Universal spin models as Turing machines and neural networks

A common theme throughout many disciplines is the ease of generating complexity: Simple systems and simple rules are rich enough to reproduce many complex phenomena. Why is this so? I suggest that this is due to the phenomenon of universality: Fairly simple systems are already universal, meaning that they can explore all complexity in their domain. This explains why everything is so complicated. We understand the phenomenon of universality in three domains: In computer science, universal Turing machines can run any algorithm. In machine learning, simple artificial neural networks can learn any function. In physics, universal spin models can simulate any other spin model. These three notions of universality share many similarities, both in their features and in their proofs, but the knowledge is hitherto isolated in their respective disciplines. The main goal of this project, called UniX, is to establish rigorous links between universal spin models, universal Turing machines, and notions of universality in neural networks, and to explore their implications. More specifically, UniX will connect these three well-established disciplines by transferring results on (i) universality, (ii) efficiency and approximations, and (iii) undecidability, thereby enriching them with novel concepts, methodologies, and high-impact results. As significant examples, UniX will establish rigorous relations between spin models and automata, discover the relation between universal spin models and universal Turing machines, and prove that artificial neural networks are in fact universal spin models. This will establish a new emerging field of research at the interface of spin physics, theoretical computer science and machine learning, pivoting on the multifaceted phenomenon of universality. The implications of UniX are far-reaching. Given that universal Turing machines sit at the very heart of computer science, which has revolutionised the world, and given the transformative power that neural networks are having for science, society and technology, comprehending the reach of universality and its consequences could transform our understanding of science. Furthermore, since spin models are generally used as toy models of complex systems in biology or linguistics, UniX will open the door to extending this work to other fields.