non-commutative dynamic modeling diagrams



0.1 Introduction

In an interesting report, Rosen(1987) showed that complex dynamical systems, such as biological organisms, cannot be adequately modelled through a commutativePlanetmathPlanetmathPlanetmathPlanetmath modelling diagram– in the sense of digital computer simulation–whereas the simple (‘physical’/ engineering) dynamical systems can be thus numerically simulated.

0.2 Non-commutative vs. commutative dynamic modeling diagrams

Furthermore, his modelling commutative diagramMathworldPlanetmath for a simple dynamical system included both the ‘encoding’ of the ‘real’ system 𝐍 in (𝐌) as well as the ‘decoding’ of (𝐌) back into 𝐍:

\xymatrix@C=5pc[SYSTEM]\ar[r]Encoding…\ar[d]δ&LOGICSMATHS.\ar[d]MSYSTEM&\ar[l][MATHS.MODEL]Decoding ,

where δ is the real system dynamics and is an algorithmMathworldPlanetmath implementing the numerical computation of the mathematical model (𝐌) on a digital computer. Firstly, one notes the ominous absence of the logical model, L, from Rosen’s diagram published in 1987. Secondly, one also notes the obvious presence of logical arguments and indeed (non-Boolean) ‘schemes’ related to the entailment of organismic models, such as MR-systems, in the more recent books that were published last by Robert Rosen (1994, 2001, 2004). Further mathematical details are provided in the paper by Brown, Glazebrook and Baianu (2007). Furthermore, Elsasser (1980) pointed out a fundamental, logical differencePlanetmathPlanetmath between physical systems and biosystems or organisms: whereas the former are readily represented by homogeneousPlanetmathPlanetmath logic classes, living organisms exhibit considerable variability and can only be represented by heterogeneous logic classes. One can readily represent homogeneous logic classes or endow them with ‘uniform’ mathematical structures, but heterogeneous ones are far more elusive and may admit a multiplicityMathworldPlanetmath of mathematical representations or possess variable structureMathworldPlanetmath. This logical criterion may thus be useful for further distinguishing simple systems from highly complex systems.

The importance of logic algebrasPlanetmathPlanetmath, and indeed of categories of logic algebras, is rarely discussed in modern Ontology even though categorical formulations of specific ontology domains such as biological Ontology and Neural Network ontology are being extensively developed. For a recent review of such categories of logic algebras the reader is referred to the concise presentationMathworldPlanetmathPlanetmath by Georgescu (2006); their relevance to network biodynamics was also recently assessed (Baianu, 2004, Baianu and Prisecaru, 2005; Baianu et al, 2006).

Super-complex systems, such as those supporting neurophysiological activities, are explained only in terms of non–linear, rather than linear causality. In some way then, these systems are not normally considered as part of either traditional physics or the complex, chaotic systems physics that are known to be fully deterministicMathworldPlanetmath. However, super-complex (biological) systems have the potential to manifest novel and counter–intuitive behavior such as in the manifestation of ‘emergence’, development/morphogenesis and biological evolution. The precise meaning of supercomplex systems is formally defined here in the next sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

0.3 Simple and super–complex dynamics: Closed vs. open systems

In an early report (Baianu and Marinescu, 1968), the possibility of formulating a super–categorical unitary theory of systems (that is, of both simple and complex systems, etc.) was pointed out both in terms of organizational structure and dynamics. Furthermore, it was proposed that the formulation of any model or computer simulation of a complex system– such as living organism or a society–involves generating a first–stage logical model (not-necessarily Boolean!), followed by a mathematical one, completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with structure (Baianu, 1970). Then, it was pointed out that such a modeling process involves a diagram containing the complex system, (CS) and its dynamics, a corresponding, initial logical model, L, ‘encoding’ the essential dynamic and/or structural properties of CS, and a detailed, structured mathematical model 𝐌; this initial modeling diagram may or may not be commutative, and the modeling can be iterated through modifications of L, and/or 𝐌, until an acceptable agreement is achieved between the behaviour of the model and that of the natural, complex system (Baianu and Marinescu, 1968; Comoroshan and Baianu, 1969). Such an iterative modeling process may ultimately converge to appropriate models of the complex system, and perhaps a best possible model could be attained as the categorical colimitMathworldPlanetmath of the directed family of diagrams generated through such a modelling process. The possible models 𝐋, or especially 𝐌, were not considered to be necessarily either numerical or recursively computable (that is, with an algorithm or software program) by a digital computer (Baianu, 1971b, 1986-87). The mathematician John von Neumann regarded and defined complexity as a measurable property of natural systems below the threshold of which systems behave ‘simply’, but above which they evolve, reproduce, self–organize, etc. It was claimed that any ‘natural’ system fits this profile. But the classical assumptionPlanetmathPlanetmath that natural systems are simple, or ‘mechanistic’, is too restrictive since ‘simple’ is applicable only to machinesMathworldPlanetmath, closed physicochemical systems, computers, or any system that is recursively computable. Rosen (1987) proposed a major refinement of these ideas about complexity by a more exact classification between ‘simple’ and ‘complex’. Simple systems can be characterized through representations which admit maximal models, and can be therefore re–assimilated via a hierarchy of informational levels. Besides, the duality between dynamical systems and states is also a characteristicPlanetmathPlanetmath of such simple dynamical systems. Complex systems do not admit any maximal model. On the other hand, an ultra-complex system– as applied to psychological–sociological structures– can be described in terms of variable categories or structures, and thus cannot be reasonably represented by a fixed state space for its entire lifespan. Simulations by limiting dynamical approximations lead to increasing system ‘errors’. Just as for simple systems, both super–complex and ultra-complex systems admit their own orders of causation, but the latter two types are different from the first–by inclusion rather than exclusion– of the mechanisms that control simple dynamical systems.

Title non-commutative dynamic modeling diagrams
Canonical name NoncommutativeDynamicModelingDiagrams
Date of creation 2013-03-22 18:12:52
Last modified on 2013-03-22 18:12:52
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 17
Author bci1 (20947)
Entry type Topic
Classification msc 03C98
Classification msc 03C52
Classification msc 18-00
Classification msc 03D80
Classification msc 03D15
Synonym non-abelian structures
Synonym non-commutative structuresPlanetmathPlanetmathPlanetmath
Synonym nonabelian structures
Synonym non-Abelian structures
Related topic AxiomsForAnAbelianCategory
Related topic SystemDefinitions
Related topic AxiomaticTheoryOfSupercategories
Related topic AlgebraicCategoryOfLMnLogicAlgebras
Related topic CategoricalOntology
Related topic NonCommutingGraphOfAGroup
Related topic SimilarityAndAnalogousSystemsDynamicAdjointness2
Related topic SystemDefinitions
Defines von Neumann complexity
Defines system dynamics
Defines model encoding algorithm
Defines model decoding
Defines homogeneous logic class
Defines Rosen complexity
Defines heterogeneous logic class
Defines modelling diagram
Defines complex system
Defines super-complex system