More Tight Bounds for Active Self-Assembly Using an Insertion Primitive

We prove several limits on the behavior of a model of self-assembling particles introduced by Dabby and Chen (SODA 2013), called insertion systems, where monomers insert themselves into the middle of a growing linear polymer. First, we prove that the expressive power of these systems is equal to context-free grammars, answering a question posed by Dabby and Chen. Second, we give tight bounds on the maximum length and minimum expected time of constructed polymers in systems of three increasingly restricted classes. We prove that systems of $k$ monomer types can deterministically construct polymers of length $n = 2^{\Theta(k^{3/2})}$ in $O(\log^{5/3}(n))$ expected time. We also prove that if non-deterministic construction of a finite number of polymers is permitted, then the expected construction time can be reduced to $O(\log^{3/2}(n))$ at the trade-off of decreasing the length to $2^{\Theta(k)}$. If the system is allowed to construct an infinite number of polymers, then constructing polymers of unbounded length in $O(\log{n})$ expected time is possible. We follow these positive results with a set of lower bounds proving that these are the best possible polymer lengths and expected construction times.