XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_simply_typed_lambda.mdwn;h=047ee8b368a8a6f62c3111031a3b59d4ba06c559;hp=1fedd2a3c9843afd4b0721ba26e7be798c75dac3;hb=a4d2693effe839524592f4427465ff8d97625302;hpb=967055fbde4c32649cdf8acf16ac127d43ec3118
diff git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index 1fedd2a3..047ee8b3 100644
 a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ 54,11 +54,10 @@ tells us that "The simple theory of types was suggested as a
modification of Russell's ramified theory of types by Leon Chwistek in
1921 and 1922 and by F. P. Ramsey in 1926." This footnote appears in
Church's 1940 paper [A formulation of the simple theory of
types](churchsimpletypes.pdf). In this paper, as Will Starr
mentioned in class, Church does indeed write types by simple
apposition, without the ugly angle brackets and commas used by
Montague. Furthermore, he omits parentheses under the convention that
types associated to the *left*the opposite of the modern
+types](churchsimpletypes.pdf). In this paper, Church writes types
+by simple apposition, without the ugly angle brackets and commas used
+by Montague. Furthermore, he omits parentheses under the convention
+that types associated to the *left*the opposite of the modern
convention. This is ok, however, because he also reverses the order,
so that `te` is a function from objects of type `e` to objects of type
`t`. Cool paper! If you ever want to see Church numerals in their
@@ 82,16 +81,16 @@ types `T`, the smallest set such that
* ground types, including `e` and `t`, are in `T`
* for any types σ and τ in `T`, the type σ >
+* for any types σ and τ in `T`, the type σ >
τ is in `T`.
For instance, here are some types in `T`:
e
 e > t
 e > e > t
 (e > t) > t
 (e > t) > e > t
+ e > t
+ e > e > t
+ (e > t) > t
+ (e > t) > e > t
and so on.
@@ 103,28 +102,28 @@ which is the smallest set such that
* each type `t` has an infinite set of distinct variables, {x^t}_1,
{x^t}_2, {x^t}_3, ...
* If a term `M` has type σ > τ, and a term `N` has type
+* If a term `M` has type σ > τ, and a term `N` has type
σ, then the application `(M N)` has type τ.
* If a variable `a` has type σ, and term `M` has type τ,
 then the abstract λ a M
has type σ > τ.
+ then the abstract λ a M
has type σ > τ.
The definitions of types and of typed terms should be highly familiar
to semanticists, except that instead of writing σ > τ,
+to semanticists, except that instead of writing σ > τ,
linguists write <σ, τ>. We will use the arrow notation,
since it is more iconic.
Some examples (assume that `x` has type `o`):
x o
 \x.x o > o
+ \x.x o > o
((\x.x) x) o
Excercise: write down terms that have the following types:
 o > o > o
 (o > o) > o > o
 (o > o > o) > o
+ o > o > o
+ (o > o) > o > o
+ (o > o > o) > o
#Associativity of types versus terms#
@@ 132,7 +131,7 @@ As we have seen many times, in the lambda calculus, function
application is left associative, so that `f x y z == (((f x) y) z)`.
Types, *THEREFORE*, are right associative: if `x`, `y`, and `z`
have types `a`, `b`, and `c`, respectively, then `f` has type
`a > b > c > d == (a > (b > (c > d)))`, where `d` is the
+`a > b > c > d == (a > (b > (c > d)))`, where `d` is the
type of the complete term.
It is a serious faux pas to associate to the left for types. You may
@@ 147,13 +146,13 @@ present it here; see Berendregt or Hankin.
Since Ω does not have a normal form, it follows that Ω
cannot have a type in Λ_T. We can easily see why:
 Ω = (\x.xx)(\x.xx)
+Ω = (\x.xx)(\x.xx)
Assume Ω has type τ, and `\x.xx` has type σ. Then
because `\x.xx` takes an argument of type σ and returns
something of type τ, `\x.xx` must also have type σ >
+something of type τ, `\x.xx` must also have type σ >
τ. By repeating this reasoning, `\x.xx` must also have type
(σ > τ) > τ; and so on. Since variables have
+(σ > τ) > τ; and so on. Since variables have
finite types, there is no way to choose a type for the variable `x`
that can satisfy all of the requirements imposed on it.
@@ 161,51 +160,151 @@ In general, there is no way for a function to have a type that can
take itself for an argument. It follows that there is no way to
define the identity function in such a way that it can take itself as
an argument. Instead, there must be many different identity
functions, one for each type.
+functions, one for each type. Some of those types can be functions,
+and some of those functions can be (typerestricted) identity
+functions; but a simplytypes identity function can never apply to itself.
#Typing numerals#
Version 1 type numerals are not a good choice for the simplytyped
lambda calculus. The reason is that each different numberal has a
different type! For instance, if zero has type σ, then since
one is represented by the function `\x.x false 0`, it must have type
`b > σ > σ`, where `b` is the type of a boolean. But
this is a different type than zero! Because each number has a
different type, it becomes unbearable to write arithmetic operations
that can combine zero with one, since we would need as many different
addition operations as we had pairs of numbers that we wanted to add.

Fortunately, the Church numerals are well behaved with respect to
types. They can all be given the type (σ > σ) >
σ > σ.






+The Church numerals are well behaved with respect to types.
+To see this, consider the first three Church numerals (starting with zero):
+
+ \s z . z
+ \s z . s z
+ \s z . s (s z)
+
+Given the internal structure of the term we are using to represent
+zero, its type must have the form ρ > σ > σ for
+some ρ and σ. This type is consistent with term for one,
+but the structure of the definition of one is more restrictive:
+because the first argument (`s`) must apply to the second argument
+(`z`), the type of the first argument must describe a function from
+expressions of type σ to some result type. So we can refine
+ρ by replacing it with the more specific type σ > τ.
