[Paper Review] Programming Biomolecules That Fold Greedily During Transcription

Cannon, Sarah, Erik D. Demaine+6 more|arXiv (Cornell University)|Jan 8, 2012

Modular Robots and Swarm Intelligence20 references53 citations

TL;DR

This paper compares the seeded abstract Tile Assembly Model (aTAM) and the two-handed tile assembly model (2HAM), demonstrating that 2HAM is more powerful in terms of efficiency and expressiveness. It shows that any aTAM system can be simulated in 2HAM with only a constant-factor increase in tile count, while 2HAM enables exponential savings for certain shapes and can assemble infinite structures unreachable in aTAM. Crucially, verifying unique assembly is co-NP-complete in 2HAM—unlike in aTAM, where it is polynomial-time solvable.

ABSTRACT

We study the difference between the standard seeded model of tile self-assembly, and the "seedless" two-handed model of tile self-assembly. Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.

Motivation & Objective

• To compare the theoretical power of the seeded aTAM and unseeded two-handed 2HAM models of tile self-assembly.
• To determine whether 2HAM enables more efficient or expressive self-assembly than aTAM for finite and infinite shapes.
• To analyze the computational complexity of verifying unique assembly in 2HAM, contrasting it with the polynomial-time solvability in aTAM.
• To establish separations in tile complexity between the two models, including exponential differences and busy-beaver-level savings.
• To investigate the existence of shapes that can be assembled in one model but not the other, particularly in the infinite case.

Proposed method

• Constructing simulations of aTAM systems within the 2HAM model using scaled tile sets (5×5 scaling) and temperature-preserving designs.
• Designing specialized tile gadgets to simulate Turing machine computations in both models, using path-based growth and blocking mechanisms.
• Using nondeterministic path growth and glue matching to encode computational steps, with blocking tiles representing halting configurations.
• Applying reductions from the undecidable halting problem to prove uncomputability of finite existence verification (FEV) in both models.
• Modifying tile sets to allow spontaneous nucleation in 2HAM, ensuring that incomplete simulations lead to infinite assemblies unless the full computation halts.
• Proving co-NP-completeness of unique assembly verification in 2HAM via reductions from known co-NP-complete problems, contrasting with polynomial-time solvability in aTAM.

Experimental results

Research questions

• RQ1 Can any aTAM system with temperature τ ≥ 2 be simulated in 2HAM with only a constant-factor increase in tile count?
• RQ2 Are there finite shapes for which 2HAM achieves exponential savings in tile complexity compared to aTAM?
• RQ3 Can 2HAM assemble infinite shapes that are impossible to assemble in aTAM?
• RQ4 Is there a shape that can be assembled in aTAM but not in 2HAM, or vice versa, in the infinite case?
• RQ5 What is the computational complexity of verifying unique assembly in 2HAM, and how does it compare to aTAM?

Key findings

• Any aTAM system with temperature τ ≥ 2 can be simulated in 2HAM using a 5×5 scaled tile set, with only a constant-factor increase in tile count.
• For temperature τ = 1, a specific shape requires n + 5 tile types in aTAM but 2n + 2 tile types in 2HAM, demonstrating a linear separation in tile complexity.
• There exists a shape that can be assembled in 2HAM with O(n) tile types, while any aTAM system assembling the same shape requires BB(n) tile types, where BB(n) is the busy beaver function.
• An infinite shape (an infinite staircase) can be constructed in 2HAM but not in aTAM, showing a fundamental expressiveness gap.
• An infinite shape (a blob with an infinite tail) can be constructed in aTAM but not in 2HAM, indicating that the models differ in their ability to assemble certain infinite structures.
• Verifying whether a 2HAM system uniquely assembles a given supertile is co-NP-complete, whereas the same problem is polynomial-time solvable in aTAM.