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##The simply-typed lambda calculus##

The untyped lambda calculus is pure.  Pure in many ways: nothing but
variables and lambdas, with no constants or other special symbols;
also, all functions without any types.  As we'll see eventually, pure
also in the sense of having no side effects, no mutation, just pure
computation.

But we live in an impure world.  It is much more common for practical
programming languages to be typed, either implicitly or explicitly.
Likewise, systems used to investigate philosophical or linguistic
issues are almost always typed.  Types will help us reason about our
computations.  They will also facilitate a connection between logic
and computation.

From a linguistic perspective, types are generalizations of (parts of)
programs.  To make this comment more concrete: types are to (e.g.,
lambda) terms as syntactic categories are to expressions of natural
language.  If so, if it makes sense to gather a class of expressions
together into a set of Nouns, or Verbs, it may also make sense to
gather classes of terms into a set labelled with some computational type.

To develop this analogy just a bit further, syntactic categories
determine which expressions can combine with which other expressions.
If a word is a member of the category of prepositions, it had better
not try to combine (merge) with an expression in the category of, say,
an auxilliary verb, since *under has* is not a well-formed constituent
in English.  Likewise, types in formal languages will determine which
expressions can be sensibly combined. 

Now, of course it is common linguistic practice to supply an analysis
of natural language both with syntactic categories and with semantic
types.  And there is a large degree of overlap between these type
systems.  However, there are mismatches in both directions: there are
syntactic distinctions that do not correspond to any salient semantic
difference (why can't adjectives behave syntactically like verb
phrases, since they both denote properties with (extensional) type
`<e,t>`?); and in some analyses there are semantic differences that do
not correspond to any salient syntactic distinctions (as in any
analysis that involves silent type-shifters, such as Herman Hendriks'
theory of quantifier scope, in which expressions change their semantic
type without any effect on the syntactic expressions they can combine
with syntactically).  We will consider again the relationship between
syntactic types and semantic types later in the course.

Soon we will consider polymorphic type systems.  First, however, we
will consider the simply-typed lambda calculus.  

[Pedantic on.  Why "*simply* typed"?  Well, the type system is
particularly simple.  As mentioned to us by Koji Mineshima, Church
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](church-simple-types.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
native setting--but I'm getting ahead of my story.  Pedantic off.]

There's good news and bad news: the good news is that the simply-typed
lambda calculus is strongly normalizing: every term has a normal form.
We shall see that self-application is outlawed, so &Omega; can't even
be written, let alone undergo reduction.  The bad news is that
fixed-point combinators are also forbidden, so recursion is neither
simple nor direct.

#Types#

We will have at least one ground type.  For the sake of linguistic
familiarity, we'll use `e`, the type of individuals, and `t`, the type
of truth values.

In addition, there will be a recursively-defined class of complex
types `T`, the smallest set such that

*    ground types, including `e` and `t`, are in `T`

*    for any types &sigma; and &tau; in `T`, the type &sigma; ->
     &tau; is in `T`.

For instance, here are some types in `T`:

     e
     e -> t
     e -> e -> t
     (e -> t) -> t
     (e -> t) -> e -> t

and so on.

#Typed lambda terms#

Given a set of types `T`, we define the set of typed lambda terms <code>&Lambda;_T</code>,
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 &sigma; -> &tau;, and a term `N` has type
     &sigma;, then the application `(M N)` has type &tau;.

*    If a variable `a` has type &sigma;, and term `M` has type &tau;, 
     then the abstract <code>&lambda; a M</code> has type &sigma; -> &tau;.

The definitions of types and of typed terms should be highly familiar
to semanticists, except that instead of writing &sigma; -> &tau;,
linguists write <&sigma;, &tau;>.  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) x)   o

Excercise: write down terms that have the following types:

                   o -> o -> o
                   (o -> o) -> o -> o
                   (o -> o -> o) -> o

#Associativity of types versus terms#

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
type of the complete term.

It is a serious faux pas to associate to the left for types.  You may
as well use your salad fork to stir your tea.

#The simply-typed lambda calculus is strongly normalizing#

If `M` is a term with type &tau; in &Lambda;_T, then `M` has a
normal form.  The proof is not particularly complex, but we will not
present it here; see Berendregt or Hankin.

Since &Omega; does not have a normal form, it follows that &Omega;
cannot have a type in &Lambda;_T.  We can easily see why:

<code>&Omega; = (\x.xx)(\x.xx)</code>

Assume &Omega; has type &tau;, and `\x.xx` has type &sigma;.  Then
because `\x.xx` takes an argument of type &sigma; and returns
something of type &tau;, `\x.xx` must also have type &sigma; ->
&tau;.  By repeating this reasoning, `\x.xx` must also have type
(&sigma; -> &tau;) -> &tau;; 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.

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.  Some of those types can be functions,
and some of those functions can be (type-restricted) identity
functions; but a simply-types identity function can never apply to itself.

