On the logic structure of exact rational representation in fixed-radix number systems

The inner logic structure of Exact Rational Operative Representation in arbitrary Fixed-Radix Number Representation Systems is described and related to the outer Rational Symbolic Representation, for the first time. The presented approach operates on a discrete variable domain by offering an arbitrary precision closed solution, based on formal power series, modular arithmetic and discrete group theory, able to match the tough operative requirements of higher Biomedical Engineering computational environments. In fact, contemporary techniques from the core of numerical analysis that are relevant to Biomedical Engineering, Mathematical Methods for Learning, Knowledge Discovery, Data Mining, and science computational intensive applications still use static fixed finite precision floating point arithmetic, lacking numerical result robustness and reliability for critical mission applications. In order to avoid severe degradation of the overall required accuracy to the final global solution in finite computational resource systems, computational error propagation control requires exceptional care. In fact, floating point computation performed by standard computers is not exact, but rounded off to a close rational approximation usually. But unfortunately, even rational numbers cannot be represented exactly in traditional digital computer systems. The numerical result of a computation is in general close to the true result, but often not equal. If only a numerical result is sought, imprecision may usually be still acceptable for non critical mission applications; but, if a logical decision has to be taken based on this result, the decision will be right in most cases, but may be wrong (if the value is close to zero, its sign is not guaranteed to be correct). The consequences of taking a wrong decision can be fatal to any inference algorithm or to any decision maker. Over the years many correction techniques have been studied and developed, by either pure computational point of view or numeric representation point of view. In the latter case, the main approaches ranged from factorial-base representation to residue arithmetic, from p-adic base number to interval arithmetic. Unfortunately, they all still show practical limitations and do not solve the basic problem for reliable and effective numerical computation: the use of an efficient, lossless DYNamically reconfigurABLE (DYNABLE) precision number representation system for relative precision “error-free” computation. The presented techniques can offer an optimal solution to relative precision “error-free” computation and to computational load optimization and are amenable to parallel data manipulation and to rapid lossless data compression/decompression, in both software and hardware. Furthermore, their implementation on parallel structured multiprocessor architectures and inorganic 3D memories is quite straightforward and offers advantages at every level of the memory/interconnect hierarchy: less storage space is needed on memory and disk, less transmission time is needed on networks. A numerical example is presented: an optimal computational precision level can be dynamically selected according to the required operation accuracy, with relative precision exact error knowledge.