TITLE:
Causal Groupoid Symmetries
AUTHORS:
Sergio Pissanetzky
KEYWORDS:
Hierarchies; Groupoids; Symmetry; Causality; Intelligence; Adaptation; Emergence
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.4,
March
10,
2014
ABSTRACT: Proposed here is a new framework for the analysis of
complex systems as a non-explicitly programmed mathematical hierarchy of
subsystems using only the fundamental principle of causality, the mathematics
of groupoid symmetries, and a basic causal metric needed to support measurement
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 describes 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, selforganization, adaptation, intelligence, and semantics.
The theory and a thought experiment are discussed and previous evidence is
referenced. Several predictions in the human brain are confirmed with wide
experimental bases. Applications are anticipated in many disciplines, including
Biology, Neuroscience, Computation, Artificial Intelligence, and areas of
Engineering such as system autonomy, robotics, systems integration, and image
and voice recognition.