Computing Exponentially Faster: Implementing a Non-deterministic Universal Turing Machine Using DNA

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, and 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.

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