Types and Programming Languages

Lecture 12

Simon Gay Department of Computing Science University of Glasgow 2006/07

Type Safety: Unique Use

In order to prove that every value is used exactly once, we need to define an alternative operational semantics which allows us to see values being “consumed”.

The idea is to explicitly represent every value as being stored in memory and accessed by a pointer. Then we can define reductions on Store , Term configurations so that every value is removed from the store when it is first used.

Then we will prove that when executing a well-typed term, we never get a dangling pointer, and that at the end of execution, there is nothing left in the store except the final value.

The system will look like what we would get in lambda calculus with references, if we put

ref

around every value.

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Linear Lambda Calculus: Syntax

The same as before, plus store locations (pointers) m,n,… v ::=

integer literal

| true | false |  x:T.e

| m (not top-level syntax) e ::= v | x | e + e | e == e | e & e | if e then e else e | ee 2006/07 T ::= int | bool | T  T Store S ::= m=v,… Types and Programming Languages Lecture 12 - Simon Gay 3

Linear Lambda Calculus: Semantics

First define removal of a location from the store: (S, m=v, S’) – m = S,S’ Now define reductions of the form S , e  S’ , e’ Evaluating a value creates a store location and returns it: S , true  S , false  S , v  S ,  x:T.e

 S+[m=true] , m S+[m=false] , m S+[m=v] , m S+[m=  x:T.e] , m v is a integer literal In each case, m is a fresh location.

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Linear Lambda Calculus: Semantics

Next we define reductions which consume values.

S , if m S ( m then t )  else true e  S  m , t S , if m S ( m then t )  else false e  S  m , e S ( m )  S u , m  S(n) n   v ( S w is  m  n the )  [ sum of p  w ], u and v p S , S ( m ) mn   S  x : T .

e  m , e [ n / x ] m,n different 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 5

Linear Lambda Calculus: Semantics

Finally we define reductions within expressions, as usual.

S , e S , e  f   S ' , e ' S ' , e '  f S , m S , e  e   S ' , e ' S ' , m  e ' Similarly for the other operators.

S , c S , if c then t else e   S ' , c ' S ' , if c' then t else e 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 6

Example

 , (  x:int. x+1)2 m =  x:int. x+1 , m 2 (m =  x:int. x+1 , n = 2) , m n n = 2 , n+1 (n = 2 , p = 1) , n+p q = 3 , q FINAL RESULT 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 7

Example

 , (  x:int. (x+1)+x)2 m =  x:int. (x+1)+x , m 2 (m =  x:int. (x+1)+x , n = 2) , m n n = 2 , (n+1)+n (n = 2 , p = 1) , (n+p)+n q = 3 , q+n STUCK: n is a “dangling pointer” 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 8

Exercise

We now have two different semantics for (essentially) the same language: lambda calculus. The original semantics is based on reductions of expressions. The new semantics uses a store (and destroys values after their first use).

Try to prove the following theorem, relating the semantics:

If

 , e  * S,m=v, m

semantics, then

e  * v

semantics.

in the linear lambda calculus in the standard lambda calculus

Why don’t we expect the converse to be true?

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Proving Type Safety: Unique Use

Just as in the case of lambda calculus with references, we need the idea of a

store typing

 , so that we can give a type to the expression m (store location).

The store typing must be treated in the same way as the environment  , so that in a typing judgement  |   e:T the  and  describe

exactly

the variables and locations used by e .

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Linear Lambda Calculus with Store Typings

   |  |  |     true : bool false : bool v : int if v is an integer literal x : T |   x : T (LS-Var)  | m:T  m : T (LS-Loc)  |    ,  e | : int  ,  '   e |   '  f : f int : int (LS-Plus) similarly &, ==   |   c  ,  : bool |  ,  '   if |  '  c e then : T e  else f | :  '  T f : T |  , x   : T |   x : T .

e  : e T : U  o U (LS-Abs) (LS-If)  |   e :  ,  T  o U |  ,  '   |  '  e ' : T ee ' : U (LS-App) 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 11

Well-Typed Stores

Just as we saw for references, we need the idea of a well-typed store. We write  S ::  and define it by the following rules:   ::  Empty  S  :: S,  1 m ,   2 v ::  |   1 2 ,  m : v T : T Next This is rather subtle. The store typing  describes the store locations that are available for use, i.e. not already used within other parts of the store.

