Simulation in Algorithmic Self-assembly

Winfree introduced a model of self-assembling systems called the abstract Tile Assembly Model [64] (aTAM) where square tiles with glues on their edges attach spontaneously via matching glues to form complex structures. A generalization of the aTAM called the 2HAM (two-handed aTAM) not only allows for single tiles to bind, but also for “supertile” assemblies consisting of any number of tiles to attach. We consider a variety of models based on either the aTAM or the 2HAM. The underlying commonality of the work presented here is simulation. We introduce the polyTAM, where a tile system consists of a collection of polyomino tiles, and show that for any polyomino P of size ≥3 and any Turing machine M , there exists a temperature-1 polyTAM system containing only shape-P tiles that simulates M . We introduce the RTAM (Reflexive Tile Assembly Model) that works like the aTAM except that tiles can nondeterministically flip prior to binding. We show that the temperature-1 RTAM cannot simulate a Turing machine by showing the much stronger result that the RTAM can only selfassemble “periodic” patterns. We consider a variety of models and define notions of simulation which serve as relations between two tile assembly systems (possibly belonging to different models). Using simulation as a basis of comparison, we first show that cellular automata and the class of all tile assembly systems in the aTAM are equivalent. Next, we introduce the Dupled aTAM (DaTAM) and show that the temperature-2 aTAM and the temperature-1 DaTAM are “mutually exclusive” by showing that there is an aTAM system that cannot be simulated by any DaTAM system, and vice versa. Third, we consider the restricted glues Tile Assembly Model [52] (rgTAM) and show that there is an aTAM system that cannot be simulated by any rgTAM system. We introduce the Dupled restricted glues Tile Assembly Model (DrgTAM), and show that the DrgTAM is intrinsically universal for the aTAM. Finally, we consider a variation of the Signal-passing Tile Assembly Model (STAM) [50] called the STAM and show that the STAM is intrinsically universal and that the 3-D 2HAM is intrinsically universal for the STAM. Acknowledgements First and foremost, I would like to thank my advisor, friend and co-author Matt Patitz. As a non-traditional student, I entered the PhD program with a good deal of self-doubt, and throughout my work as a PhD student, Matt has believed in me even when I did not. What I know about being a productive researcher I have learned by his example, and for this reason, this thesis owes its existence to him. I cannot say enough about what Matt has done for me; I just hope that he knows this and that someday I have a chance to repay him. When I joined the self-assembly research group at the University of Arkansas, the group consisted of Matt and Trent Rogers. As we worked on projects together, Trent and I were both grappling with the same concepts, and in doing so, Trent has been indispensable in my understanding of theoretical self-assembly. When it looks as though Trent and I are trading mumblings coupled with a few hand-gestures, we are, in fact, working on a new paper. He’s a remarkable student with a very bright future who is always trying to push research further as well as trying to push my buttons. This thesis would not exist without him. I would also like to thank my friend and fellow graduate student Tyler Fochtman. The self-assembly group wouldn’t be complete with the “wild card”. I would also like to thank Scott Summers, a friend and co-author, who has been an excellent example for me. Scott’s guidance in seeking employment was especially helpful. I look forward to working with Matt, Trent and Scott on future projects. Dedication I would like to thank my loving wife for her unconditional support and patience while I was working on the projects that eventually led to this thesis.