I'm asking in place of several my colleagues and myself of course. The question is almost off topic. It is from lambda calculus definition, in particular, definition of alpha reduction (and others as well).
Alpha reduction definition: a lambda expression (\v.e) can be transformed (reduced) to (\v'.e[v'/v]) if the substitution e[v'/v] is valid.
Beta reduction definition: a lambda expression (e1 e2) can be reduced to the expression e[e2/v] if e1 is of the form (\v.e) and if the substitution e[e2/v] is valid.
Eta reduction definition: a lambda expression e can be reduced to a lambda expression (\v.e v) if v is not free in e.
OK. If we have these two expressions: 1) (\x.x b x) 2) (\x.x c x)
The question is, are they equal? (They are not identical, of course.) For answer "no", there is a strong argument - there is no reduction sequence that can make these identical. On the other hand, their "meaning" expresses the same operation.
Well, what is the answer? I will be lucky with any link to WWW resource or your opinion. Nevertheless, the more formal and precise your answer will be the more I will be lucky. ;-)
If b and c are free, then no, they can't be considered equal, and i don't see how you can find a common "meaning" in this case either. Those two are equivalent: (\b.\x.x b x) = (\c.\x.x c x).
Yes, those of yours are equal of no doubt. Those of mine are not, that's even my opinion, on the other hand, I was not precise enough in my explanation. Those of mine have the same behavior unless you mean something else by variables b and c. Otherwise the behavior is the same, isn't it? If the behavior is the same, they can be interchanged and, thus, they are equal... OK, I agree this may be a more "philosophical" question. ;-) Thanks, Dusan