#Typing numerals#

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 &rho; -> &sigma; -> &sigma; for
some &rho; and &sigma;.  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 &sigma; to some result type.  So we can refine
&rho; by replacing it with the more specific type &sigma; -> &tau;.
At this point, the overall type is (&sigma; -> &tau;) -> &sigma; ->
&sigma;.  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 &tau; must be suitable to serve as
arguments to functions of type &sigma; -> &tau;, 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 &tau; &equiv; &sigma;.  So at this
point, we have the overall type of (&sigma; -> &sigma;) -> &sigma;
-> &sigma;.

<!-- Make sure there is talk about unification and computation of the
principle type-->

## Predecessor and lists are not representable in simply typed lambda-calculus ##

As Oleg Kiselyov points out, [[predecessor and lists can't be
represented in the simply-typed lambda
calculus|http://okmij.org/ftp/Computation/lambda-calc.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 &tau; -> &tau;, where &tau;
is our type for Church numbers (i.e., (&sigma; -> &sigma;) -> &sigma;
-> &sigma;).  (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 simply-typed.  We'll illustrate with our
implementation of the predecessor function, based on the discussion in
Pierce 2002:547:

    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 shift = \p. p (\a b. pair (succ a) a)
    let pred = \n. n shift (pair zero zero) snd in

Note that `shift` applies its argument p ("p" for "pair") to a
function that ignores its second argument---why does it do that?  In
order to understand what this code is doing, it is helpful to go
through a sample computation, the predecessor of 3:

    pred (\s z.s(s(s z)))
    (\s z.s(s(s z))) (\n.n shift (\f.f 0 0) snd)
    shift (shift (shift (\f.f 0 0))) snd
    shift (shift ((\f.f 0 0) (\a b.pair(succ a) a))) snd
    shift (shift (\f.f 1 0)) snd
    shift (\f. f 2 1) snd
    (\f. f 3 2) snd
    2

At each stage, `shift` sees an ordered pair that contains two numbers
related by the successor function.  It can safely discard the second
element without losing any information.  The reason we carry around
the second element at all is that when it comes time to complete the
computation---that is, when we finally apply the top-level ordered
pair to `snd`---it's the second element of the pair that will serve as
the final result.

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.,
<code>N &equiv; (&sigma; -> &sigma;) -> &sigma; -> &sigma;</code> for
some &sigma;, `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 `shift` has type
`pair -> pair`, where `pair` is the type of an ordered pair of
numbers, namely, <code>pair &equiv; (N -> N -> N) -> N</code>.  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
`shift` function.  Since `n` is a number, its type is <code>(&sigma;
-> &sigma;) -> &sigma; -> &sigma;</code>.  This means that the type of
`shift` has to match <code>&sigma; -> &sigma;</code>. But we
concluded above that the type of `shift` also had to be `pair ->
pair`.  Putting these constraints together, it appears that
<code>&sigma;</code> 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
<code>&sigma;</code>.  If <code>&sigma;</code> turns out to depend on
`N`, and `N` depends in turn on <code>&sigma;</code>, then
<code>&sigma;</code> is a proper subtype of itself, which is not
allowed in the simply-typed lambda calculus.

The way we got here is that the `pred` function relies on the built-in
right-fold 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 `shift` operation.  And since `shift` 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 simply-typeable?  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
simply-typed lambda calculus.

This result is not obvious, to say the least.  It illustrates how
recursion is built into the structure of the Church numbers (and
lists).  Most importantly for the discussion of the simply-typed
lambda calculus, it demonstrates that even fairly basic recursive
computations are beyond the reach of a simply-typed system.


## Montague grammar is based on a simply-typed lambda calculus

Systems based on the simply-typed lambda calculus are the bread and
butter of current linguistic semantic analysis.  One of the most
influential modern semantic formalisms---Montague's PTQ
fragment---included a simply-typed 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 two-sorted
type theory.  *Journal of Symbolic Logic* ***54.1***: 65--77 for a
precise characterization of the correspondence between IL and
two-sorted Ty2.]

We'll need three base types: `e`, for individuals, `t`, for truth
values, and `s` for evaluation indicies (world-time pairs).  The set
of types is defined recursively:

    the base types e, t, and s are types
    if a and b are types, <a,b> is a type

So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types.  As we have mentioned,
this paper is the source for the convention in linguistics that a type
of the form `<a, b>` corresponds to a functional type that we will
write here as `a -> b`.  So the type `<a,b>` 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 *&alpha;* is an expression of type *<a, b>*, and *&beta;* is an
expression of type b, then *&alpha;(&beta;)* has type *b*.  

* If *&alpha;* is an expression of type *a*, and *u* is a variable of type *b*, then *&lambda;u&alpha;* has type <code><b, a></code>.

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 simply-typed lambda calculus the baseline
semantic analysis for linguistics.