An Axiomatic Study of Leveraging Blockers to Self-Assemble Arbitrary Shapes via Temperature Programming

In this paper we present a theoretical design for tile-based self-assembling systems where individual tile sets that are universal for various tasks (e.g. building shapes or encoding arbitrary data for algorithmic systems) can be provided their input solely using sequences of variations in temperatures. Although there is prior theoretical work in so-called "temperature programming," the results presented here are based upon recent experimental work with "blocked tiles" that provides plausible evidence for their practical implementation. We develop and present an abstract mathematical model, the BlockTAM, for such systems that is based upon the experimental work, and provide constructions within it for universal number encoding systems and a universal shape building construction. These results mirror previous results in temperature programming, covering the two ends of the spectrum that seek to balance the scale factors of assemblies with the number of temperature phases required, while leveraging the features of the BlockTAM that we hope provide a pathway for future experimental implementations.