Structural Collapse in Rewrite Systems: A Study of Generalized Normalization

Many classical normalization procedures in computer algebra—Gaussian elimination, canonical
forms for term algebras, Gröbner bases—can be viewed as instances of a unified process of structural
collapse E↓, and structural entropy Hstruct in which a potentially vast rewrite space is compressed by
eliminating those rewrite rules that do not alter any normal form. This is motivated by utilizing how
unique something is relative to adjacent systems to understand its relative complexity. By counting
the number of distinct terms reachable for a given set of rules and an encoded system of terms, we
can quantify its inherent redundancy and compare it to other configurations. Iteratively modifying
rules within the available structure allows us to track the differential effects and follow a path to the
smallest possible count of terms.