Complexity of compact proofreading for self-assembled patterns

Fault-tolerance is a critical issue for biochemical computation. Recent theoretical work on algorithmic self-assembly has shown that error correcting tile sets are possible, and that they can achieve exponential decrease in error rates with a small increase in the number of tile types and the scale of the construction [24, 4]. Following [17], we consider the issue of applying similar schemes to achieve error correction without any increase in the scale of the assembled pattern. Using a new proofreading transformation, we show that compact proofreading can be performed for some patterns with a modest increase in the number of tile types. Other patterns appear to require an exponential number of tile types. A simple property of existing proofreading schemes - a strong kind of redundancy - is the culprit, suggesting that if general purpose compact proofreading schemes are to be found, this type of redundancy must be avoided.