Complexity analysis of the SAT engine: DNA algorithms as probabilistic algorithms

Abstract

Taking advantage of the power of DNA molecules to spontaneously form hairpin structures, Sakamoto et al. designed a molecular algorithm to solve instances of the satisfiability problem on Boolean expressions in clausal form (the SAT problem), and by developing new experimental techniques for molecular biology, they succeeded in solving a 6-variable, 10-clause instance of the 3-SAT problem (Sakamoto et al., Science 288 (2000) 1223). Sakamoto et al. call this computational architecture the SAT Engine. In this paper, we analyze the complexity of the SAT Engine as a probabilistic algorithm. We first estimate the time dependence of the probability of hairpin formation using standard chemical kinetics and the Jacobson–Stockmayer expression. We then estimate the number of DNA molecules required to solve the satisfiability problem with a given error probability. By taking the number of DNA molecules into account, we finally estimate the minimum total time and number of strands, respectively, required to achieve combined error rates of < ε 1 (the probability of a false positive) and ε 2 (the probability of a false negative). If the number of clauses is n, then the time required for solving the problem is proportional to n 1.5( ln(1/ε 1)+ ln( ln(1/ε 2)))+n 2.5 ln(3+α) , and the number of necessary DNA molecules is proportional to (3+α) n ln(1/ε 2) with arbitrarily small α>0.