Date of Graduation

5-2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Computer Science (PhD)

Degree Level

Graduate

Department

Computer Science & Computer Engineering

Advisor

Matthew Patitz

Committee Member

Jin-Woo Kim

Second Committee Member

Wing-Ning Li

Third Committee Member

Xintao Wu

Keywords

bit-reading, Computational complexity, Models of computation, Self-assembly, Universal Simulator

Abstract

In nature there are a variety of self-assembling systems occurring at varying scales which give rise to incredibly complex behaviors. Theoretical models of self-assembly allow us to gain insight into the fundamental nature of self-assembly independent of the specific physical implementation. In Winfree's abstract tile assembly model (aTAM), the atomic components are unit square "tiles" which have "glues" on their four sides. Beginning from a seed assembly, these tiles attach one at a time during the assembly process in an asynchronous and nondeterministic manner.

We can gain valuable insights into the nature of self-assembly by comparing different models of self-assembly which use fundamentally different mechanisms for local interactions. A powerful notion which allows us to compare models of self-assembly is simulation. The first result of this thesis examines the role of non-determinism in simulation. It shows that the universal simulation of directed aTAM systems requires undirectedness. A tile assembly model is said to be directed if it always assembles the same final assembly.

We distinguish between two types of aTAM systems: cooperative systems and non-cooperative systems. In cooperative aTAM systems, we are able to enforce that in order for a tile to attach to an assembly, the glues of a tile must match two or more glues of neighboring tiles. On the other hand, in non-cooperative aTAM systems, tiles are able to attach to an assembly provided that one of the tile's glues match an exposed glue on the assembly. It is well known that the cooperative aTAM is computationally universal, and it is conjectured that the non-cooperative aTAM is not computationally universal. For our second result, we show that if we allow tiles to be polygons with six or more sides, then the class of non-cooperative systems is capable of universal computation. On the other hand, we show that the class of systems consisting of polygons with six or less sides is not capable of computing using any of the currently known methods.

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