A Mathematical Structure for Emergent Computation

On an irreducible theory of complex systems

“…To understand a complex system we consider the dynamics of the elementary parts and focus on the correla- * Electronic address: v.korotkikh@cqu.edu.au † Electronic address: g.korotkikh@cqu.edu.au tions preserving certain quantities of the complex system [2], [3]. Let I be an integer alphabet and…”

“…It turns out that the second integrals are also equal f [2] (t8) = g [2] (t8), but the third integrals are not f [3] (t8) = g [3] (t8). Thus, two quantities remain invariant and…”

“…This fact is important in order to prove that C(x, x ′ ) < N [2]. The equations (3) present a special type of correlations that do not have reference to the distances between parts, local times and physical signals.…”

“…Through the hierarchical structures a new type of processes, i.e., the self-organization processes of prime integer relations, is revealed [2]. Starting with integers as the elementary building blocks and following a single ′ ) = 3.…”

“…By using the integer code series [4] the prime integer relations can be geometrized as two-dimensional geometric patterns and the self-organization processes can be isomorphically expressed by transformations of the patterns [2]. As it becomes possible to measure a prime integer relation by an isomorphic geometric pattern, quantities of the prime integer relation and a complex system it describes can be defined by quantities of the geometric pattern such as the area and the length of its boundary curve ( Figure 2).…”