Negative Interactions in Irreversible Self-assembly

Abstract

This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate an s-space-bounded, t-time-bounded Turing machine, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s⋅t) as required by the standard Turing machine simulation with tiles. In addition to the space-bounded Turing machine simulation, we show another example application of negative glues: reducing the number of tile types required to assemble "thin" (n × o(logn/loglogn)) rectangles.

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