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=cee3f916d991985d7b7d07aaec537a0fd137086d;hb=a4d2693effe839524592f4427465ff8d97625302;hpb=5e9e42542f2f56e0236af240519eda4cdde09a52
diff git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index cee3f916..047ee8b3 100644
 a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ 81,16 +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.
@@ 102,28 +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#
@@ 131,7 +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
@@ 150,9 +150,9 @@ cannot have a type in Λ_T. We can easily see why:
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.
@@ 174,47 +174,137 @@ To see this, consider the first three Church numerals (starting with zero):
\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 > $sigma; for
+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 other 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 >
+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 examinine the definition of two, we
+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. The most general
+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.

+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]].
The reason is that ...

Need to digest the following, which is quoted from Oleg's page:


The predecessor of a Churchencoded numeral, or, generally, the encoding of a list with the car and cdr operations are both impossible in the simply typed lambdacalculus. Henk Barendregt's ``The impact of the lambdacalculus in logic and computer science'' (The Bulletin of Symbolic Logic, v3, N2, June 1997) has the following phrase, on p. 186:

Even for a function as simple as the predecessor lambda definability remained an open problem for a while. From our present knowledge it is tempting to explain this as follows. Although the lambda calculus was conceived as an untyped theory, typeable terms are more intuitive. Now the functions addition and multiplication are defineable by typeable terms, while [101] and [108] have characterized the lambdadefineable functions in the (simply) typed lambda calculus and the predecessor is not among them [the story of the removal of Kleene's four wisdom teeth is skipped...]
Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 7381.

Since list is a generalization of numeral  with cons being a successor, append being the addition, tail (aka cdr) being the predecessor  it follows then the list cannot be encoded in the simply typed lambdacalculus.

To encode both operations, we need either inductive (generally, recursive) types, or System F with its polymorphism. The first approach is the most common. Indeed, the familiar definition of a list

 data List a = Nil  Cons a (List a)

gives an (iso) recursive data type (in Haskell. In ML, it is an inductive data type).

Lists can also be represented in System F. As a matter of fact, we do not need the full System F (where the type reconstruction is not decidable). We merely need the extension of the HindleyMilner system with higherranked types, which requires a modicum of type annotations and yet is able to infer the types of all other terms. This extension is supported in Haskell and OCaml. With such an extension, we can represent a list by its fold, as shown in the code below. It is less known that this representation is faithful: we can implement all list operations, including tail, drop, and even zip.


+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.