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[lambda.git] / topics / _week5_system_F.mdwn

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@@ -1,3 +1,5 @@
+[[!toc]]
+

 # System F and recursive types
 
 In the simply-typed lambda calculus, we write types like <code>&sigma;
 # System F and recursive types
 
 In the simply-typed lambda calculus, we write types like <code>&sigma;


@@ -24,9 +26,13 @@ Reynolds, who called his version the *polymorphic lambda calculus*.
 continuations.)  
 
 System F enhances the simply-typed lambda calculus with abstraction
 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
-notational convention that "<code>x:&alpha;</code>" represents an
-expression `x` whose type is <code>&alpha;</code>.
+over types.  Normal lambda abstraction abstracts (binds) an expression
+(a term); type abstraction abstracts (binds) a type.
+
+In order to state System F, we'll need to adopt the
+notational convention (which will last throughout the rest of the
+course) 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):


@@ -36,15 +42,19 @@ match up with usage in O'Caml, whose type system is based on System F):
        types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
        expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
 
        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`, 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 types, "`c`" is a type constant.  Type
+constants play the role in System F that base types play in the
+simply-typed lambda calculus.  So in a lingusitics context, type
+constants might include `e` and `t`.  "`'a`" is a type variable.  The
+tick mark just indicates that the variable ranges over types rather
+than over values; in various discussion below and later, type variable
+can be distinguished by using letters from the greek alphabet
+(&alpha;, &beta;, etc.), or by using capital roman letters (X, Y,
+etc.).  "`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
 
 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


@@ -63,17 +73,22 @@ 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
 <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 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:
+the <code>&lambda;</code> abstract.  Of course, as long as type
+variables are carefully distinguished from expression variables (by
+tick marks, Grecification, or capitalization), there is no need to
+distinguish expression abstraction from type abstraction by also
+changing the shape of the lambda.
+
+The expression immediately below is a polymorphic 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 this identity function to, say, a variable of type
+boolean, just do this:

 
 <code>(&Lambda; 'a (&lambda; x:'a . x)) [t]</code>    
 
 This type application (where `t` is a type constant for Boolean truth
 
 <code>(&Lambda; 'a (&lambda; x:'a . x)) [t]</code>    
 
 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
+values) specifies the value of the type variable `'a`.  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:
 


@@ -87,7 +102,7 @@ of type `e -> e`:
 
 Clearly, for any choice of a type `'a`, the identity function can be
 instantiated as a function from expresions of type `'a` to expressions
 
 Clearly, for any choice of a type `'a`, the identity function can be
 instantiated as a function from expresions of type `'a` to expressions

-of type `'a`.  In general, then, the type of the unapplied
+of type `'a`.  In general, then, the type of the uninstantiated

 (polymorphic) identity function is
 
 <code>(&Lambda; 'a (&lambda; x:'a . x)): (&forall; 'a . 'a -> 'a)</code>
 (polymorphic) identity function is
 
 <code>(&Lambda; 'a (&lambda; x:'a . x)): (&forall; 'a . 'a -> 'a)</code>


@@ -96,10 +111,93 @@ 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
+relevant evaluator is called "fullpoly"):
+
+    N = All X . (X->X)->X->X;
+    Pair = (N -> N -> N) -> N;
+    let zero = lambda X . lambda s:X->X . lambda z:X. z in 
+    let fst = lambda x:N . lambda y:N . x in
+    let snd = lambda x:N . lambda y:N . y in
+    let pair = lambda x:N . lambda y:N . lambda z:N->N->N . 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 . pair (suc (p fst)) (p fst) in
+    let pre = lambda n:N . n [Pair] shift (pair zero zero) 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 expressive power provided by System F is evident
+in the typing imposed by the definition of `pre`.  The variable `n` is
+typed as a Church number, i.e., as `All X . (X->X)->X->X`.  The type
+application `n [Pair]` instantiates `n` in a way that allows it to
+manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`.  In
+other words, the instantiation turns a Church number into a
+pair-manipulating function, which is the heart of the strategy for
+this version of predecessor.  
+
+Could we try to build a system for doing Church arithmetic in which
+the type for numbers always manipulated ordered pairs?  The problem is
+that the ordered pairs we need here are pairs of numbers.  If we tried
+to replace the type for Church numbers with a concrete (simple) type,
+we would have to replace each `X` with the type for Pairs, `(N -> N ->
+N) -> N`.  But then we'd have to replace each of these `N`'s with the
+type for Church numbers, `(X -> X) -> X -> X`.  And then we'd have to
+replace each of these `X`'s with... ad infinitum.  If we had to choose
+a concrete type built entirely from explicit base types, we'd be
+unable to proceed.
+ 

 [See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
 [See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT

-Press, pp. 350--353, for `tail` for lists in System F.]
+Press, chapter 23.]
+
+Typing &omega;
+--------------
+
+In fact, unlike in the simply-typed lambda calculus, 
+it is even possible to give a type for &omega; in System F. 
+
+<code>&omega; = lambda x:(All X. X->X) . x [All X . X->X] x</code>
+
+In order to see how this works, we'll apply &omega; to the identity
+function.  
+
+<code>&omega; id ==</code>
+
+    (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+
+Since the type of the identity function is `(All X . X->X)`, it's the
+right type to serve as the argument to &omega;.  The definition of
+&omega; instantiates the identity function by binding the type
+variable `X` to the universal type `All X . X->X`.  Instantiating the
+identity function in this way results in an identity function whose
+type is (in some sense, only accidentally) the same as the original
+fully polymorphic identity function.
+
+So in System F, unlike in the simply-typed lambda calculus, it *is*
+possible for a function to apply to itself!
+
+Does this mean that we can implement recursion in System F?  Not at
+all.  In fact, despite its differences with the simply-typed lambda
+calculus, one important property that System F shares with the
+simply-typed lambda calculus is that they are both strongly
+normalizing: *every* expression in either system reduces to a normal
+form in a finite number of steps.  
+
+Not only does a fixed-point combinator remain out of reach, we can't
+even construct an infinite loop.  This means that although we found a
+type for &omega;, there is no general type for &Omega; &equiv; &omega;
+&omega;.  Furthermore, it turns out that no Turing complete system can
+be strongly normalizing, from which it follows that System F is not
+Turing complete.

 
 
 Types in OCaml
 
 
 Types in OCaml