[om] semantics of n-ary xor?
Andreas Strotmann
Strotmann at rrz.uni-koeln.de
Wed Sep 17 13:07:27 CEST 2003
What exactly is the semantics of an n-ary xor?
I'm not kidding -- I really don't know. Let me explain why.
At first glance, an obvious definition is xor(a,b,c):=xor(a,xor(b,c))
-- i.e. n-ary xor is something like a parity predicate.
It struck me that for infinite index sets I, xor_{i\in I}(P(i)) is
always undefined and thus not very useful, which of course bugged me
since I was the one who brought up the topic of "big" versions of n-ary
operators, and I fell to wondering if the "obvious" semantics of n-ary
xor is really the correct one.
And I realized that textbooks tend to explain the meaning of binary xor
as "either...or...(but not both)" -- and that the n-ary version of that
phrase (either ... or... or... or...) does *not* mean parity -- it means
"only one of these choices". Thus, a "natural" (as opposed to
"obvious") semantics of n-ary xor is "true if exactly one of the
arguments is true, false otherwise" -- which very nicely generalizes to
a well-known "big-xor" operator, namely the "exists-uniquely"
quanitifier, which I suspect retains a well-defined semantics even in
arbitrary transfinite contexts.
Now I ask you: what exactly *is* the meaning of n-ary xor in MathML (or
OpenMath, for that matter)?
-- Andreas
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