Re: Invertible nondeterminism?

From: Wolfgang Lux <>
Date: Fri, 17 Dec 2010 08:25:56 +0100

Sebastian Fischer wrote:

> On Thu, 2010-12-16 at 20:22 +0900, Sebastian Fischer wrote:
>> It is an equivalence class of pairs of (multi-)sets for the
>> equivalence relation
>> (A,X) ~ (B,Y) := A \union Y = X \union B
> This was incorrect. The correct definition is (writing u for union)
> (A,X) ~ (B,Y) := exists S : A u Y u S = X u B u S

Maybe I'm a bit dense, but using multisets this definition looks
equivalent to the previous one. On the other hand, using sets this
definition is quite useless, since the right hand side can always be
satisfied with S = D, where D is the underlying domain and thus (A,X)
is equivalent to any other value.

> As a consequence, if we use sets and not multisets, then every element
> is equivalent to failure. Writing E for empty set:
> (A,X) ~ (E,E) because for S = A u X : A u E u S = S = X u E u S

Yes, its equivalent to failure, but this is so because (A,X) is
equivalent to any other value as well (see above).


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