A New Quantitative Definition of the Complexity of Organized Matters

One of the most fundamental problems in science is to define the complexity of organized matters quantitatively, that is, organized complexity. Although many definitions have been proposed toward this aim in previous decades (e.g., logical depth, effective complexity, natural complexity, thermodynamics depth, effective measure complexity, and statistical complexity), there is no agreed-upon definition. The major issue of these definitions is that they captured only a single feature among the three key features of complexity, descriptive, computational, and distributional features, for example, the effective complexity captured only the descriptive feature, the logical depth captured only the computational, and the statistical complexity captured only the distributional. In addition, some definitions were not computable; some were not rigorously specified; and any of them treated either probabilistic or deterministic forms of objects, but not both in a unified manner. This paper presents a new quantitative definition of organized complexity. In contrast to the existing definitions, this new definition simultaneously captures all of the three key features of complexity for the first time. In addition, the proposed definition is computable, is rigorously specified, and can treat both probabilistic and deterministic forms of objects in a unified manner or seamlessly. The proposed definition is based on circuits rather than Turing machines and ɛ-machines. We give several criteria required for organized complexity definitions and show that the proposed definition satisfies all of them.