Abstract
Data complexity is an important concept in the natural sciences and related areas, but lacks a rigorous and computable definition. This paper focusses on a particular sense of complexity that is high if the data is structured in a way that could serve to communicate a message. In this sense, human speech, written language, drawings, diagrams and photographs are high complexity, whereas data that is close to uniform throughout or populated by random values is low complexity. I describe a general framework for measuring data complexity based on dividing the shortest description of the data into a structured and an unstructured portion, and taking the size of the former as the complexity score. I outline an application of this framework in statistical mechanics that may allow a more objective characterisation of the macrostate and entropy of a physical system. Then, I derive a more precise and computable definition geared towards human communication, by proposing local compositionality as an appropriate specific structure. Experimental evaluation shows that this method can distinguish meaningful signals from noise or repetitive signals in auditory, visual and text domains, and could potentially help determine whether an extra-terrestrial signal contained a message.