+At this point, the overall type is (σ > τ) > σ >
+σ. Note that this refined type remains compatible with the
+definition of zero. Finally, by examinining the definition of two, we
+see that expressions of type τ must be suitable to serve as
+arguments to functions of type σ > τ, since the result of
+applying `s` to `z` serves as the argument of `s`. The most general
+way for that to be true is if τ ≡ σ. So at this
+point, we have the overall type of (σ > σ) > σ
+> σ.
+
+
+
+## Predecessor and lists are not representable in simply typed lambdacalculus ##
+
+As Oleg Kiselyov points out, [[predecessor and lists can't be
+represented in the simplytyped lambda
+calculushttp://okmij.org/ftp/Computation/lambdacalc.html#predecessor]].
+This is not because there is any difficulty typing what the functions
+involved do "from the outside": for instance, the predecessor function
+is a function from numbers to numbers, or τ > τ, where τ
+is our type for Church numbers (i.e., (σ > σ) > σ
+> σ). (Though this type will only be correct if we decide that
+the predecessor of zero should be a number, perhaps zero.)
+
+Rather, the problem is that the definition of the function requires
+subterms that can't be simplytyped. We'll illustrate with our
+implementation of the predecessor, sightly modified in inessential
+ways to suit present purposes:
+
+ let zero = \s z. z in
+ let snd = \a b. b in
+ let pair = \a b. \v. v a b in
+ let succ = \n s z. s (n s z) in
+ let collect = \p. p (\a b. pair (succ a) a)
+ let pred = \n. n collect (pair zero zero) snd in
+
+Let's see how far we can get typing these terms. `zero` is the Church
+encoding of zero. Using `N` as the type for Church numbers (i.e.,
+N ≡ (σ > σ) > σ > σ
for some
+σ, `zero` has type `N`. `snd` takes two numbers, and returns
+the second, so `snd` has type `N > N > N`. Then the type of `pair`
+is `N > N > (type(snd)) > N`, that is, `N > N > (N > N > N) >
+N`. Likewise, `succ` has type `N > N`, and `collect` has type `pair
+> pair`, where `pair` is the type of an ordered pair of numbers,
+namely, pair ≡ (N > N > N) > N
. So far so good.
+
+The problem is the way in which `pred` puts these parts together. In
+particular, `pred` applies its argument, the number `n`, to the
+`collect` function. Since `n` is a number, its type is (σ
+> σ) > σ > σ
. This means that the type of
+`collect` has to match σ > σ
. But we
+concluded above that the type of `collect` also had to be `pair >
+pair`. Putting these constraints together, it appears that
+σ
must be the type of a pair of numbers. But we
+already decided that the type of a pair of numbers is `(N > N > N)
+> N`. Here's the difficulty: `N` is shorthand for a type involving
+σ
. If σ
turns out to depend on
+`N`, and `N` depends in turn on σ
, then
+σ
is a proper subtype of itself, which is not
+allowed in the simplytyped lambda calculus.
+
+The way we got here is that the `pred` function relies on the builtin
+rightfold structure of the Church numbers to recursively walk down
+the spine of its argument. In order to do that, the argument had to
+apply to the `collect` operation. And since `collect` had to be the
+sort of operation that manipulates numbers, the infinite regress is
+established.
+
+Now, of course, this is only one of myriad possible implementations of
+the predecessor function in the lambda calculus. Could one of them
+possibly be simplytypeable? It turns out that this can't be done.
+See the works cited by Oleg for details.
+
+Because lists are (in effect) a generalization of the Church numbers,
+computing the tail of a list is likewise beyond the reach of the
+simplytyped lambda calculus.
+
+This result is surprising. It illustrates how recursion is built into
+the structure of the Church numbers (and lists). Most importantly for
+the discussion of the simplytyped lambda calculus, it demonstrates
+that even fairly basic recursive computations are beyond the reach of
+a simplytyped system.
+
+
+## Montague grammar is based on a simplytyped lambda calculus
+
+Systems based on the simplytyped lambda calculus are the bread and
+butter of current linguistic semantic analysis. One of the most
+influential modern semantic formalismsMontague's PTQ
+fragmentincluded a simplytyped version of the Predicate Calculus
+with lambda abstraction.
+
+Montague called the semantic part of his PTQ fragment *Intensional
+Logic*. Without getting too fussy about details, we'll present the
+popular Ty2 version of the PTQ types, roughly as proposed by Gallin
+(1975). [See Zimmermann, Ede. 1989. Intensional logic and twosorted
+type theory. *Journal of Symbolic Logic* ***54.1***: 6577 for a
+precise characterization of the correspondence between IL and
+twosorted Ty2.]
+
+We'll need three base types: `e`, for individuals, `t`, for truth
+values, and `s` for evaluation indicies (worldtime pairs). The set
+of types is defined recursively:
+
+ the base types e, t, and s are types
+ if a and b are types, is a type
+
+So `>` and `,t>>` are types. As we have mentioned,
+this paper is the source for the convention in linguistics that a type
+of the form `` corresponds to a functional type that we will
+write here as `a > b`. So the type `` is the type of a function
+that maps objects of type `a` onto objects of type `b`.
+
+Montague gave rules for the types of various logical formulas. Of
+particular interest here, he gave the following typing rules for
+functional application and for lambda abstracts:
+
+* If *α* is an expression of type **, and *β* is an
+expression of type b, then *α(β)* has type *b*.
+
+* If *α* is an expression of type *a*, and *u* is a variable of type *b*, then *λuα* has type
.
+
+When we talk about monads, we will consider Montague's treatment of
+intensionality in some detail. In the meantime, Montague's PTQ is
+responsible for making the simplytyped lambda calculus the baseline
+semantic analysis for linguistics.