On complexity of colloid cellular automata

TL;DR

This work probes the space-time complexity of colloid cellular automata formed from Boolean functions mined from ZnO and proteinoid colloids under electrical stimulation. By applying all $2$-, $4$-, and $8$-bit input configurations and mapping outputs to one-dimensional CA rules, the authors analyze resulting dynamics with metrics such as Lempel-Ziv complexity ($LZ$), Shannon entropy ($H$), Simpson diversity ($S$), and expressiveness ($E$) to build arity-specific complexity hierarchies. They find that $f_7$ and $f_8$—corresponding to XOR-like functions—tend to maximize several complexity measures in the 2-ary case, while 4- and 8-ary functions exhibit distinct hierarchies; expressiveness can diverge from global complexity, underscoring metric-dependent interpretations. The study demonstrates the potential of colloid-based substrates for unconventional computing and provides a framework to compare complexity across physical implementations, while acknowledging limitations such as the 1D restriction and the need for broader metrics and higher-dimensional models.

Abstract

The colloid cellular automata do not imitate the physical structure of colloids but are governed by logical functions derived from the colloids. We analyse the space-time complexity of Boolean circuits derived from the electrical responses of colloids: ZnO (zinc oxide, an inorganic compound also known as calamine or zinc white, which naturally occurs as the mineral zincite), proteinoids (microspheres and crystals of thermal abiotic proteins), and combinations thereof to electrical stimulation. To extract Boolean circuits from colloids, we send all possible configurations of two-, four-, and eight-bit binary strings, encoded as electrical potential values, to the colloids, record their responses, and thereby infer the Boolean functions they implement. We map the discovered functions onto the cell-state transition rules of cellular automata (arrays of binary state machines that update their states synchronously according to the same rule) -- the colloid cellular automata. We then analyse the phenomenology of the space-time configurations of the automata and evaluate their complexity using measures such as compressibility, Shannon entropy, Simpson diversity, and expressivity. A hierarchy of phenomenological and measurable space-time complexity is constructed.

Figures (6)

• Figure 1: a) A scheme of the experiments. PC –- laptop for generating sequences; CU -- control unit, the dashed section is a breakdown of a single channel; ADC –- analogue to digital converter roberts2023mining. b) experimental setup. c) A schematic of the inside of the unit control box. d) A close-up photo of the colloid dish and the electrodes interfacing it. From roberts2023logical.
• Figure 2: Functions with two-arguments and those functions with four or eight arguments which produce alike patterns. Space (1D CA array) states are horizontal, and time (progressing from top to bottom) is vertical: $x^0_1 x^0_2 \ldots x^0_{500}$$x^1_1 x^1_2 \ldots x^1_{500}$$\ldots$$x^{500}_1 x^{500}_2 \ldots x^{500}_{500}$
• Figure 3: Functions with four-arguments and those functions with eight arguments which produce alike patterns. Space (1D CA array) states are horizontal, and time (progressing from top to bottom) is vertical.
• Figure 4: Functions with eight-arguments. Space (1D CA array) states are horizontal, and time (progressing from top to bottom) is vertical.
• Figure 5: (a) Shannon entropy $H$ vs expressiveness $E$, linear approximation $E=0.30874 + 2.5032*H$. (b) Shannon entropy $H$ vs Simpson index $S$, linear approximation $S=0.058092 + 0.43781*H$. (c) Shannon entropy $H$ vs space filling $D$, linear approximation $D=0.56455 + (-0.073223)*H$.