X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=0dcb0957803ffa0855c51e792a0c0979a4590683;hp=d7a2cf12e68708f389b99b3d55cb60d3dd956fc4;hb=4d9fe4645917a4966694c7c4bf8edde8b243745f;hpb=8a2e3fac5b5faf8eebef6b4c24db35dfc101c07c
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index d7a2cf12..0dcb0957 100644
--- 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 @@ -32,35 +24,34 @@ 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):
+Then System F can be specified as follows:
System F:
---------
- types Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï
- expressions e ::= x | λx:Ï. e | e1 e2 | Î'a. e | e [Ï]
+ 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`. "`'a`" is a type variable. The
+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
+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.). "`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
+type `Ï1` to expressions of type `Ï2`. And "`âα.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.
+`'a`. 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
+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
+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
@@ -68,9 +59,9 @@ 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)
+Λ Î± (λ x:α. x)
-the Λ
binds the type variable `'a` that occurs in
+the Λ
binds the type variable `α` 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
@@ -83,27 +74,27 @@ 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]
This type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `'a`. Not
+values) specifies the value of the type variable `α`. Not
surprisingly, the type of 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 uninstantiated
+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)): (∀ 'a . 'a -> 'a)
+(Λα (λx:α . x)): (∀α. α-α)
Pred in System F
----------------
@@ -115,15 +106,16 @@ however. Here is one way, coded in
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
+ N = âα.(α->α)->α->α;
+ Pair = (N->N->N)->N;
+
+ 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 suc = λn:N. Îα. 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)));
@@ -136,7 +128,7 @@ 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
+typed as a Church number, i.e., as `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
@@ -149,8 +141,8 @@ 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
+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.
@@ -163,19 +155,19 @@ 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
+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 ==
- (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+ (λx:(âα.α->α). x [âα.α->α] x) (Îα.λx:α.x)
-Since the type of the identity function is `(All X . X->X)`, it's the
+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 `X` to the universal type `All X . X->X`. Instantiating the
+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.
@@ -198,8 +190,89 @@ be strongly normalizing, from which it follows that System F is not
Turing complete.
-Types in OCaml
---------------
+## 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 = λl:t. λr:t. l r false
+ and = Îα.Îβ.λl:α->β.λr:α->β.λx: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
+
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.