Date of Graduation

12-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Engineering (PhD)

Degree Level

Graduate

Department

Electrical Engineering and Computer Science

Advisor/Mentor

Patitz, Matthew J.

Committee Member

Gauch, Susan E.

Second Committee Member

Rieck, Yo’av

Third Committee Member

Zhang, Lu

Keywords

Algorithms; Computational Complexity; Fractals; Self-Assembly

Abstract

Tile-assembly studies abstract models of computation inspired by advancements in the emerging field of DNA-nanotechnology, where synthetic strands of DNA are used as building blocks for microscopic structures. These DNA strands can be made to combine into structural units with selectively sticky sides that abstractly resemble Wang tiles. However, unlike Wang tiles, an assembly process is modeled where tiles combine one-by-one to form larger assemblies according to matching rules based on their sticky sides. While in practice, these tile-assembly models have seen use in designing DNA nano-structures, the mathematical study of their theory has revealed an exciting interplay between geometric and computational dynamics. Many tile-assembly models have been shown to exhibit not only computational universality, the ability to simulate arbitrary Turing machines, but also intrinsic universality, a notion borrowed from the study of cellular automata, where one set of tiles is capable of simulating the full dynamics of all others in some class at scale. Unlike the purely symbolic notion of simulation exhibited between Turing machines, intrinsic simulation requires that the simulating tile system resemble the simulated system geometrically by "zooming out". Much like when comparing traditional models of computation, intrinsic simulation allows for a direct comparison between tile-assembly models whereby a model can be said to be more powerful than another if it is capable of intrinsically simulating all systems within the other model. This dissertation explores the use of intrinsic simulation as a means to compare various tile-assembly models and evaluate the shape-building capabilities of the abstract Tile-Assembly Model (aTAM).

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