Flipping Tiles: Concentration Independent Coin Flips in Tile Self-Assembly

In this paper we introduce the robust coin flip problem in which one must design an abstract tile assembly system aTAM system whose terminal assemblies can be partitioned such that the final assembly lies within either partition with exactly probability 1/2, regardless of what relative concentration assignment is given to the tile types of the system. We show that robust coin flipping is possible within the aTAM, and that such systems can guarantee a worst case $$\\mathcal {O}1$$O1 space usage. As an application, we then combine our coin-flip system with the result of Chandran, Gopalkrishnan, and Reif﾿[3] to show that for any positive integer n, there exists a $$\\mathcal {O}\\log n$$Ologn tile system that assembles a constant-width linear assembly of expected length n that works for all concentration assignments. We accompany our primary construction with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility, and also prove some negative results. Further, we consider the harder scenario in which tile concentrations change arbitrarily at each assembly step and show that while this is not solvable in the aTAM, this version of the problem can be solved by more exotic tile assembly models from the literature.