A pumping lemma for non-cooperative self-assembly (1610.07908v2)

Abstract: We prove a result which strongly hints at the computational weakness of a model of tile assembly that has so far resisted many attempts of formal analysis or positive constructions. Specifically, we prove that, in Winfree's abstract Tile Assembly Model, when restricted to use only noncooperative bindings, any long enough path starting from the seed that can grow in all terminal assemblies is pumpable, meaning that this path can be extended into an infinite, ultimately periodic path. This result can be seen as a geometric generalization of the pumping lemma of finite state automata, and closes the question of what can be computed deterministically in this model. Moreover, this question has motivated the development of a new method called visible glues. We believe that this method can also be used to tackle other long-standing problems in computational geometry, in relation for instance with self-avoiding paths. Tile assembly (including non-cooperative tile assembly) was originally introduced by Winfree and Rothemund in STOC 2000 to understand how to program shapes. The non-cooperative variant, also known as temperature 1 tile assembly, is the model where tiles are allowed to bind as soon as they match on one side, whereas in cooperative tile assembly, some tiles need to match on several sides in order to bind. Previously, exactly one known result (SODA 2014) showed a restriction on the assemblies general non-cooperative self-assembly could achieve, without any implication on its computational expressiveness. With non-square tiles (like polyominos, SODA 2015), other recent works have shown that the model quickly becomes computationally powerful.

• The paper proves that any sufficiently long path in non-cooperative tile assembly systems is pumpable, highlighting a key computational constraint analogous to limitations in finite automata.
• The paper introduces novel geometric techniques, including the concept of visible glues, to analyze planarity and tile interactions in self-assembly.
• The paper distinguishes non-cooperative models from cooperative ones, offering critical insights into the inherent computational boundaries of programmable matter.

An Analysis of the Pumping Lemma for Non-Cooperative Self-Assembly

The paper, "A pumping lemma for non-cooperative self-assembly," by Pierre-Étienne Meunier and Damien Regnault is a theoretical investigation into the computational expressiveness of the abstract Tile Assembly Model (aTAM), particularly focusing on its non-cooperative variant, commonly known as temperature 1 tile assembly. The paper extends upon previous work by proposing a pumping lemma for non-cooperative self-assembly, analogous to the geometrical generalization of the pumping lemma used in finite state automata theory.

The paper's primary contribution is proving that any sufficiently long path in non-cooperative tile assembly systems, which can grow within all terminal assemblies, is pumpable. That is, such paths can be extended into infinitely repeating, periodic paths. This result implies a fundamental computational limitation of the non-cooperative tile assembly model, echoing the known restrictions in deterministic finite automata.

In proving their principal theorem, the authors tackle an open problem in computational geometry concerning the role of planarity and tile shape in non-cooperative tile assemblies. They introduce a framework to understand how information is communicated via geometric interactions within planar spaces. The methodology includes novel techniques like visible glues, which could address longstanding problems in related fields such as self-avoiding paths and polymer chemistries.

Key Findings and Implications

1. Pumping Lemma Extension: The core result that paths are pumpable signifies a substantial restriction to the computational power of non-cooperative systems, highlighting their inherent inability to perform deterministic computations beyond certain constraints.
2. Model Limitations: By only allowing non-cooperative bindings, it becomes evident that a deterministic computation is improbable in these settings. This insight complements earlier findings that cooperative models (temperature ≥ 2) can simulate Turing machines, emphasizing a dichotomy in computational power when cooperative interactions are minimized or absent.
3. Geometric Influences: The introduction of geometric components within the theoretical framework provides a deeper understanding of the model's limitations, suggesting an essential connection between geometric properties and the computational bounds of assembly systems.
4. Generalization Potential: The paper suggests that insights gained and methods developed might extend to other problems in computational geometry that involve similar non-cooperative constraints.

Future Directions

The research opens multiple pathways for further exploration:

• Investigation of whether non-cooperative systems can achieve semi-linear sets of assemblies, which could transform our understanding of their computational simplicity.
• Exploration of three-dimensional generalizations and alternative tile shapes beyond polyominoes, potentially leading to different computational results or universal behaviors.
• Applying the geometrical approaches and newly introduced concepts like visible glues to broader issues in computational science and geometry, possibly impacting algorithm design in robotics and synthetic biology where similar interaction-based constraints exist.

The authors' contribution forms a critical bridge in the tile assembly literature, proposing a coherent and robust link between geometry and computational limits. Their results point toward a compelling conclusion: while non-cooperative self-assembly configurations depict creative intrigue, they are circumscribed substantially in computational performance when harnessed alone, versus more intricately cooperative methods. This distinction provides clarity to the boundaries of programmable matter, shaping the next decade of research endeavors in nano-technology and programmable material science.