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
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