A linear bound for the size of the finite terminal assembly of a directed non-cooperative tile assembly system
Sergiu Ivanov, Damien Regnault
TL;DR
The paper analyzes directed non-cooperative tile assembly systems (aTam) to bound the size of finite terminal assemblies. By introducing a rigorous path-centric framework with cuts, visibility, and canonical paths, it shows that if the terminal assembly is finite, its horizontal width and vertical height are bounded linearly in the seed size $|\sigma|$ and the tile set size $|T|$, specifically by $7|\sigma|+58|T|+30$ in each dimension. The main method combines tools from path analysis, spans and arcs decompositions, and a shield-based argument to prevent unbounded growth, ultimately ruling out generalized efficient-path constructions in the directed setting. This yields asymptotic optimality results, such as the square of width $n$ requiring $2n-1$ tile types, and provides a framework potentially extendable to non-directed cases. The result advances understanding of computation in non-cooperative tile assemblies by showing that, in the directed finite-terminal regime, growth is inherently constrained and cannot exploit broad-path nondeterminism to achieve unbounded constructions.
Abstract
The abstract tile assembly model (aTam) is a model of DNA self-assembly. Most of the studies focus on cooperative aTam where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings where assemblies do not need to cover the whole plane and some mismatches may occur). Introduced by P.E. Meunier and D. Regnault, efficient paths are a non-trivial construction for non-cooperative aTam. These paths of width nlog(n) are designed with n different tile types. Assembling them relies heavily on a form of ``non-determinism''. Indeed, the set of tiles may produced different finite terminal assemblies but they all contain the same efficient path. Directed non-cooperative aTam does not allow this non-determinism as only one assembly may be produced by a tile assembly system. This variant of aTam is the only one who was shown to be decidable. In this paper, we show that if the terminal assembly of a directed non-cooperative tile assembly system is finite then its width and length are of linear size according to the size of the tile assembly system. This result implies that the construction of efficient paths cannot be generalized to the directed case and that some computation must rely on a competition between different paths. It also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.