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.

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.

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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