Limitations on counting in Boolean circuits and self-assembly

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 $2^n$ 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 $2^n$. This explains why the experimentally implemented iterated Boolean circuit model of tile assembly can not count to $2^n$, 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.

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