Limitations on counting in Boolean circuits and self-assembly

TL;DR

This paper proves fundamental limitations on the ability of certain Boolean circuit models and self-assembly systems to count up to 2^n, highlighting the features needed for maximal counting capabilities.

Contribution

It introduces a new Boolean circuit model and shows its limitations in counting, then applies these results to self-assembly systems, explaining their counting constraints.

Findings

The n-wire local railway circuit model cannot count to 2^n.

Self-assembly systems within this model cannot implement certain bijective functions.

Experimentally-implemented systems can count to 2^n, but the studied model cannot.

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 $2^n$ 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.