pred in system F

diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index 047ee8b..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

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index fe451a0..cd1b617 100644 (file)
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -96,11 +96,53 @@ Pred in System F
 ----------------
 
 We saw that the predecessor function couldn't be expressed in the
 ----------------
 
 We saw that the predecessor function couldn't be expressed in the

-simply-typed lambda calculus.  It can be expressed in System F, however.
+simply-typed lambda calculus.  It can be expressed in System F,
+however.  Here is one way, coded in
+[[Benjamin Pierce's type-checker and evaluator for
+System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
+part you want is called "fullpoly"):
+
+    N = All X . (X->X)->X->X;
+    Pair = All X . (N -> N -> X) -> X;
+    let zero = lambda X . lambda s:X->X . lambda z:X. z in 
+    let snd = lambda x:N . lambda y:N . y in
+    let pair = lambda x:N . lambda y:N . lambda X . lambda z:N->N->X . z x y in
+    let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in
+    let shift = lambda p:Pair . p [Pair] (lambda a:N . lambda b:N . pair (suc a) a) in
+    let pre = lambda n:N . n [Pair] shift (pair zero zero) [N] snd in
+
+    pre (suc (suc (suc zero)));
+
+We've truncated the names of "suc(c)" and "pre(d)", since those are
+reserved words in Pierce's system.  Note that in this code, there is
+no typographic distinction between ordinary lambda and type-level
+lambda, though the difference is encoded in whether the variables are
+lower case (for ordinary lambda) or upper case (for type-level
+lambda).
+
+The key to the extra flexibility provided by System F is that we can
+instantiate the `pair` function to return a number, as in the
+definition of `pre`, or we can instantiate it to return an ordered
+pair, as in the definition of the `shift` function.  Because we don't
+have to choose a single type for all uses of the pair-building
+function, we aren't forced into a infinite regress of types.

 
 [See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
 Press, pp. 350--353, for `tail` for lists in System F.]
 
 
 [See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
 Press, pp. 350--353, for `tail` for lists in System F.]
 

+Typing ω
+--------------
+
+In fact, it is even possible to give a type for &omeage; in System F. 
+
+    omega = lambda x:(All X. X->X) . x [All X . X->X] x in
+    omega;
+
+Each time the internal application is performed, the type of the head
+is chosen anew.  And each time, we choose the same type as before, the
+type of a function that takes an argument of any type and returns a
+result of the same type...
+

 
 Types in OCaml
 --------------
 
 Types in OCaml
 --------------