# Re: Invertible nondeterminism?

From: Sebastian Fischer <sebf_at_informatik.uni-kiel.de>
Date: Thu, 16 Dec 2010 22:59:53 +0900

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

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

So, for this construction to be useful, one should consider
multiplicities of results such that (a ? a) is different from a.

We can also show that every element equals failure if (?) is idempotent
using the laws:

a = a ? failure
= a ? (a ? anti a)
= (a ? a) ? anti a
= a ? anti a
= failure

Sebastian

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Received on Do Dez 16 2010 - 15:39:13 CET

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