added discussion of Montague's PTQ

[lambda.git] / topics / _week5_simply_typed_lambda.mdwn

diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index a7917b2..4b1bde5 100644 (file)
--- a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ -220,7 +220,7 @@ ways to suit present purposes:
 
 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.,
 
 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 == (&sigma; -> &sigma;) -> &sigma; -> &sigma;</code> for some
+<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) ->
 &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) ->


@@ -230,24 +230,24 @@ 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
 
 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 `(&sigma; ->
-&sigma;) -> &sigma; -> &sigma;`.  This means that the type of
-`collect` has to match `&sigma; -> &sigma;`. But we concluded above
-that the type of `collect` also had to be `pair -> pair`.  Putting
-these constraints together, it appears that `&sigma;` 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 `&sigma;`.  If `&sigma;` turns out to
-depend on `N`, and `N` depends in turn on `&sigma;`, then `&sigma;` 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 right-fold
-structure of the Church numbers to recursively walk down the spine of
-its argument.  In order to do that, the argument number had to take
-the operation in question as its first argument.  And the operation
-required in order to build up the predecessor must be the sort of
-operation that manipulates numbers, and the infinite regress is
+`collect` function.  Since `n` is a number, its type is <code>(&sigma;
+-> &sigma;) -> &sigma; -> &sigma;</code>.  This means that the type of
+`collect` has to match <code>&sigma; -> &sigma;</code>. But we
+concluded above that the type of `collect` 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 `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
 established.
 
 Now, of course, this is only one of myriad possible implementations of


@@ -255,3 +255,49 @@ 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.
 
 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 surprising.  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 a simply-typed
+
+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---involved a simply-typed version of the Predicate Calculus
+with lambda abstraction.  More specifically, Montague called the
+semantic part of the PTQ fragment `Intensional Logic'.  Montague's IL
+had three base types: `e`, for individuals, `t`, for truth values, and
+`s` for evaluation indicies (world-time pairs).  The set of types was
+defined recursively:
+
+    e, t, s are types
+    if a and b are types, <a,b> is a type
+    if a is a type, <s,a> is a type
+
+So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types, but `<e,s>` is not a
+type.  As 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 `a -> 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*, 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>.
+
+In future discussions about monads, we will investigate Montague's
+treatment of intensionality in some detail.  In the meantime,
+Montague's PTQ fragment is responsible for making the simply-typed
+lambda calculus the baseline semantic analysis for linguistics.
+