2016/08/21 by Hector Zenil, Héctor Zenil, Narsis A. Kiani +2 · 1 voice · 1 citation
Computer Science · Engineering · Mathematics · #Artificial Immune Systems Applications #Computability, Logic, AI Algorithms #Evolutionary Algorithms and Applications #cs.CC #cs.IT #math.CO
paper · pdf · doi:10.1103/physreve.96.012308
arxiv published 2016/08/21 · arxiv updated 2017/05/10 · openalex publication_date 2017/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In estimating the complexity of objects, in particular, of graphs, it is common practice to rely on graph- and information-theoretic measures. Here, using integer sequences with properties such as Borel normality, we explain how these measures are not independent of the way in which an object, such as a graph, can be described or observed. From observations that can reconstruct the same graph and are therefore essentially translations of the same description, we see that when applying a computable measure such as the Shannon entropy, not only is it necessary to preselect a feature of interest where there is one, and to make an arbitrary selection where there is not, but also more general properties, such as the causal likelihood of a graph as a measure (opposed to randomness), can be largely misrepresented by computable measures such as the entropy and entropy rate. We introduce recursive and nonrecursive (uncomputable) graphs and graph constructions based on these integer sequences, whose different lossless descriptions have disparate entropy values, thereby enabling the study and exploration of a measure's range of applications and demonstrating the weaknesses of computable measures of complexity.