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.

中文翻译：

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布尔电路和自组装中计数的限制

在自组装中，$k$-counter 是一个瓦片集，它从左到右增加一个水平标尺，包含 $k$ 列，每列编码一个不同的二进制字符串。计数器一直是瓷砖组装、分子机器人和基于热力学的自组装的广泛理论模型的基本研究对象，因为它们使用少量瓷砖类型的构建能力、生长的时间效率和组合结构。在这里，我们定义了一个布尔电路模型，称为 $n$-wire 本地铁路电路，其中 $n$ 平行线被布尔门跨越，每个匹配的扇入/扇出严格小于 $n$，并且我们证明了这样的模型不能算到 $2^n$ 也不能实现任何所谓的奇双射或准双射函数。然后，我们定义了一类自组装系统，其中包括理论上有趣和实验实现的系统，这些系统计算 $n$-bit 函数并逐层计数。我们应用我们的布尔电路结果来表明那些自组装系统不能计数到 $2^n$。这解释了为什么实验实现的瓦片组装迭代布尔电路模型不能计算到 $2^n$，而一些以前研究过的瓦片系统却可以。我们的工作指出了理解自组装和布尔电路实现最大计数器所需的各种特征的方法。这解释了为什么实验实现的瓦片组装迭代布尔电路模型不能计算到 $2^n$，而一些以前研究过的瓦片系统却可以。我们的工作指出了理解自组装和布尔电路实现最大计数器所需的各种特征的方法。这解释了为什么实验实现的瓦片组装迭代布尔电路模型不能计算到 $2^n$，而一些以前研究过的瓦片系统却可以。我们的工作指出了理解自组装和布尔电路实现最大计数器所需的各种特征的方法。