[[!toc levels=2]]
-# System F and recursive types
+# System F: the polymorphic lambda calculus
+
+The simply-typed lambda calculus is beautifully simple, but it can't
+even express the predecessor function, let alone full recursion. And
+we'll see shortly that there is good reason to be unsatisfied with the
+simply-typed lambda calculus as a way of expressing natural language
+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
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 variables
-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.
+constants might include `e` and `t`. "α" is a type variable. In
+various discussions, type variables are distinguished by using letters
+from the greek alphabet (α, β, etc.), as we do here, or by
+using capital roman letters (X, Y, etc.), or by adding a tick mark
+(`'a`, `'b`, etc.), as in OCaml. "`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 `α`. 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
<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:
+the <code>λ</code> abstract.
+
+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:
<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 `α`. Not
-surprisingly, the type of this type application is a function from
-Booleans to Booleans:
+surprisingly, the type of the expression that results from this type
+application is a function from Booleans to Booleans:
<code>((Λα (λ x:α . x)) [t]): (b->b)</code>
of type `α`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
-<code>(Λα (λx:α . x)): (∀α. α-α)</code>
+<code>(Λα (λx: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 = Λα.λs:α->α.λz:α.z in
- 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
+however. Here is one way:
+
+ let N = ∀α.(α->α)->α->α in
+ let Pair = (N->N->N)->N in
+
+ let zero = Λα. λs:α->α. λz:α. z in
+ 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
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).
+[If you want to run this code 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", and you'll need to
+truncate the names of "suc(c)" and "pre(d)", since those are
+reserved words in Pierce's system.]
+
+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
In fact, unlike in the simply-typed lambda calculus,
it is even possible to give a type for ω in System F.
-<code>ω = λx:(∀α.α->α).x [∀α.α->α] x</code>
+<code>ω = λx:(∀α.α->α). x [∀α.α->α] x</code>
In order to see how this works, we'll apply ω to the identity
function.