Examples:

 m=2, n=true :: m:int, n:bool  m=2, n=  x:int. x+m :: n:int  int 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 12

Well-Typed Stores

We will need the following fact about well-typed stores.

Lemma:

If  S,m=v ::  ,m:T and then  S ::  ,  ’  |  ’  v : T This might seem trivial, but the effect is that we can use rule

Next

(previous slide) in reverse even when

m

location to be added. was not the last It can be proved (

exercise

) by induction on the derivation of  S,m=v ::  ,m:T . The base case is trivial and the inductive case breaks into two sub-cases, depending on whether or not

m

is the last location added. 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 13

Substitution Lemma

As usual we need a substitution lemma. Because of the way the operational semantics is defined, we only need to consider substituting a store location for a variable.

Lemma:

If  , x:T |   e:U then  |  ,n:T  e[n/x] : U

Proof (outline):

e cannot be a boolean or integer literal or a store location (why?) If e is a variable then it must be be  x (why?) and  (why?) so the desired conclusion is  and | n:T   must n:T which follows from rule

LS-Loc

.

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Substitution Lemma

Proof (continued):

The other cases use the induction hypothesis in a similar way to the Substitution Lemma for simply typed lambda calculus. The difference is that the substitution only goes into one part of the expression.

If e is t+u then we have  1 |   , x  : e T : | int  ,  2  '  e |   ' f  : f int : int where  1 ,  2   , x : T and we consider two cases, depending on whether x is used in e or in f .

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Type Preservation

Theorem:

If  |   e:T then there exists  ’ and  S ::  such that  and |  ’  S , e e’:T  and S’ , e’  S’ ::  ’ .

Proof:

By induction on the derivation of S , e  S’ , e’ .

1. S , true  S+[m=true] , m We have  |   true:bool so  =  (why?) and S=  (why?).

Therefore S’ = (m=true) .

Taking  ’ = m:bool as required. gives  |  ’  m:bool and  S’ ::  ’ The cases of the other values are similar. For  y:U.e

say that  =  and S=  because e may use locations.

we can’t 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 16

Type Preservation

2. S , if m then e else e’  S-m , e because S(m)=true .

We have  |  1  m  | : bool  1 ,  2  |  2  if m  then e : T e else  |  2 e ' : T  e ' : T therefore Taking  1  m : bool  '  (  1 ,  2 )  m   2 and we just need  S-m ::  ’ we have  |  ’  e:T which follows from the Lemma on slide 13. 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 17

Type Preservation

3. S , mn  S-m , e[n/x] because S(m)=  x:T.e

.

We have  | m : T   o U  | m : T m :  o T  U , n o : U T  | n :  mn : U T  n : T and  S :: m:T —o U, n:T By the lemma on slide 13,  S-m ::  ,n:T where  |    x:T.e : T  U This typing is justified by x:T |   e : U from which the Substitution Lemma gives  |  ,n:T  e[n/x] : U as required.

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Type Preservation

4. The remaining cases, such as S , e S , e  f   S ' , e ' S ' , e '  f S , m S , e  e   S ' , e ' S ' , m  e ' follow from straightforward uses of the induction hypothesis.

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Progress

Finally we can also prove

Progress Theorem

If  S ::  and  (for some S’) or e |   e:T then either is a store location.

S , e  S’ , e’ very easily, by checking that for every potentially reducing term (e.g. if m then t else e ), the typing and the store typing mean that one of the reduction rules applies.

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The Final Value

Combining Progress and Preservation, we see that reduction of a typed term in a typed store terminates with S , m and one of the following cases applies: 1. S 2. S is is m=true m=false 3. S is m=v for some integer literal v 4. S(m) =  x:T.e

and  S-m ::  and x:T |   e : U i.e. S contains just m and the locations referred to by e

Exercise:

work out the complete reduction sequence for (  x:int.

 y:int.x+y)3 2006/07 Types and Programming Languages Lecture 12 - Simon Gay 21