non-commutative dynamic modeling diagrams

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0.1 Introduction

In an interesting report, Rosen(1987) showed that complex dynamical systems,such as biological organisms, cannot be adequately modelled througha commutative modelling diagram– in the sense of digitalcomputer 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 diagram for a simple dynamicalsystem included both the ‘encoding’ of the ‘real’ system$𝐍$ in ($𝐌$) as well as the ‘decoding’ of($𝐌$) back into $𝐍$:

where $\delta$ is the real system dynamics and $\mathrm{\aleph }$ is analgorithm implementing the numerical computation of themathematical 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 notesthe obvious presence of logical arguments and indeed (non-Boolean)‘schemes’ related to the entailment of organismic models, such asMR-systems, in the more recent books that were published last byRobert Rosen (1994, 2001, 2004). Further mathematical details are provided in the paper byBrown, Glazebrook and Baianu (2007). Furthermore, Elsasser (1980) pointed out a fundamental, logical difference between physical systems and biosystems or organisms: whereas the former are readily represented by homogeneous logic classes, living organisms exhibit considerable variability and can only be representedby heterogeneous logic classes. One can readily representhomogeneous logic classes or endow them with ‘uniform’ mathematicalstructures, but heterogeneous ones are far more elusive and may admita multiplicity of mathematical representations or possess variablestructure. This logical criterion may thus be useful for furtherdistinguishing simple systems from highly complex systems.

The importance of logic algebras, 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 presentation 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 aspart of either traditional physics or the complex, chaotic systems physicsthat are known to be fully deterministic. However, super-complex (biological) systems have the potential to manifest novel and counter–intuitive behavior such as in the manifestation of ‘emergence’, development/morphogenesis andbiological evolution. The precise meaning of supercomplex systems is formally defined herein the next section.

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

In an early report (Baianu and Marinescu, 1968), the possibilityof formulating a super–categorical unitary theory of systems(that is, of both simple and complex systems, etc.) was pointed out both interms of organizational structure and dynamics. Furthermore, itwas proposed that the formulation of any model or computer simulation ofa complex system– such as living organism or a society–involvesgenerating a first–stage logical model (not-necessarilyBoolean!), followed by a mathematical one, completewith structure (Baianu, 1970). Then, it was pointed out thatsuch a modeling process involves a diagram containingthe 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 mathematicalmodel $𝐌$; this initial modeling diagram may or maynot be commutative, and the modeling can be iterated throughmodifications of L, and/or $𝐌$, until anacceptable agreement is achieved between the behaviour of themodel and that of the natural, complex system (Baianu and Marinescu, 1968;Comoroshan and Baianu, 1969). Such an iterative modeling processmay ultimately converge to appropriate models of the complex system,and perhaps a best possible model could be attained as the categoricalcolimit of the directed family of diagrams generated through such a modellingprocess. The possible models $𝐋$, or especially$𝐌$, were not considered to be necessarily eithernumerical or recursively computable (that is, with an algorithm orsoftware program) by a digital computer (Baianu, 1971b, 1986-87).The mathematician John von Neumann regarded and defined complexity as ameasurable property of natural systems below the threshold ofwhich 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 assumption that natural systems are simple, or ‘mechanistic’, is too restrictive since ‘simple’ is applicable only to machines, 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 characteristic 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 toincreasing system ‘errors’. Just as for simple systems, both super–complex andultra-complex systems admit their own orders of causation,but the latter two types are different from the first–by inclusionrather than exclusion– of the mechanisms that control simple dynamicalsystems.