Specification of Reactive Hardware/Software Systems - Electronic ...

432 Proofs of Propositions and Transformations
Encoding
of
Turing Machine
Halt if
loops
Then the following two cases apply:
Loop if
(1) Magic does not eventually halt on ¡ (Magic). Then NoComChange(¡ (Magic)) is true
and SSpecA(¡ (Magic)) is transformation equivalent with SSpecB(¡ (Magic)). But
then there exists a proof of equivalence using complete proof system and thus
Magic must eventually terminate. This is a contradiction.
(2) Magic does eventually halt on ¡ (Magic). Then NoComChange(¡ (Magic)) is false.
Furthermore, it is not hard to verify that SSpecA(¡ (Magic)) and SSpecB(¡ (Magic))
are in that case not transformation equivalent. Hence, since is sound, there
exists no equivalence proof and thus Magic does not terminate. Again we have a
contradiction.
In any case we end up with a contradiction. Clearly the Magic machine cannot exist.
Hence we must conclude that our initial assumption about the existence of was wrong.
There does not exist a proof system for transformation equivalence that is complete with
respect to all transformations that are allowed by transformation 6. This concludes
the proof of Proposition 7. ¨
Proposition 8
Let SSpec be a POOSL specification. Then there exists no sound proof system for transformation
equivalence that is complete with respect to all specifications that are transformation equivalent
to SSpec. ¨
BSpec¥ ¥ Sys
Proof of Proposition 8
Let SSpec
p Sys be a POOSL specification and assume that is a sound
proof system for ¢
¡ that is complete with respect to all specification that are transfor-
mation equivalent to SSpec. We have to derive a contradiction. To this end let A(¡ (TM))
be a process that behaves as follows 4 . It starts emulating Turing Machine TM on input
¡ (TM). If TM eventually halts on its own input ¡ (TM) then process A(¡ (TM)) sends a
message m on a channel ch. Here it is assumed that neither m nor ch is used in behaviour
specification BSpec.
4 See also the proof of Proposition 7.
halts