13th International Conference on Membrane Computing - MTA Sztaki

<strong>13th</strong> <strong>International</strong> <strong>Conference</strong> on Membrane Computing, CMC13,
Budapest, Hungary, August 28 - 31, 2012. Proceedings, pages 63 - 65.
Membrane Systems and Hypercomputation
Mike Stannett ⋆
Department of Computer Science, University of Sheffield,
Regent Court, 211 Portobello, Sheffield S1 4DP, UK
Extended Abstract
Throughout this paper, the term hypercomputation refers to systems whose
behaviour cannot be simulated by any Turing program. Many systems have
been proposed in the literature which appear to have hypercomputational power.
While these are mostly theoretical in nature, some of the more recent models
may arguably be implementable (by suitably advanced civilisations); we review
this evidence below, and explain why this is not incompatible with Turing’s
original (and compelling) analysis of what constitutes real-world computation.
A related topic – called fypercomputation in [Pău11, Pău12] – concerns systems
which operate exponentially faster than Turing machines, in the strong sense
that they allow NP-complete (and possibly harder) problems to be solved by
polynomial means. It should be noted that this is an extremely strong property,
which may well go beyond what is possible even using quantum computation. For
example, although Shor’s algorithm [Sho97] can famously solve the traditionally
hard problem of integer factorisation in polynomial time, it is not known whether
factorisation is itself an NP-complete problem. Indeed, it is suspected that no
problem in BQP (problems soluble with high probability in polynomial time
using a quantum computer) is NP-complete [NC00].
The simplest way to achieve (theoretical) hypercomputation is via accelerating
speed-up. If each instruction in a program can be executed in half the time
of its predecessor, even an infinite program can be executed in finite time, but
Thomson’s Lamp [Tho54] reminds us that we need to provide clear semantics
concerning the program’s output, since this may need to be defined as the limit
of successive approximations, and there is no a priori guarantee that any meaningful
limit exists. Accelerating P systems have been considered by Calude and
Păun [CP04], and can easily be used to solve Σ 1 problems, which are those
problems (including the Halting Problem) which can be expressed in the form
∃x.R(x), where R is a recursive decision procedure (we place a no token in the
output compartment, and then dovetail successive instances of R(n). If any R(n)
returns true, activate a rule that replaces the no token with yes. After two seconds
(say) the entire procedure has run to completion, so we can simply check
the output container to see whether the token it contains is yes or no).
⋆ The author is partially supported under the Royal Society <strong>International</strong> Exchanges
Scheme (ref. IE110369). This work was partially undertaken whilst the author was
a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences in the
programme Semantics & Syntax: A Legacy of Alan Turing.
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