Capabilities and Limits of Compact Error Resilience Methods for Algorithmic Self-assembly in Two and Three Dimensions

Winfree’s pioneering work led the foundations in the area of error-reduction in algorithmic self-assembly[26], but the construction resulted in increase of the size of assembly. Reif et. al. contributed further in this area with compact error-resilient schemes [15] that maintained the original size of the assemblies, but required certain restrictions on the Boolean functions to be used in the algorithmic self-assembly. It is a critical challenge to improve these compact error resilient schemes to incorporate arbitrary Boolean functions, and to determine how far these prior results can be extended under different degrees of restrictions on the Boolean functions. In this work we present a considerably more complete theory of compact error-resilient schemes for algorithmic self-assembly in two and three dimensions. First we consider two-dimensional algorithmic self-assembly. We present an error correction scheme for reduction of errors from e to e2 for arbitrary Boolean functions in two dimensional algorithmic self-assembly. Then we characterize the class of Boolean functions for which the error reduction can be done from e to e3, and present an error correction scheme that achieves this reduction. Then we prove ultimate limits on certain classes of compact error resilient schemes: in particular we show that they can not provide reduction of errors from e to e4 is for any Boolean functions. Further, we develop the first provable compact error resilience schemes for three dimensional tiling self-assemblies. We also extend the work of Winfree on self-healing in two-dimensional self-assembly[25] to obtain a self-healing tile-set for three-dimensional self-assembly.