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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 71856f6..4caeb55 100644 (file)
--- a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ -208,32 +208,55 @@ 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 &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 `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 <code>(&sigma;
+`shift` function.  Since `n` is a number, its type is <code>(&sigma;

 -> &sigma;) -> &sigma; -> &sigma;</code>.  This means that the type of
 -> &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 ->
+`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)
 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)


@@ -246,7 +269,7 @@ 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
 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
+apply to the `shift` operation.  And since `shift` had to be the

 sort of operation that manipulates numbers, the infinite regress is
 established.
 
 sort of operation that manipulates numbers, the infinite regress is
 established.
 


@@ -259,11 +282,11 @@ 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.
 
 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.
+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
 
 
 ## Montague grammar is based on a simply-typed lambda calculus


@@ -272,33 +295,39 @@ 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
 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.  
+with lambda abstraction.

 
 Montague called the semantic part of his PTQ fragment *Intensional
 
 Montague called the semantic part of his 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 base types
+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
     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 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 `<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
 
 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>.
+expression of type b, then *&alpha;(&beta;)* has type *b*.  

 
 

-When we talk 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.
+* 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.