Neural Cellular Automata (NCAs) have been proven effective in simulating morphogenetic processes, the continuous construction of complex structures from very few starting cells. Recent developments in NCAs lie in the 2D domain, namely reconstructing target images from a single pixel or infinitely growing 2D textures. In this work, we propose an extension of NCAs to 3D, utilizing 3D convolutions in the proposed neural network architecture. Minecraft is selected as the environment for our automaton since it allows the generation of both static structures and moving machines. We show that despite their simplicity, NCAs are capable of growing complex entities such as castles, apartment blocks, and trees, some of which are composed of over 3,000 blocks. Additionally, when trained for regeneration, the system is able to regrow parts of simple functional machines, significantly expanding the capabilities of simulated morphogenetic systems.

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Biological systems are very robust to morphological damage, but artificial systems (robots) are currently not. In this paper we present a system based on neural cellular automata, in which locomoting robots are evolved and then given the ability to regenerate their morphology from damage through gradient-based training. Our approach thus combines the benefits of evolution to discover a wide range of different robot morphologies, with the efficiency of supervised training for robustness through differentiable update rules. The resulting neural cellular automata are able to grow virtual robots capable of regaining more than 80% of their functionality, even after severe types of morphological damage.

Zipf's law predicts a power-law relationship between word rank and frequency in language communication systems and has been widely reported in a variety of natural language processing applications. However, the emergence of natural language is often modeled as a function of bias between speaker and listener interests, which lacks a direct way of relating information-theoretic bias to Zipfian rank. A function of bias also serves as an unintuitive interpretation of the communicative effort exchanged between a speaker and a listener. We counter these shortcomings by proposing a novel integral transform and kernel for mapping communicative bias functions to corresponding word frequency-rank representations at any arbitrary phase transition point, resulting in a direct way to link communicative effort (modeled by speaker/listener bias) to specific vocabulary used (represented by word rank). We demonstrate the practical utility of our integral transform by showing how a change from bias to rank results in greater accuracy and performance at an image classification task for assigning word labels to images randomly subsampled from CIFAR10. We model this task as a reinforcement learning game between a speaker and listener and compare the relative impact of bias and Zipfian word rank on communicative performance (and accuracy) between the two agents.