Negative Interactions in Irreversible Self-Assembly

TL;DR

This paper investigates the capabilities of negative interactions in irreversible tile assembly, proving limitations on tile reuse and demonstrating efficient Turing machine simulation with bounded intermediate assemblies.

Contribution

It provides an impossibility theorem for tile detachment and constructs a tile set that simulates Turing machines with space bounds, improving efficiency over standard methods.

Findings

Negative interactions do not enable unlimited tile reuse.

A tile set can simulate Turing machines with bounded intermediate assembly size.

Intermediate assembly size can be limited to O(s) during simulation.

Abstract

This paper explores the use of negative (i.e., repulsive) interaction the abstract Tile Assembly Model defined by Winfree. Winfree postulated negative interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu, and Yin explored their power in the context of reversible attachment operations. We explore the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Omega(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate a Turing machine with space bound s and time bound t, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s * t) as required by the standard Turing machine simulation with tiles.