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

Proofs of Propositions and Transformations 431
c c
A (p(TM)) B
a b
T 6
A (p(TM)) B
c
a b
Process A(¡ (TM)) behaves as follows. It starts emulating Turing Machine TM on input
¡ (TM) 1 . If TM eventually halts on its own input ¡ (TM), then process A(¡ (TM))
sends message m on channel a. Process B only tries to receive message m from
channel b. Now transformation condition NoComChange(¡ (TM)) is true if and only
if l¡ for (TM))¥ p Sys £ all ,
Sys l¥ ¡ A(¡ and ¥ £
B¥ Sysp £ ¥ (l¡ Sys £
implies c© a¥ c© l NoComChange(¡ , is, l¡ (TM)) is true if and
and f (l) f )
where f b. That
only if A(¡ (TM))¥ Sysp £ ¥ Sys a!m
£ and B¥ Sysp £ ¥ Sys b?m
£
¡
and c!m c?m implies
a!m b?m. Since a!m b?m, c!m c?m and B¥ Sysp £ ¥ Sys b?m
£
NoComChange(¡
, we thus have that
(TM)) is true if and only if not A(¡ (TM))¥ Sysp £ ¥ Sys a!m
£
A(¡ NoComChange(¡
¡
, i.e. if process
(TM)) does not send message m on channel a. Hence, (TM))
if and only if Turing Machine TM does not halt on its own input (TM).
true
Using proof system we will construct a Turing Machine Magic that behaves as follows.
When fed with an encoding of a Turing Machine ¡ (TM), it starts generating POOSL specifications
SSpecA(¡ (TM)) and SSpecB(¡ (TM)) 2 . Then it tries to prove the transformation
equivalence of SSpecA(¡ (TM)) and SSpecB(¡ (TM)) using proof system . This is done by
first examining all proofs of length 1, then all proofs of length 2, etcetera. Since only
has a finite amount of transformations, all possible proofs 3 will eventually be examined
this way. If SSpecA(¡ (TM)) and SSpecB(¡ (TM)) are transformation equivalent, Turing
Machine Magic will eventually terminate. Otherwise it will loop forever.
Now let us feed Turing Machine Magic with its own encoding ¡ (Magic).
1Process A can be considered a variant of the universal Turing Machine [LP81]. It can easily be
expressed in POOSL.
2This boils down to a very simple copy operation.
3Note that a proof is a finite sequence of successively applied transformations.
l
l ¢