We prove two 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 prove that systems of k monomer types can deterministically construct polymers of length n = 2 Θ(k 3/2 ) in O(log 5/3 (n)) expected time, and that this is optimal in both the number of monomer types and expected time.

We prove two limits on the behavior of a model of self-assembling particles introduced by Dabby and Chen (Proceedings of 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1526-1536, 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 prove that systems of k monomer types can deterministically construct polymers of length n = 2 Θ(k 3/2 ) in O(log 5/3 (n)) expected time, and that this is optimal in both the number of monomer types and expected time.

No abstract

No abstract

No abstract