[[!toc levels=2]]
-# System F and recursive types
+# System F: the polymorphic lambda calculus
-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.
+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.
System F was discovered by Girard (the same guy who invented Linear
Logic), but it was independently proposed around the same time by
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 . λlambda s:α->α . λz:α. s (n [α] s z) in
+ 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
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
+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 fact, unlike in the simply-typed lambda calculus,
it is even possible to give a type for ω in System F.
-<code>ω = λlambda x:(∀ 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.
<code>ω id ==</code>
- (λx:(∀α. α->α) . x [∀α.α->α] x) (Λα.λx:α. x)
+ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx: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