Computing Real Numbers using DNA Self-Assembly

TL;DR

This paper advances DNA self-assembly computational models to efficiently compute rational numbers, square roots, and an approximation of pi, enhancing the versatility of DNA-based computation.

Contribution

It introduces modifications to existing DNA tile division methods for rational numbers and proposes a new tile-based square root computation, enabling broader computational capabilities.

Findings

Rational numbers computed with O(1) and O(h) tile complexity.

Square root computed with O(n) tile complexity.

Approximation of pi using infinite series with DNA tiles.

Abstract

DNA Self-Assembly has emerged as an interdisciplinary field with many intriguing applications such DNA bio-sensor, DNA circuits, DNA storage, drug delivery etc. Tile assembly model of DNA has been studied for various computational primitives such as addition, subtraction, multiplication, and division. Xuncai et. al. gave computational DNA tiles to perform division of a number but the output had integer quotient. In this work, we simply modify their method of division to improve its compatibility with further computation and this modification has found its application in computing rational numbers, both recurring and terminating, with computational tile complexity of $\mathcal{O} (1)$ and $\mathcal{O} (h)$ respectively. Additionally, we also propose a method to compute square-root of a number with computational tile complexity of $\mathcal{O} (n)$ for an n bit number. Finally, after combining tiles of division and square-root, we propose a simple way to compute the ubiquitously used irrational number, $\pi$, using its infinite series.