Partitions, Rational Partitions, and Characterization of Complexity

Abstra ct. Th e quantitat ive characterizat ion of complexity for several dynami c processes is approached by a mathematical scheme based on th e met ric and combinatori al prop ert ies of the space Z of finite measur able par titions . Precisely, Z is embed ded in a larger spa ce R of new objects, "rational part it ions," describin g the anti similarity between two prob abilistic experiments . Rational par titions consist of properly reduced couples of ordinary par titions, and th eir mai n features are briefly reviewed. An ext ended ent ropy functional in R allows the introdu ction of indicators sensitive to different possible aspects of complexity for cellular auto mat a, shifts, mapp ings, etc. Th ese indicators appear to be accessible to numerical experiments in many nontrivial situations .