Re: Avoiding variable clashes

[Hans Aberg]

On 13 Apr 2011, at 18:25, Andy Wingo wrote:

>>> Sorry, I don't know what you mean.  References?
>> 
>> There is an article here:
>> http://en.wikipedia.org/wiki/Variable_binding_operator
> 
> I still don't understand.  What are you trying to do?

The beta rule is in denotational semantics something like
 ((lambda x . E_1) E_2) => [E_2/x]E_1, E_2 free for x in in E_1
where [E_2/x]E_1 means substituting all free occurrences of x with E_2.

In addition, one has the alpha rule
 (lambda x . E) => (lambda y . [y/x]E), y free for x in E

Now, if one would want to implement say a quantifier 'forall', one would still 
need the alpha rule, and substitution, but instead of the beta rule have
 (forall x . E) => [t/x]E, t free for x in E
where t is any term. One would have to write additional stuff to manipulate it. 
But the alpha rule is tricky to implement.

The generalization would be to admit defining operators b satisfying the alpha 
rule
 (b x . E) => (b y . [y/x]E), y free for x in E
where one can add additional evaluation rules.

Hans