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

Abstract

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 engineer unconventional computing systems in a systematic way, we need guidance from a formal theory that is different from the classical symbolic-algorithmic Turing machine theory. 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 a physical computing system. This leads to an understanding of computing as the structuring of processes, while classical models of computing systems describe the processing of structures.

Fig. 1. Modeling an algorithmic and a cybernetic computing system on three levels of abstraction.

a–c Digital computing system are typically modelled as algorithmic. The overall functionality of such a system is to transform input data structures u⁽³⁾ into output data structures y⁽³⁾. An algorithmic model thus represents, on the global task level, a mathematical function f⁽³⁾ from inputs to outputs (a). To model and implement how this global task function is realized on a digital computing machine, it it stepwise broken down (compiled) into finer-grained models, where data structures (vertical colour bars) and functions (arrows) become hierarchically dissected until at a machine-interface level (c), both can be straightforwardly mapped to the digital circuits of the underlying microchip. d–f Cybernetic computing systems, like brains or analogue processing chips for signal processing and control, transform a continually arriving input signal u⁽³⁾ into an output signal y⁽³⁾ by a continually ongoing nonlinear dynamical coupling F⁽³⁾. Like digital data structures and processes, signals and their couplings can be hierarchically broken down from a global task-level specification (d) to machine-interfacing detail (f). Symbols x represent intermediate data structures (in algorithmic models) and signals (in cybernetic models). Three modeling levels are shown, but there may be more or fewer.

Fig. 2. Dynamical re-configuration effects in FC modeling.

Observers and their associated chronicles can dynamically bind and unbind in and from compounds during the execution of an FC model, leading to a variety of structural re-organization effects that have obvious analogues in algorithmic computing. From top to bottom: merging (a) and splitting (copying, b) of observers; termination and creation (c); binding and unbinding (d). The central segment in d (dashed orange outline) shows the temporary presence of a compound observer made from two primitive observers and one compound observer, which in turn is a binding of three primitive ones. The compound observers have activation histories of their own, which are not shown in the graphic. Coupling dynamics is symbolized by the circular arrow.