Toward a formal theory for computing machines made out of whatever physics offers: extended version

Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on unconventional physical effects, we need guidance from a formal theory that is different from the symbolic-algorithmic theory of today's computer science textbooks. We propose a general strategy for developing such a theory, and within that general view, a specific approach that we call "fluent computing". In contrast to Turing, who modeled computing processes from a top-down perspective as symbolic reasoning, we adopt the scientific paradigm of physics and model physical computing systems bottom-up by formalizing what can ultimately be measured in any physical substrate. This leads to an understanding of computing as the structuring of processes, while classical models of computing systems describe the processing of structures. -- This is an extended version of a perspective article with the same title that appeared in Nature Communications on August 16, 2023