Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness

TL;DR

This paper proves that a universal simulator for directed self-assembling systems in the abstract Tile Assembly Model cannot be both universal and always directed, highlighting the essential nondeterminism in self-assembly processes.

Contribution

It demonstrates that directed systems in the aTAM are not intrinsically universal, establishing the fundamental nondeterminism in self-assembly.

Findings

Directed systems cannot be universally simulated by always directed systems.

Nondeterminism is essential for universality in self-assembly.

New techniques for designing systems with disjoint computational powers.

Abstract

As a mathematical model of self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. Capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a seed structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the universal simulator previously given makes use of nondeterminism in terms of tile types placed in several key locations when different assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This provides insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process. To achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.