# An introduction to abstract mathematical systems by David M. Burton

By David M. Burton

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An assembly system evolves as a continuous time Markov process. The rate for each rule application, and for each agitation step, is 1. e. -L. Chen et al.                                 Fig. 3. A nubots system that slowly grows a length n line in O(n) time, n monomer line = {ri | ri = (i, empty, null, x) → states, and using space n × 1. (a) Rule set: Rslow n (0, i − 1, rigid, x), where n > i > 0}. (b) Starting from an initial conﬁguration with a single monomer in state n, the system generates a length n line.

There is a set of nubots rules Nline , such that for any > 0, for suﬃciently large n ∈ N, starting from a line of log2 n + 1 monomers, each in state 0 or 1, Nline in the agitation nubots model assembles an n × 1 line in O(n1/3+ ) expected time, n × 5 space and O(1) monomer states. The proof is in Section 5. Lines and squares are examples of fundamental components for the self-assembly of arbitrary computable shapes and patterns in nubots [11,2,3] and other self-assembly models [5,8]. Our work here suggests that random agitations applied in an uncontrolled fashion throughout the grid are a powerful resource.

17], where Zavattaro and Cardelli showed that the following question is uncomputable: Will a given CRN with probability 1 reach a state where no further reactions are possible? Although their construction relied on repeated simulations of a Turing machine, it did not use the BorelCantelli Lemma, and could not be directly applied to computation with output. 2 Preliminaries Computability theory. a. language, decision problem) interchangeably to mean a subset L ⊆ Nk , or equivalently a function φ : Nk → {0, 1}, such that φ(x) = 1 ⇐⇒ x ∈ L.