That "all information is not of equal value" implies meaning, or
semantics. The AIC and "depth" measures are strictly syntactic.
Semantics basically involves categorization and context.
In the most general terms, it seems to me that complexity could be
measured by the number of different functional relationships a system
can maintain, and might further be weighted by the number of functions
it can perform simultaneously.
As for getting a handle on this mathematically, set theory seems to be
helpful for analyzing relationships between things other than numbers.
You might enjoy Luis Rocha’s essay "Evidence Sets: Contextual
Categories" at http://ssie.binghamton.edu/~rocha/es_contx.html which
describes how
<Evidence Sets are set formalisms which extend fuzzy sets and interval
valued fuzzy sets through the Dempster-Shafer theory of evidence (DST)
[and are] well positioned to model human linguistic categorization
processes because they offer a mechanism to explicitly contextualize
uncertainty. Each of the graded intervals of membership is associated
with a particular context, which can then be reduced or eliminated as
more evidence is gained through an extended theory of approximate
reasoning or through the incorporation of more knowledge into the
model by pragmatic, evolutionary, strategies.>
JKC:
<I also think it would be a mistake to equate complexity with virtue,
in fact all things being equal, the opposite is true. A simple program
is better than a complex one that does the same thing. A simple
ecosystem such as a temperate forest of oaks and conifers is more
resilient than a very complex one like a tropical rain forest.>
The temperate forest is more resilient than the tropical forest by
whose measure (survivability against current human agricultural
practices?) What is "the same thing" that these two types of forests
do (provide a green coloring on the map?) I’m sure ecologists have
better measures of forest complexity, I’m just not familiar with them.
As for programs: I can write a terse program with lots of hardcoded
constants and clever optimized algorithms that is simpler than another
program performing the same fixed function, and probably much more
efficient in its execution speed; but, the more complex program, which
might have much more parameterization and be more data-driven by
looking up values in external files, is going to be more flexible and
possibly more maintainable.
Simplicity IS virtuous at a certain level of abstraction and
operation, but monotonic functions usually tend to get assembled (or
gobbled up) into more complex organizations.
Bruce Edmonds argues in his "What is Complexity?" essay at
http://www.fmb.mmu.ac.uk/~bruce/evolcomp/evolcomp.txt
that
<Complexity . . . is unlikely to have any useful value as applied to
"real" objects or systems. Further that even relativising it to an
observer has problems. It is proposed that complexity can usefully be
applied only to constructions within a given language. It is argued
that complexity is usefully differentiated from the concepts of size,
ignorance, variety, minimum description length and order. A definition
of complexity is proposed which can be summarised as "that property of
a language expression which makes it difficult to formulate its
overall behaviour even when given almost complete information about
its atomic components and their inter-relations." Some of the
consequences of this definition are discussed.>
Perhaps you are really saying, as Edmonds put it, that "the reduction
in complexity of our interaction with any part of reality is an
essential component of our existence"; or, the more simplicity we can
achieve in our interactions the more fit we will be (Kauffman’s
"number of conflicting constraints" notion of complexity). But, it’s
also wise to remember that this makes us more dependent on all these
‘black boxes’ or infrastructure that we’ve built to simplify things
for us.
Mark Crosby
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