A Computational Theory of Modelling
Axel G. Rossberg · AIP conference proceedings · 2003
A metatheory is developed which characterizes the relationship between a modelled system, which complies with some “basic theory”, and a model, which does not, and yet reproduces important aspects of the modelled system. A model is represented by an (in a certain sense, s.b.) optimal algorithm which generates data that describe the model’s state or evolution complying with a “reduced theory”. Theories are represented by classes of (in a similar sense, s.b.) optimal algorithms that test if their input data comply with the theory. The metatheory does not prescribe the formalisms (data structure, language) to be used for the description of states or evolutions. Transitions to other formalisms and loss of accuracy, common to theory reduction, are explicitly accounted for. The basic assumption of the theory is that resources such as the code length (≈ programming time) and the computation time for modelling and testing are costly, but the relative cost of each recourse is unknown. Thus, if there is an algorithm a for which there is no other algorithm b solving the same problem but using less of each recourse, then a is considered optimal. For tests (theories), the set X of wrongly admitted inputs is treated as another resource. It is assumed that X1 is cheaper than X2 when X1 ⊂ X2 (X1 ≠ X2). Depending on the problem, the algorithmic complexity of a reduced theory can be smaller or larger than that of the basic theory. The theory might help to distinguish actual properties of complex systems from mere mental constructs. An application to complex spatio‐temporal patterns is discussed.