[Paper Review] Intrinsic universality in tile self-assembly requires cooperation

Pierre-Étienne Meunier, Matthew J. Patitz+4 more|arXiv (Cornell University)|Jan 5, 2014

DNA and Biological Computing43 references66 citations

TL;DR

This paper proves that Winfree's abstract Tile Assembly Model is not intrinsically universal under noncooperative (temperature-1) binding, where tiles bind if they match on at least one side. In contrast, the model is intrinsically universal with cooperative binding requiring two or more matching sides. The result reveals that cooperation fundamentally enhances the computational expressiveness of algorithmic self-assembly, even though noncooperative systems can still simulate Turing machines.

ABSTRACT

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as temperature 1, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides.Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

Motivation & Objective

• To determine whether the abstract Tile Assembly Model can intrinsically simulate all algorithmic self-assembly processes under noncooperative binding.
• To resolve a long-standing open problem regarding the computational power of noncooperative tile self-assembly.
• To clarify the qualitative difference between cooperative and noncooperative binding in enabling universal computation within self-assembly models.
• To investigate whether noncooperative systems can simulate any other noncooperative systems, despite lacking intrinsic universality.

Proposed method

• The authors use a proof by contradiction to show that no noncooperative tile assembly system can simulate all other noncooperative systems.
• They analyze the structural and dynamic limitations of noncooperative binding, particularly the lack of binding threshold requirements.
• They construct a specific three-dimensional noncooperative tile set that can simulate any two-dimensional noncooperative system, demonstrating a restricted form of intrinsic universality.
• They compare the expressive power of noncooperative and cooperative models, emphasizing the role of multi-side binding in enabling complex behaviors.
• They extend their results to both two and three dimensions, showing that the non-universality result holds even in 3D, where noncooperative systems are already known to be Turing universal.

Experimental results

Research questions

• RQ1 Can noncooperative tile self-assembly systems intrinsically simulate all algorithmic self-assembly processes?
• RQ2 Does the ability to simulate Turing machines imply intrinsic universality in noncooperative tile assembly?
• RQ3 What is the role of cooperative binding in enabling intrinsic universality in the abstract Tile Assembly Model?
• RQ4 Can a single noncooperative tile set simulate all other noncooperative systems in a restricted sense?
• RQ5 Why does the addition of cooperative binding significantly increase the expressive power of tile self-assembly models?

Key findings

• The abstract Tile Assembly Model is not intrinsically universal under noncooperative binding, despite being Turing universal.
• Cooperative binding—requiring at least two matching sides—is essential for intrinsic universality in the model.
• The result holds in both two and three dimensions, with the 3D case being particularly surprising given the known Turing universality of 3D noncooperative systems.
• A three-dimensional noncooperative tile set can simulate any two-dimensional noncooperative self-assembly system, establishing a restricted form of intrinsic universality.
• Turing universality does not imply the ability to simulate all algorithmic self-assembly processes, demonstrating a qualitative gap between computational power and expressive universality.