On Uncertainty of Dynamic Systems via State Aggregation Coarse-Graining and State Decomposition Fine-Graining Ways

Uncertainty is an important feature of dynamic systems, and entropy has been widely used to measure this attribute. In this Letter, we prove that state aggregation and decomposition can decrease and increase the entropy, respectively, of dynamic systems. More than 20 popular entropies in the literature are summarized and analyzed, and it is noted that none of them breaks this property. Finally, pertinent proofs are given for four cases.