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diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
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--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -1,21 +1,13 @@
-# System F and recursive types
+[[!toc levels=2]]
-In the simply-typed lambda calculus, we write types like σ
--> τ. 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:
+# System F: the polymorphic lambda calculus
-
-Expression Type Implication
------------------------------------
-fn α -> β α ⊃ β
-arg α α
------- ------ --------
-(fn arg) β β
-
-
-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
@@ -24,77 +16,266 @@ Reynolds, who called his version the *polymorphic lambda calculus*.
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 "x:α" represents a
-expression whose type is α.
-
-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 Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï
- expressions e ::= x | λx:Ï. e | e1 e2 | Î'a. e | e [Ï]
-
-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). "`Ï1 -> Ï2`" is
-the type of a function from expressions of type `Ï1` to expressions of
-type `Ï2`. And "`â'a. Ï`" is called a universal type, since it
-universally quantifies over the type variable `'a`.
-
-In the definition of the expressions, we have variables "`x`".
-Abstracts "`λx:Ï. e`" are similar to abstracts in the simply-typed lambda
+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 "x:α" represents an expression `x`
+whose type is α.
+
+Then System F can be specified as follows:
+
+ System F:
+ ---------
+ types Ï ::= c | α | Ï1 -> Ï2 | âα.Ï
+ expressions e ::= x | λx:Ï.e | e1 e2 | Îα.e | e [Ï]
+
+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. 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. "`Ï1 -> Ï2`" is the type of a
+function from expressions of type `Ï1` to expressions of type `Ï2`.
+And "`âα.Ï`" is called a universal type, since it universally
+quantifies over the type variable `α`. You can expect that in
+`âα.Ï`, the type `Ï` 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:Ï.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: "`Î'a. e`" is a type
-abstraction, and "`e [Ï]`" is a type application. The idea is that
-Λ is a capital λ. Just like
-the lower-case λ, Λ binds
-variables in its body; unlike λ,
-Λ binds type variables. So in the expression
+additional ways of forming expressions: "`Îα.e`" is called a *type
+abstraction*, and "`e [Ï]`" is called a *type application*. The idea
+is that Λ is a capital λ: just
+like the lower-case λ, Λ binds
+variables in its body, except that unlike λ,
+Λ binds type variables instead of expression
+variables. So in the expression
+
+Λ Î± (λ x:α. x)
-Λ 'a (λ x:'a . x)
+the Λ binds the type variable `α` that occurs in
+the λ abstract.
-the Λ binds the type variable `'a` that occurs in
-the λ 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
-boolean, just do this:
+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:
-(Λ 'a (λ x:'a . x)) [t]
+(Λ Î± (λ x:α. x)) [t]
-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
-surprisingly, the type of this type application is a function from
-Booleans to Booleans:
+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 the expression that results from this type
+application is a function from Booleans to Booleans:
-((Λ 'a (λ x:'a . x)) [t]): (b -> b)
+((Λα (λ x:α . x)) [t]): (b->b)
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`:
-((Λ 'a (λ x:'a . x)) [e]): (e -> e)
+((Λα (λ x:α. x)) [e]): (e->e)
-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
-(Λ 'a (λ x:'a . x)): (\forall 'a . 'a -> 'a)
-
-
-
-
-##
-
+(Λα (λx:α . x)): (∀α. α->α)
+
+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:
+
+ let N = âα.(α->α)->α->α in
+ let Pair = (N->N->N)->N in
+
+ let zero = Îα. 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 succ = λn:N. Îα. 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)));
+
+[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 `pred`. The variable `n`
+is typed as a Church number, i.e., as `N ≡ âα.(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 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.]
+
+Typing ω
+--------------
+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
+
+ω = λx:(âα.α->α). x [âα.α->α] x
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+ω id ≡ (λx:(âα.α->α). x [âα.α->α] x) (Îα.λx:α.x)
+
+Since the type of the identity function is `âα.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 `âα.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. 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.
+
+
+## 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.)
+
+ 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
+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 = λl:t. λr:t. l r false
+ gen_and = Îα.Îβ.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
+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.
+
+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
+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.