For the most important class of problem in computer science,
non-deterministic polynomial complete problems, non-deterministic UTMs (NUTMs)
are theoretically exponentially faster than both classical UTMs and quantum
mechanical UTMs (QUTMs). This design is based on Thue string rewriting systems,
and thereby avoids the limitations of most previous DNA computing schemes: all
the computation is local (simple edits to strings) so there is no need for
communication, and there is no need to order operations. The design exploits
DNA’s ability to replicate to execute an exponential number of computational
paths in P time. Each Thue rewriting step is embodied in a DNA edit implemented
using a novel combination of polymerase chain reactions and site-directed
mutagenesis. We demonstrate that the design works using both computational
modeling and in vitro molecular
biology experimentation: the design is thermodynamically favorable,
microprogramming can be used to encode arbitrary Thue rules, all classes of
Thue rule can be implemented and non-deterministic rule implementation. In an
NUTM, the resource limitation is space, which contrasts with classical UTMs and
QUTMs where it is time. This fundamental difference enables an NUTM to trade
space for time, which is significant for both theoretical computer science and
physics. It is also of practical importance, for to quote Richard Feynman
‘there’s plenty of room at the bottom’. This means that a desktop DNA NUTM
could potentially utilize more processors than all the electronic computers in
the world combined, and thereby outperform the world’s current fastest
supercomputer, while consuming a tiny fraction of its energy.

 

However, we acknowledge that further experimentation is
required to complete the physical construction of a fully working NUTM. Indeed,
we are unaware of any fully working molecular implementation of a UTM, far less
an NUTM. The key point about implementing a UTM compared with special purpose
hardware is that special purpose hardware typically needs to be redesigned for
each new problem. By contrast, in a UTM only the software needs to be changed
for a new problem, and the hardware stays fixed. The situation for molecular
UTMs is currently similar to that of QUTMs where hardware prototypes have
executed significant computation, but no full physical implementation of a QUTM
exists.

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The greatest challenge in developing a working NUTM is
control of ‘noise’. Noise was a serious problem in the early days of electronic
computers however; the problem has now essentially been solved. Noise is also
the most serious hindrance to the physical implementation of QUTMs, and may
actually make QUTMs physically impossible. By contrast, in an NUTM, well-understood
classical approaches can be employed to deal with noise. These classical
methods enable unreliable components to be combined together to form extremely
reliable overall systems.

The way in NUTM for noise reduction is that the use of
error-correcting codes. These codes are used ubiquitously in electronic computers,
and are also essential for QUTMs. Classical error-correcting code methods can
be directly ported to NUTMs. Another way is the
repetition of computations. The most basic way to reduce noise is
to repeat computations, either spatially or temporally. The use of a polynomial
number of repetitions does not affect the fundamental speed advantage of NUTMs
over classical UTMs or QUTMs.

Most
effort on non-standard computation has focused on developing QUTMs. Steady
progress is being made in theory and implementation, but no QUTM currently
exists. Although abstract QUTMs have not been proven to outperform classical
UTMs, they are thought to be faster for certain problems. The best evidence for
this is Shor’s integer factoring algorithm, which is exponentially faster than
the current best classical algorithm. While integer factoring is in NP, it is
not thought to be NP complete, and it is generally believed that the class of
problems solvable in P time by a QUTM (BQP) is not a superset of NP.

NUTMs
and QUTMs both utilize exponential parallelism, but their advantages and
disadvantages seem distinct. NUTMs utilize general parallelism, but this takes
up physical space. In a QUTM, the parallelism is restricted, but does not
occupy physical space (at least in our Universe). In principle therefore, it
would seem to be possible to engineer an NUTM capable of utilizing an
exponential number of QCs in P time.

Advocates of the many-worlds interpretation of quantum
mechanics argue that QUTMs work through exploitation of the hypothesized
parallel universes. Intriguingly, if the multiverse were an NUTM this would
explain the profligacy of worlds.

In an NUTM, the resource limitation is space,
which contrasts with classical UTMs and QUTMs where it is time. This
fundamental difference enables an NUTM to trade space for time, which is
significant for both theoretical computer science and physics.

NUTM m 1: n relation possible h but QUTM m 1:1 hta h

NUTMs are much faster than QUTMs in terms of speeds

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