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Proposed here is a new framework for the analysis ofcomplex systems as a non-explicitly programmed mathematical hierarchy ofsubsystems using only the fundamental principle of causality, the mathematicsof groupoid symmetries, and a basic causal metric needed to support measurementin Physics. The complex system is described as a discrete set S of state variables. Causality isdescribed by an acyclic partial order w on S, and is considered as aconstraint on the set of allowed state transitions. Causal set S, wis the mathematical model of the system. The dynamics it describes isuncertain. Consequently, we focus on invariants, particularly group-theoreticalblock systems. The symmetry of S byitself is characterized by its symmetric group, which generates a trivial blocksystem over S. The constraint ofcausality breaks this symmetry and degrades it to that of a groupoid, which mayyield 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 acausal set with its own, smaller block system. Recursion yields a multilevelhierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariantbeing sought. The finding hints at a deep connection between the principle ofcausality and a class of poorly understood phenomena characterized by theformation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics.The theory and a thought experiment are discussed and previous evidence isreferenced. Several predictions in the human brain are confirmed with wideexperimental bases. Applications are anticipated in many disciplines, includingBiology, Neuroscience, Computation, Artificial Intelligence, and areas ofEngineering such as system autonomy, robotics, systems integration, and imageand voice recognition.


Hierarchies; Groupoids; Symmetry; Causality; Intelligence; Adaptation; Emergence

Cite this paper

Pissanetzky, S. 2014 Causal Groupoid Symmetries. Applied Mathematics, 5, 628-641. doi: 10.4236-am.2014.54059.

Autor: Sergio Pissanetzky



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