Information Theory Laws: A Recollection

Information theory underpins modern communication, computation and complex systems, yet the structure of its governing inequalities remains an active area of research. This paper revisits the concept and mathematical foundations of Information Theory Laws, that is, constraints applied to the entropy function. Starting from Shannon’s seminal framework, we review the evolution from basic linear inequalities– i.e., polymatroid axioms– to the discovery of non-Shannon-type inequalities, which revealed that the Shannon region does not overlap with its closure region for n ≥ 4. We outline the geometric and algebraic representation of entropy spaces, discuss the critical threshold where complexity escalates, and highlight the role of machine-assisted verification in uncovering new inequalities that are not of Shannon-type. By tracing historical milestones and computational advances, this work provides a structured recollection of the spectrum of information inequalities and their implications for converse coding theorems and related applications. The study emphasizes the interplay between the theoretical developments and computational tools of the field in shaping the landscape of information theory.