Improved DNA-sticker arithmetic: tube-encoded-carry, Logarithmic Number System and Monte-Carlo methods

Abstract

The sticker model of computation, implemented using robotic processing of DNA, manipulates in parallel many bitstrings, called strands, that are contained in a limited number of tubes. Prior sticker arithmetic algorithms, patterned on digital-electronics, generate carry bits in the strand, either wasting bits or using a clear operation (with problematic biochemical implementation). The novel addition algorithm here does not need to record the carry. Instead, which tube holds a particular strand implicitly encodes the carry. The speed and number of tubes are half that of prior approaches. Further encoding data in the Logarithmic Number System (LNS) allows such integer operations to perform cost-effective real multiplications, divisions and roots. An example LNS Euclidian norm is more efficient than prior methods, assuming perfect operations. Unfortunately, DNA-stickers are unreliable. This paper uses sticker unreliability as a source of randomness to implement Monte-Carlo (MC) arithmetic (previously fabricated in silicon at the cost of pseudo-random generators). With stickers, the randomness is free. MC engineering mimics natural systems using unreliable but redundant components. Here, MC randomness is only useful in low-order bits. Multiple re-testing of the same bit (“refinement”) trades improved reliability for slower operation using more tubes. Simulations (with different sizes, probabilities and refinement) show that increasing refinement as a function of bit position allows imperfect implementations to achieve suitable MC strand-error distributions, predicting 1000x speed-mass advantage of sticker-MCLNS over conventional supercomputers.