Self-assembly is the process whereby relatively simple components autonomously combine to form more complex objects. Nature exhibits self-assembly to form everything from microscopic crystals to living cells to galaxies. With a desire to both form increasingly sophisticated products and to understand the basic components of living systems, scientists have developed and studied artificial self-assembling systems. One such framework is the Tile Assembly Model introduced by Erik Winfree in 1998. In this model, simple two-dimensional square `tiles' are designed so that they self-assemble into desired shapes. The work in this thesis consists of a series of results which build toward the future goal of designing an abstracted, high-level programming language for designing the molecular components of self-assembling systems which can perform powerful computations and form into intricate structures.

The first two sets of results demonstrate self-assembling systems which perform infinite series of computations that characterize computably enumerable and decidable languages, and exhibit tools for algorithmically generating the necessary sets of tiles. In the next chapter, methods for generating tile sets which self-assemble into complicated shapes, namely a class of discrete self-similar fractal structures, are presented. Next, a software package for graphically designing tile sets, simulating their self-assembly, and debugging designed systems is discussed. Finally, a high-level programming language which abstracts much of the complexity and tedium of designing such systems, while preventing many of the common errors, is presented.

The summation of this body of work presents a broad coverage of the spectrum of desired outputs from artificial self-assembling systems and a progression in the sophistication of tools used to design them. By creating a broader and deeper set of modular tools for designing self-assembling systems, we hope to increase the complexity which is attainable. These tools provide a solid foundation for future work in both the Tile Assembly Model and explorations into more advanced models.