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[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..4caeb55 100644 (file)
--- a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ -208,46 +208,69 @@ 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
 
 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, sightly modified in inessential
-ways to suit present purposes:
+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 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 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.,
 
 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
-&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 `collect` 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.
+<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
 
 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
+`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
 established.
 
 Now, of course, this is only one of myriad possible implementations of


@@ -255,3 +278,56 @@ 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 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.