Dealing with Complexity. Peter Andras Department of Psychology University of Newcastle [email protected] Overview. What is complexity ? What is chaos ? Linking chaos and complexity Handling complexity: Information External and internal complexity Generating matching complexity.
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The buzz word ‘complexity’:
‘complexity of an NHS trust’ (Guardian, February 12, 2002)
‘increasing complexity in natural resource management’ (Conservation Ecology, January 2002)
‘citizens add an additional level of complexity’ (Political Behavior, March 2001)
The system of functional protein interaction clusters in the yeast (www.cellzome.com).
C. Elegans (devbio-mac1.ucsf.edu)
C. Elegans ventral ganglion transverse-section (www.wormbase.org)
Task: calculate the total area of the shape
Question: how many operations does it take to find the total area
The complexity of a computational problem is given by the length of the sequence of operations that are need to solve the problem.
In principle there is a universal way to to find out the complexity of a problem, which is the finding of the shortest program that a Turing machine requires to solve the problem. In practice this cannot be applied.
Problem: to describe a phenomenon, an object, a solution of some other problem
Description complexity: the length of the description in language units (words)
Description complexity depends on the language.
The buzz word ‘chaos’:
‘managing at the edge of chaos’ (Guardian, February 19, 2002)
‘the brain as a system near the edge of chaos’ (Journal of Consciousness Studies, July, 2000)
‘chaos theory is used metaphorically to address aspect of creative process’ (Creativity Research Journal, no.3-4, 2000)
Light reflection from four spheres (www-chaos.umd.edu)
Equational description of the phenomena or object:
E.g. xt+1 = xt * xt – yt * yt + a
yt+1 = 2 * xt * yt + b
It is chaotic if shows sensitive dependency on the initial conditions (on x0 and y0 in the case above).
Sensitive dependency: small initial changes may lead to large changes later.
Key feature of mathematical chaos:
unpredictability, due to the sensitive dependency on initial conditions
Practical unpredictability links the deterministic chaos to the randomness.
z = (x,y)
z0 = z
zt+1 = zt * zt + z0
n(z) is the first t for which ||zt|| > 4
Different languages can be used to describe the same chaotic phenomena. These languages may differ in the length of descriptions that are required for a given level of precision.
Mathematical chaos appears complex if we use a description language which requires long descriptions to achieve a desired level of precision.
The unpredictability of chaos guarantees that the description of the chaos requires long descriptive sequences in most of the languages.
Finding the appropriate language that allows compact description of a given chaotic phenomena is far from trivial, and the search for the appropriate language may be itself very complex (in the sense of Turing computability).
How to survive within an environment ?
How to find the appropriate response to a given environmental situation ?
First step: DIFFERENTIATING between states of the environment.
Being able to differentiate between environmental states means that information can be gathered about the environment.
Having the criteria for differentiation is not enough.
It is important that the observer is able to make a choice, and use the criteria for differentiation to choose between the possible states of the environment. On the basis of this choice the observer provides the appropriate response.
Information is the difference that makes a difference. (Bateson, 1971)
The environment is described by a sequence of choices for the observer.
The choices are the description language units for the observer.
The length of the choice sequences gives the perceived complexity of the environment for the observer.
The level of inappropriateness of the selected responses shows to what extent the description language of the observer captures the real complexity of the environment.
The observer always ignores those features of the environment that cannot be evaluated using its differentiation criteria. The ignored features create the mismatch between the perceived and real complexity of the environment.
Information processing means the effectuation of choices.
Information processing structures of the observer perform this choices by selecting their own appropriate action.
The information processing structures deal with a restricted complexity environment that is the real environment filtered through an appropriate part of the description language
Such structures are specialized on processing of information that can be gathered by applying a restricted set of the differentiation options and related choices.
From the point of view of the legal system the single issue is whether something is legal or illegal.
If an action or a state cannot be assessed from this point of view, it is just outside of the interest of the legal system, it is ignored.
In all cases when an action or state falls in the interest of the legal system, the single question about it is whether it is legal or not, and all the investigations consider only the definitions, guidelines, rules and other formal components of the legal system in deciding the legality.
The external complexity is the complexity of the environment.
Environment: the Earth, a city, the university, a cave, a skin patch of an animal, the programs that are running in a computer, etc.
The internal complexity is the complexity of the information processing structures of a system that acts in an environment.
Internal complexity: system of adaptive protein folding, sensory discrimination and adaptive response generation system of an animal, the internal structure and functional rules of an organisation, the input interpretation and output generation system of a computer program
The information processing structures and processes can be viewed as a description language of the external complexity.
This language gives the perceived complexity of the environment.
The system performs well in an environment (its adaptive actions fit to the environmental conditions) if its internal complexity matches the external complexity of the environment.
The information processing sub-system of the system can be described in many ways, using many description languages. The key is that using at least one of these languages the internal complexity fits to a good extent the external complexity.
In a world of co-existing systems (e.g., individual animals, animal species, cars, etc.) those have better survival chances that gather and process information more efficiently.
Those systems that develop internal complexity that matches better their external complexity perform better, their adaptive responses fit better to the environmental challenges.
If these systems reproduce (directly or indirectly), those will become dominant that have better information processing sub-systems.
The evolution of information processing sub-systems can be viewed as an evolution of world descriptions.
The better world descriptions capture better the real complexity of the environment (the relevant part of the world for some systems inhabiting that part).
Generating better descriptions of the world the systems are able to deal more efficiently with the complexity of their environment.
Mutant genes may encode proteins that are able to transform environmental energy resources into internal energy storage of the cell.
Cells having these mutant genes have a better description of their environment, and will conquer the living space of those which do not have the mutant genes.
The governing parties change at elections. Those get more votes who provide a better description of the world, which fits better the perceived complexity of the world, as it is perceived by the voting public.
Those politicians who use a better language to address the real world problems are more likely to be voted and more likely to put in practice their views as members of the government.
How to build / generate information processing sub-systems that capture the complexity of the environment to a large extent ?
How to generate internal complexity that matches the external complexity ?
Deterministic chaos offers the advantage that it has a very simple and a very complex description.
Generating it is simple, by using the simple description language.
It can be used to capture high complexity descriptions by using the complex description of the deterministic chaos.
A simple way to generate complex deterministic chaos is by the application of self-similar expansions at smaller scales.
Self-reference is the extension of the self-similarity concept to generic systems.
Systems operating by self-referential expansion are able to generate high complexity deterministic chaos that can capture high levels of environmental complexity.
Self-referential expansion is possible in effective way if the simple description language is available.
Highly specialized components of the information processing sub-system can develop their internal language that is highly standardized and communicates information efficiently (with low ambiguity and in a compressed form).
Such standardized internal languages can constitute the basis for effective self-referential expansion.
Two description languages have matching complexity if their information encoding capacity is close.