Digital Arithmetic in Nature: Continuous-Digit RNS

Abstract

It has been reported in the literature on computational neuroscience that a rat's uncanny ability to dash back to a home position in the absence of any visual clues (or in total darkness, for that matter) may stem from its distinctive method of position representation. More specifically, it is hypothesized that the rat uses a multimodular method akin to residue number system (RNS), but with continuous residues or digits, to encode position information. After a brief review of the evidence in support of this hypothesis, and how it relates to RNS, we discuss the properties of continuous-digit RNS, and derive results on the dynamic range, representational accuracy and factors affecting the choice of the moduli, which are themselves real numbers. We conclude with suggestions for further research on important open problems concerning the process of selection, or evolutionary refinement, of the set of moduli in such a representation.

Towards Robustness in Residue Number Systems

“…It is shown in [34] that the remainder error bound may be above the quarter of the gcd of all the moduli. By relaxing the assumption that the dynamic range is fixed to the maximum, i.e., the lcm of all the moduli, another method of position representation on the remainder plane was proposed for solving the robust remaindering problem with only two moduli m 1 , m 2 and m 1 < m 2 in [36]. Different from the robust CRT, all the nonnegative integers less than the dynamic range are connected by the slanted lines with the slope of 1 on the two dimensional remainder plane and a robust reconstruction is obtained by finding the closest point to the erroneous remainders on one of the slanted lines in [36].…”

“…By relaxing the assumption that the dynamic range is fixed to the maximum, i.e., the lcm of all the moduli, another method of position representation on the remainder plane was proposed for solving the robust remaindering problem with only two moduli m 1 , m 2 and m 1 < m 2 in [36]. Different from the robust CRT, all the nonnegative integers less than the dynamic range are connected by the slanted lines with the slope of 1 on the two dimensional remainder plane and a robust reconstruction is obtained by finding the closest point to the erroneous remainders on one of the slanted lines in [36]. As the dynamic range increases, the number of the slanted lines increases, and thereby, the distance between the slanted lines decreases, that is, the remainder error bound becomes small.…”

“…As the dynamic range increases, the number of the slanted lines increases, and thereby, the distance between the slanted lines decreases, that is, the remainder error bound becomes small. In [36], an exact dynamic range was first presented, provided that the remainder error bound is smaller than a quarter of the remainder of m 2 modulo m 1 (see Proposition 3 in Section II). When the remainder of m 2 modulo m 1 does not equal the gcd of m 1 and m 2 , an extension with a smaller remainder error bound and a larger dynamic range was further obtained, and as the dynamic range increases to the lcm of the moduli, the remainder error bound will decrease to the quarter of the gcd of the moduli (see Proposition 4 in Section II).…”

“…When the remainder of m 2 modulo m 1 does not equal the gcd of m 1 and m 2 , an extension with a smaller remainder error bound and a larger dynamic range was further obtained, and as the dynamic range increases to the lcm of the moduli, the remainder error bound will decrease to the quarter of the gcd of the moduli (see Proposition 4 in Section II). In [36], however, no closed-form reconstruction algorithms were proposed, and in the extension result, only lower and upper bounds of the dynamic range were provided, while the exact one was not derived or given. In some practical applications, considering that an unknown is real-valued in general, the robust remaindering problem and the above two different solutions were naturally generalized to real numbers in [31], [37], [38] and [36].…”

“…In [36], however, no closed-form reconstruction algorithms were proposed, and in the extension result, only lower and upper bounds of the dynamic range were provided, while the exact one was not derived or given. In some practical applications, considering that an unknown is real-valued in general, the robust remaindering problem and the above two different solutions were naturally generalized to real numbers in [31], [37], [38] and [36].…”