13th International Conference on Membrane Computing - MTA Sztaki

M. Stannett
The question arises, whether accelerating speed-up of this kind is physically
feasible. Calude and Păun’s construction was based on the observation
that smaller is faster, but this analogy has obvious physical limitations (in any
event, it is not the volume of reagents that matters, but their concentration).
Nonetheless, a careful analysis of accelerating systems shows that certain cosmological
implementations may indeed be possible. Our system clearly contains
two agents: an observer A and a computing device B whose output is observed
by A. Write t for the spacetime trajectory followed by B, and p for the point
in spacetime at which A makes its observation. For the system to be implementable,
we require two things: t should be infinitely long from B’s point of
view (to allow all instances of R(n) to be executed in the case they all return
false); and it must be possible for a signal to be sent from any point on t to p. We
call p a Malament-Hogarth (MH) event (Fig. 1). Perhaps surprisingly, sensible
spacetimes containing MH-events (MH-spacetimes) can be defined, and their use
[EN02] provides perhaps the most hopeful approach to demonstrating the feasibility
of physical hypercomputation since they are known to be associated with
slowly rotating massive black holes of the kind considered to have been observed
experimentally at the centre of our own galaxy [G + 09]. It should nonetheless
be noted that the suitability of such black holes for computational purposes is
the subject of ongoing debate; it is as yet an open question whether other, less
controversial, examples of physically relevant MH-spacetimes can be identified.
A says "yes" if a note
arrived, "no" if not
A receives
the note
here, if one
was sent
p
If the program
halts, B sends a
note saying so
B has infinite
proper time available
A sends the program
to B, who runs it
Does the following
program ever halt?
Fig. 1. Using Malament-Hogarth spacetime to solve the halting problem
More complicated spacetime configurations (involving chains of MH-events)
can be exploited to solve problems at all levels of the arithmetic hierarchy
[Hog04], and this naturally raises the question whether accelerating P systems
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