X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=sidebyside;f=topics%2F_week5_system_F.mdwn;h=ff0b341bfb465da312e0fb2530627b10558da40f;hb=b32335b5eb7ec1092798f3f77f4316b3709bfb8b;hp=684f42be4087594c9c97768e7f7ef66bcca0bc5b;hpb=e786697f406f29139f4116d13687b37e42594f81;p=lambda.git
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 684f42be..ff0b341b 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -1,27 +1,547 @@
+[[!toc levels=2]]
+
# System F and recursive types
In the simply-typed lambda calculus, we write types like σ
--> τ
. This looks like logical implication. Let's take
-that resemblance seriously. Then note that types respect modus
-ponens: given two expressions fn:(σ -> τ)
and
-arg:σ
, the application of `fn` to `arg` has type
-(fn arg):τ
.
+-> 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:
+
+
+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. + +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*. +(Reynolds was also an early player in the development of +continuations.) + +System F enhances the simply-typed lambda calculus with abstraction +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 (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. 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`. "`'a`" 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.). "`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`. You can expect that in `â'a. Ï`, the type `Ï` will usually
+have at least one free occurrence of `'a` 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 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 'a (λ x:'a . x)
+
+the Λ
binds the type variable `'a` that occurs in
+the λ
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:
+
+(Λ 'a (λ x:'a . x)) [t]
+
+This type application (where `t` is a type constant for Boolean truth
+values) specifies the value of the type variable `'a`. Not
+surprisingly, the type of this type application is a function from
+Booleans to Booleans:
+
+((Λ 'a (λ x:'a . 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)
+
+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 uninstantiated
+(polymorphic) identity function is
+
+(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)
+
+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 = All X . (X->X)->X->X;
+ Pair = (N -> N -> N) -> N;
+ let zero = lambda X . lambda s:X->X . lambda z:X. 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 X . lambda s:X->X . lambda z:X . s (n [X] 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 `All X . (X->X)->X->X`. 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, `(X -> X) -> X -> X`. And then we'd have to
+replace each of these `X`'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 = lambda x:(All X. X->X) . x [All X . X->X] x
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+ω id ==
+
+ (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+
+Since the type of the identity function is `(All X . X->X)`, 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 `X` to the universal type `All X . X->X`. 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 treatment of *and* 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 'a . lambda 'b .
+ lambda l:'a->'b . lambda r:'a->'b .
+ lambda x:'a . and:'b (l x) (r x)
+
+The idea is that the basic *and* conjoins expressions of type `t`, and
+when *and* conjoins functional types, the result is a function that
+distributes its argument across the two conjuncts and conjoins the
+result. So `Ann left and slept` will evaluate to `(\x.and(left
+x)(slept x)) ann`. Following 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 several 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 just one 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 metageneralization
+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
+
+
+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.
+
+For instance, if we type
+
+ # let f x = x + 3;;
+
+The system replies with
+
+ val f : int -> int = x:α
means that a term `x`
-is an expression with type α
.
+ let rec blackhole x = blackhole x in (blackhole 1) + 2;;
-This is a special case of a general pattern that falls under the
-umbrella of the Curry-Howard correspondence. We'll discuss
-Curry-Howard in some detail later.
+ let rec blackhole x = blackhole x in (blackhole 1) || false;;
-System F is due (independently) to Girard and Reynolds.
-It enhances the simply-typed lambda calculus with quantification over
-types. In System F, you can say things like
+ let rec blackhole x = blackhole x in 2 :: (blackhole 1);;
-Γ α (\x.x):(α -> α)
+By the way, what's the type of this:
-This says that the identity function maps arguments of type α to
-results of type α, for any choice of α. So the Γ is
-a universal quantifier over types.
+ let rec blackhole (x:'a) : 'a = blackhole x in blackhole
+Back to thunks: the reason you'd want to control evaluation with
+thunks is to manipulate when "effects" happen. In a strongly
+normalizing system, like the simply-typed lambda calculus or System F,
+there are no "effects." In Scheme and OCaml, on the other hand, we can
+write programs that have effects. One sort of effect is printing.
+Another sort of effect is mutation, which we'll be looking at soon.
+Continuations are yet another sort of effect. None of these are yet on
+the table though. The only sort of effect we've got so far is
+*divergence* or non-termination. So the only thing thunks are useful
+for yet is controlling whether an expression that would diverge if we
+tried to fully evaluate it does diverge. As we consider richer
+languages, thunks will become more useful.