Causal Groupoid Symmetries

Abstract

Proposed here is a new framework for the analysis of complex systems as a non-explicitly pro-grammed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measure-ment in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it de-scribes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, self-organization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are con-firmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as sys-tem autonomy, robotics, systems integration, and image and voice recognition.