Limitations on counting in Boolean circuits and self-assembly
Abstract
In self-assembly, a k-counter is a tile set that grows a horizontal ruler from left to right, containing k columns each of which encodes a distinct binary string. Counters have been fundamental objects of study in a wide range of theoretical models of tile assembly, molecular robotics and thermodynamics-based self-assembly due to their construction capabilities using few tile types, time-efficiency of growth and combinatorial structure. Here, we define a Boolean circuit model, called n-wire local railway circuits, where n parallel wires are straddled by Boolean gates, each with matching fanin/fanout strictly less than n, and we show that such a model can not count to 2n nor implement any so-called odd bijective nor quasi-bijective function. We then define a class of self-assembly systems that includes theoretically interesting and experimentally-implemented systems that compute n-bit functions and count layer-by-layer. We apply our Boolean circuit result to show that those self-assembly systems can not count to 2n. This explains why the experimentally implemented iterated Boolean circuit model of tile assembly can not count to 2n, yet some previously studied tile system do. Our work points the way to understanding the kinds of features required from self-assembly and Boolean circuits to implement maximal counters.