Complexity and Scientific Modelling

1 Overview

There have been many attempts at formulating measures of complexity*1 of physical processes (or more usually of the data models composed of sequences of measurements made on them). Here we reject this direct approach and attribute complexity only to models of these processes in a given language, to reflect its "difficulty". This means that it is a highly abstract construct relative to the language of representation and the type of difficulty that concerns one.

A framework for modelling is outlined which includes the language of modelling, the complexity of models in that language, the error in the model's predictions and the specificity of the model (which roughly corresponds to its refutability or to the information it gives about the process).

Many previous formulations of complexity can be seen as either:

a special case of this framework using particular modelling languages and focusing on particular types of difficulty;
attempts to "objectify" complexity by considering only minimally complex models or its asymptotic behaviour;
relativising it to a fixed mathematical structure in the absence of noise;
misnamed in that they capture the specificity rather than the complexity.

Such a framework makes sense of a number of aspects of scientific modelling.

As a result complexity is not situated between order and disorder, as several authors have assumed, but rather such judgements arise given certain natural assumptions about the language of modelling and the desirable trade-offs between the model complexity, its specificity and its error rate.
Noise can be seen as that which is unpredictable given the available resources of the modeller. In this way noise is distinguished from randomness. Different ways of practically distinguishing noise can thus be seen as resulting from different trade-offs between complexity, error, specificity and the choice of modelling language.
Complexity is distinguished from concerns of specificity such as refutability, entropy and information. Complexity is thus seen to have context-dependent relations with such measures but in general is independent from them.
Less complex models are not a priori likely to be more accurate, but rather that given the typical structure of expressive modelling languages and our limitations in searching through such languages, choosing the simpler model can be a useful heuristic.

Complexity and Scientific Modelling - 04 APR 97

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