Toehold DNA languages are regular: (Extended Abstract)

Abstract

We explore a method of designing algorithms using two types of DNA strands, namely rule strands (rules) and input strands. Rules are fixed in advance, and their task is to bind with the input strands in order to produce an output. We present algorithms for divisibility and primality testing as well as for square root computation. We measure the complexity of our algorithms in terms of the necessary rule strands. Our three algorithms utilize a super-constant amount of complex rules. Can one solve interesting problems using only few—or at least simple—rule strands? Our main result proves that restricting oneself to a constant number of rule strands is equivalent to deciding regular languages. More precisely, we show that an algorithm (possibly using infinitely many rule strands of arbitrary length) can merely decide regular languages if the structure of the rules themselves is simple, i.e., if the rule strands constitute a regular language.