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The Need for Seed (in the Abstract Tile Assembly Model)

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Abstract

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the “seed” assembly, which is a pre-formed assembly from which all growth propagates. Arbitrary amounts of information can be encoded into seed assemblies by both (1) the types and patterns of glues exposed on their exteriors, and (2) their shapes. Since a common metric by which aTAM systems are measured is their tile complexity (i.e. the number of unique types of tiles they utilize), in order to provide a fair basis for comparison, systems are often designed with seed assemblies consisting of only a single seed tile, a.k.a. single-tile seeds. (For instance, in STOC 2000 and 2001 information theoretically optimal tile complexity was shown possible for the self-assembly of squares.) This requires the transferring of any information that may be encoded in a multi-tile seed assembly into tile complexity. In this paper, we explore this process to show when and how such transformations are possible while ensuring that a derived system with a single-tile seed faithfully replicates the behaviors of the original system. We first show that a trivial transformation, in which the locations of a multi-tile seed are tiled by “hard-coded” tiles that can grow to represent that seed from a single tile, can succeed only if (1) there are not tile locations in the seed such that there exist growth sequences where those locations could block future growth, or (2) an ordering of growth can be enforced for the growth of the seed from a single tile to ensure that such blocking locations are tiled before collisions are possible. However, we show that knowing if this is the case is uncomputable. Therefore, we examine what is possible if the scale factor of the original system is increased and show that all systems with multi-tile seeds can be transformed into systems with single-tile seeds at scale factor 3 (i.e. each tile of the original system is replaced by a \(3 \times 3\) square of tiles), such that the transformed systems faithfully replicate the dynamics of the original systems. We also prove that this scale factor is optimal, and that in fact there exist systems with multi-tile seeds for which no systems at scale factors 1 or 2 (or scale factor 3 when a more restrictive form of simulation is required) with single-tile seeds exist that can even produce the same sets of terminal output shapes. Since the scale 3 transformation results in a tile complexity which is proportional to the size of the original tile set plus the size of the multi-tile seed multiplied by the scale factor, we then also provide a transformation that yields an asymptotically optimal tile complexity proportional to the Kolmogorov complexity of the original system and which is based on the IU construction from FOCS 2012. Additionally, we are able to make simple modifications to that construction to provide a single aTAM system which simultaneously and in parallel simulates all aTAM systems, and provide a connection between that system and the existence of systems within models other than the aTAM which are IU for the aTAM. This set of results provides a full characterization of the tradeoffs between systems with multi-tile seeds and those with single-tile seeds, which is fundamental to the measure of complexity of aTAM systems.

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Notes

  1. Note that \(R^*\) is a total function since every assembly of S represents some assembly of T; the functions R and \(\alpha \) are partial to allow undefined points to represent empty space.

  2. An alternative, but also common, interpretation of the aTAM model is that, similar to the behavior of nondeterministic versions of automata such as Turing machines, whenever a nondeterministic choice occurs the system splits into a separate instance to follow each option. In this interpretation, it is considered that there is only a single copy of the seed, and then for each assembly to which more than one tile can attach, a new instance of the assembly is created for each possible attachment. Thus (possibly in the limit) all assemblies which can form from a seed assembly do so. In this interpretation, this construction also validly simulates all systems in parallel.

  3. The set of all aTAM systems is countably infinite since it is clearly infinite (e.g. for every \(i \in {\mathbb {N}}\) there exists an aTAM system with i tile types that has a single-tile seed and self-assembles an \(i \times 1\) line at temperature 1), but every component of an aTAM system must be finite by definition of the aTAM, and therefore the set of systems must be countably infinite. Since any countably infinite set can be enumerated, there exists some enumeration of all aTAM systems.

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Acknowledgements

The authors would like to thank Trent Rogers for helpful comments and creative brainstorming, and credit him for the idea of Claim 6.1. Additionally, we thank the anonymous SODA23 reviewers who gave extremely helpful comments and suggestions that have greatly improved this version of the paper.

Funding

The research leading to these results received funding from the National Science Foundation under Grant Agreement No CAREER-1553166.

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

    Andrew Alseth & Matthew J. Patitz

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All authors contributed equally to the manuscript.

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Correspondence to Matthew J. Patitz.

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The authors have no relevant financial or non-financial interests to disclose. The authors have no conflicts of interest to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.The authors have no financial or proprietary interests in any material discussed in this article.

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This work was originally presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA23) held in Florence, Italy on January 22–25, 2023.

Andrew Alseth and Matthew J. Patitz: Supported in part by National Science Foundation Grant CAREER-1553166.

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Alseth, A., Patitz, M.J. The Need for Seed (in the Abstract Tile Assembly Model). Algorithmica 86, 218–280 (2024). https://doi.org/10.1007/s00453-023-01160-w

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