non-commutative dynamic modeling diagrams

non-commutative dynamic modeling diagrams

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curve

0.1

Introduction

In an interesting report, Rosen(1987) showed that

complex

dynamical systems

,
such as biological organisms, cannot be adequately modelled through
a

commutative

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 diagram

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

=

5

p

c

[

S

Y

S

T

E

M

]

\ar

[

r

]

Encoding…

↪

\ar

[

d

]

δ

&

L

O

G

I

C

S

⊕

M

A

T

H

S

.

\ar

[

d

]

ℵ

M

S

Y

S

T

E

M

&

\ar

[

l

]

[

M

A

T

H

S

.

□

M

O

D

E

L

]

Decoding

↩

…

,

where

δ

is the real

system dynamics

and

ℵ

is an

algorithm

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

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 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

multiplicity

of mathematical representations or possess

variable

structure

. This logical criterion may thus be useful for further
distinguishing 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 as
part of either traditional physics or the complex, chaotic systems physics
that 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 and
biological evolution. The precise meaning of supercomplex systems is formally defined here
in the next

section

.

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,

complete

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

colimit

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

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 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 structures

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

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