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The Simulation Powers and Limitations of Hierarchical Self-Assembly Systems

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Machines, Computations, and Universality (MCU 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9288))

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Abstract

In this paper, we extend existing results about simulation and intrinsic universality in a model of tile-based self-assembly. Namely, we work within the 2-Handed Assembly Model (2HAM), which is a model of self-assembly in which assemblies are formed by square tiles that are allowed to combine, using glues along their edges, individually or as pairs of arbitrarily large assemblies in a hierarchical manner, and we explore the abilities of these systems to simulate each other when the simulating systems have a higher “temperature” parameter, which is a system wide threshold dictating how many glue bonds must be formed between two assemblies to allow them to combine. It has previously been shown that systems with lower temperatures cannot simulate arbitrary systems with higher temperatures, and also that systems at some higher temperatures can simulate those at particular lower temperatures, creating an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. These previous results relied on two different definitions of simulation, one (strong simulation) seemingly more restrictive than the other (standard simulation), but which have previously not been proven to be distinct. Here we prove distinctions between them by first fully characterizing the set of pairs of temperatures such that the high temperature systems are intrinsically universal for the lower temperature systems (i.e. one tile set at the higher temperature can simulate any at the lower) using strong simulation. This includes the first impossibility result for simulation downward in temperature. We then show that lower temperature systems which cannot be simulated by higher temperature systems using the strong definition, can in fact be simulated using the standard definition, proving the distinction between the types of simulation.

J. Hendricks, M.J. Patitz, and T.A. Rogers—Supported in part by National Science Foundation Grant CCF-1422152.

T.A. Rogers—This author’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079.

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Notes

  1. 1.

    with the convention that \(\infty = \infty + 1 = \infty - 1\).

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Authors and Affiliations

  1. Deptartment of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Jacob Hendricks, Matthew J. Patitz & Trent A. Rogers

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  1. Jacob Hendricks
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  2. Matthew J. Patitz
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  3. Trent A. Rogers
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Correspondence to Jacob Hendricks .

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Editors and Affiliations

  1. Université d'Orléans, Orléans, France

    Jerome Durand-Lose

  2. Eastern Mediterranean University, Famagusta, Cyprus

    Benedek Nagy

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Hendricks, J., Patitz, M.J., Rogers, T.A. (2015). The Simulation Powers and Limitations of Hierarchical Self-Assembly Systems. In: Durand-Lose, J., Nagy, B. (eds) Machines, Computations, and Universality. MCU 2015. Lecture Notes in Computer Science(), vol 9288. Springer, Cham. https://doi.org/10.1007/978-3-319-23111-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-23111-2_10

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