edits

[lambda.git] / topics / _week5_system_F.mdwn

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 2c37ae3..559135e 100644 (file)
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -25,49 +25,55 @@ continuations.)
 
 System F enhances the simply-typed lambda calculus with abstraction
 over types.  In order to state System F, we'll need to adopt the
 
 System F enhances the simply-typed lambda calculus with abstraction
 over types.  In order to state System F, we'll need to adopt the

-notational convention that "<code>x:&alpha;</code>" represents a
-expression whose type is <code>&alpha;</code>.
+notational convention that "<code>x:&alpha;</code>" represents an
+expression `x` whose type is <code>&alpha;</code>.

 
 Then System F can be specified as follows (choosing notation that will
 match up with usage in O'Caml, whose type system is based on System F):
 
 
 Then System F can be specified as follows (choosing notation that will
 match up with usage in O'Caml, whose type system is based on System F):
 

-        System F:
+       System F:
+       ---------

        types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
        expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
 
 In the definition of the types, "`c`" is a type constant (e.g., `e` or
        types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
        expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
 
 In the definition of the types, "`c`" is a type constant (e.g., `e` or

-`t`).  "`'a`" is a type variable (the tick mark just indicates that
-the variable ranges over types rather than values).  "`τ1 -> τ2`" is
-the type of a function from expressions of type `τ1` to expressions of
-type `τ2`.  And "`∀'a. τ`" is called a universal type, since it
-universally quantifies over the type variable `'a`.
-
-In the definition of the expressions, we have variables "`x`".
+`t`, or in arithmetic contexts, `N` or `Int`).  "`'a`" is a type
+variable (the tick mark just indicates that the variable ranges over
+types rather than over values).  "`τ1 -> τ2`" is the type of a
+function from expressions of type `τ1` to expressions of type `τ2`.
+And "`∀'a. τ`" is called a universal type, since it universally
+quantifies over the type variable `'a`.  (You can expect that in
+`∀'a. τ`, the type `τ` will usually have at least one free occurrence
+of `'a` somewhere inside of it.)
+
+In the definition of the expressions, we have variables "`x`" as usual.

 Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda
 calculus, except that they have their shrug variable annotated with a
 type.  Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
 Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda
 calculus, except that they have their shrug variable annotated with a
 type.  Applications "`e1 e2`" are just like in the simply-typed lambda calculus.

+

 In addition to variables, abstracts, and applications, we have two
 In addition to variables, abstracts, and applications, we have two

-additional ways of forming expressions: "`Λ'a. e`" is a type
-abstraction, and "`e [τ]`" is a type application.  The idea is that
-<code>&Lambda;</code> is a capital <code>&lambda;</code>.  Just like
-the lower-case <code>&lambda;</code>, <code>&Lambda;</code> binds
-variables in its body; unlike <code>&lambda;</code>,
-<code>&Lambda;</code> binds type variables.  So in the expression
+additional ways of forming expressions: "`Λ'a. e`" is called a *type
+abstraction*, and "`e [τ]`" is called a *type application*.  The idea
+is that <code>&Lambda;</code> is a capital <code>&lambda;</code>: just
+like the lower-case <code>&lambda;</code>, <code>&Lambda;</code> binds
+variables in its body, except that unlike <code>&lambda;</code>,
+<code>&Lambda;</code> binds type variables instead of expression
+variables.  So in the expression

 
 <code>&Lambda; 'a (&lambda; x:'a . x)</code>
 
 the <code>&Lambda;</code> binds the type variable `'a` that occurs in
 the <code>&lambda;</code> abstract.  This expression is a polymorphic
 
 <code>&Lambda; 'a (&lambda; x:'a . x)</code>
 
 the <code>&Lambda;</code> binds the type variable `'a` that occurs in
 the <code>&lambda;</code> abstract.  This expression is a polymorphic

-version of the identity function.  It says that this one general
-identity function can be adapted for use with expressions of any
-type. In order to get it ready to apply to, say, a variable of type
-boolean, just do this:
+version of the identity function.  It defines one general identity
+function that can be adapted for use with expressions of any type. In order
+to get it ready to apply to, say, a variable of type boolean, just do
+this:

 
 <code>(&Lambda; 'a (&lambda; x:'a . x)) [t]</code>    
 
 
 <code>(&Lambda; 'a (&lambda; x:'a . x)) [t]</code>    
 

-The type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `&alpha;`, which is
-the type of the variable bound in the `&lambda;` expression.  Not
+This type application (where `t` is a type constant for Boolean truth
+values) specifies the value of the type variable &alpha;, which is
+the type of the variable bound in the &lambda; expression.  Not

 surprisingly, the type of this type application is a function from
 Booleans to Booleans:
 
 surprisingly, the type of this type application is a function from
 Booleans to Booleans:
 


@@ -84,13 +90,16 @@ instantiated as a function from expresions of type `'a` to expressions
 of type `'a`.  In general, then, the type of the unapplied
 (polymorphic) identity function is
 
 of type `'a`.  In general, then, the type of the unapplied
 (polymorphic) identity function is
 

-<code>(&Lambda; 'a (&lambda; x:'a . x)): (\forall 'a . 'a -> 'a)
-
-
+<code>(&Lambda; 'a (&lambda; x:'a . x)): (&forall; 'a . 'a -> 'a)

 
 

+Pred in System F
+----------------

 
 

-## 
+We saw that the predecessor function couldn't be expressed in the
+simply-typed lambda calculus.  It can be expressed in System F, however.

 
 

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

 
 
 Types in OCaml
 
 
 Types in OCaml