurs in
-the <code>λ</code> abstract. This expression is a polymorphic
-version of the identity function. It says that this one general
-identity function can be adapted for use with expressions of any
-type. In order to get it ready to apply to, say, a variable of 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
+variables in its body, except that unlike <code>λ</code>,
+<code>Λ</code> binds type variables instead of expression
+variables. So in the expression
+
+<code>Λ α (λ x:α . x)</code>
+
+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.
+
+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>
-The type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `α`, which is
-the type of the variable bound in the `λ` expression. Not
+This type application (where `t` is a type constant for Boolean truth
+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 unapplied
+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)): (\forall 'a . 'a -> 'a)
-
-
-
-
-##
-
+<code>(Λ α (λ x:α . x)): (∀ α . α -> α)</code>
+
+Pred in System F
+----------------
+
+We saw that the predecessor function couldn't be expressed in the
+simply-typed lambda calculus. It *can* be expressed in System F,
+however. Here is one way, coded in
+[[Benjamin Pierce's type-checker and evaluator for
+System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
+relevant evaluator is called "fullpoly"):
+
+ N = ∀ α . (α->α)->α->α;
+ Pair = (N -> N -> N) -> N;
+ let zero = lambda α . lambda s:α->α . lambda z:α. z in
+ let fst = lambda x:N . lambda y:N . x in
+ let snd = lambda x:N . lambda y:N . y in
+ let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in
+ let suc = lambda n:N . lambda α . lambda s:α->α . lambda z:α . s (n [α] s z) in
+ let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in
+ let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
+
+ pre (suc (suc (suc zero)));
+
+We've truncated the names of "suc(c)" and "pre(d)", since those are
+reserved words in Pierce's system. Note that in this code, there is
+no typographic distinction between ordinary lambda and type-level
+lambda, though the difference is encoded in whether the variables are
+lower case (for ordinary lambda) or upper case (for type-level
+lambda).
+
+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
+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.
+
+[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
+Press, chapter 23.]
+
+Typing ω
+--------------
+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
+
+<code>ω = lambda x:(∀ α. α->α) . x [∀ α . α->α] x</code>
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+<code>ω id ==</code>
+
+ (lambda x:(∀ α . α->α) . x [∀ α . α->α] x) (lambda α . lambda x:α . x)
+
+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
+ω instantiates the identity function by binding the type
+variable `α` to the universal type `∀ α . α->α`. Instantiating the
+identity function in this way results in an identity function whose
+type is (in some sense, only accidentally) the same as the original
+fully polymorphic identity function.
+
+So in System F, unlike in the simply-typed lambda calculus, it *is*
+possible for a function to apply to itself!
+
+Does this mean that we can implement recursion in System F? Not at
+all. In fact, despite its differences with the simply-typed lambda
+calculus, one important property that System F shares with the
+simply-typed lambda calculus is that they are both strongly
+normalizing: *every* expression in either system reduces to a normal
+form in a finite number of steps.
+
+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
+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 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 α . lambda β .
+ lambda l:α->β . lambda r:α->β .
+ lambda x: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.
+
+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
-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.