edits

diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index 4caeb55..1a0425f 100644 (file)
--- a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ -26,7 +26,7 @@ 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,
 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
+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. 
 
 in English.  Likewise, types in formal languages will determine which
 expressions can be sensibly combined. 
 


@@ -41,7 +41,7 @@ phrases, since they both denote properties with (extensional) type
 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
 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
+type without any effect on the expressions they can combine

 with syntactically).  We will consider again the relationship between
 syntactic types and semantic types later in the course.
 
 with syntactically).  We will consider again the relationship between
 syntactic types and semantic types later in the course.
 


@@ -61,7 +61,7 @@ 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
 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.]
+native setting--but we're getting ahead of our 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.
 
 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.


@@ -125,6 +125,26 @@ Excercise: write down terms that have the following types:
                    (o -> o) -> o -> o
                    (o -> o -> o) -> o
 
                    (o -> o) -> o -> o
                    (o -> o -> o) -> o
 

+#A first glipse of the connection between types and logic
+
+In the simply-typed lambda calculus, we write types like <code>&sigma;
+-> &tau;</code>.  This looks like logical implication.  We'll take
+that resemblance seriously when we discuss the Curry-Howard
+correspondence.  In the meantime, note that types respect modus
+ponens: 
+
+<pre>
+Expression    Type      Implication
+-----------------------------------
+fn            &alpha; -> &beta;    &alpha; &sup; &beta;
+arg           &alpha;         &alpha;
+------        ------    --------
+(fn arg)      &beta;         &beta;
+</pre>
+
+The implication in the right-hand column is modus ponens, of course.
+
+

 #Associativity of types versus terms#
 
 As we have seen many times, in the lambda calculus, function
 #Associativity of types versus terms#
 
 As we have seen many times, in the lambda calculus, function


@@ -156,13 +176,16 @@ something of type &tau;, `\x.xx` must also have type &sigma; ->
 finite types, there is no way to choose a type for the variable `x`
 that can satisfy all of the requirements imposed on it.
 
 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.
+In fact, we can't even type the parts of &Omega;, that is, `&omega;
+\equiv \x.xx`.  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#
 
 
 #Typing numerals#
 


@@ -196,9 +219,7 @@ principle type-->
 
 ## Predecessor and lists are not representable in simply typed lambda-calculus ##
 
 
 ## 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;
 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;


@@ -212,24 +233,27 @@ implementation of the predecessor function, based on the discussion in
 Pierce 2002:547:
 
     let zero = \s z. z 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 fst = \x y. x in
+    let snd = \x y. y in
+    let pair = \x y . \f . f x y in

     let succ = \n s z. s (n s z) in
     let succ = \n s z. s (n s z) in

-    let shift = \p. p (\a b. pair (succ a) a)
+    let shift = \p. pair (succ (p fst)) (p fst) in

     let pred = \n. n shift (pair zero zero) snd in
 
     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:
+Note that `shift` takes a pair `p` as argument, but makes use of only
+the first element of the pair.  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)
+    pred 3
+    3 shift (pair zero zero) snd
+    (\s z.s(s(s z))) shift (pair zero zero) snd

     shift (shift (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 (pair (succ ((\f.f 0 0) fst)) ((\f.f 0 0) fst))) snd

     shift (shift (\f.f 1 0)) snd
     shift (\f. f 2 1) snd
     (\f. f 3 2) snd
     shift (shift (\f.f 1 0)) snd
     shift (\f. f 2 1) snd
     (\f. f 3 2) snd

+    snd 3 2

     2
 
 At each stage, `shift` sees an ordered pair that contains two numbers
     2
 
 At each stage, `shift` sees an ordered pair that contains two numbers


@@ -276,7 +300,10 @@ 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.
 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.
+See Oleg Kiselyov's discussion and works cited there for details:
+[[predecessor and lists can't be represented in the simply-typed
+lambda
+calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].

 
 Because lists are (in effect) a generalization of the Church numbers,
 computing the tail of a list is likewise beyond the reach of the
 
 Because lists are (in effect) a generalization of the Church numbers,
 computing the tail of a list is likewise beyond the reach of the


@@ -313,14 +340,15 @@ of types is defined recursively:
     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,
     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's 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
 
 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:
+functional application and for lambda abstracts, which match the rules
+for the simply-typed lambda calculus exactly:

 
 * 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, b>*, and *&beta;* is an
 expression of type b, then *&alpha;(&beta;)* has type *b*.  

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index a7b4bb9..4bde11e 100644 (file)
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -1,6 +1,13 @@
 [[!toc levels=2]]
 
 [[!toc levels=2]]
 

-# System F and recursive types
+# System F: the polymorphic lambda calculus
+
+The simply-typed lambda calculus is beautifully simple, but it can't
+even express the predecessor function, let alone full recursion.  And
+we'll see shortly that there is good reason to be unsatisfied with the
+simply-typed lambda calculus as a way of expressing natural language
+meaning.  So we will need to get more sophisticated about types.  The
+next step in that journey will be to consider System F.

 
 In the simply-typed lambda calculus, we write types like <code>&sigma;
 -> &tau;</code>.  This looks like logical implication.  We'll take
 
 In the simply-typed lambda calculus, we write types like <code>&sigma;
 -> &tau;</code>.  This looks like logical implication.  We'll take