Integrate renaming-based equivalence into the layered framework

Develop a method to integrate the renaming-of-letters equivalence reduction used to avoid representing multiple regular expressions that are equivalent modulo alphabet renaming (as in Kahrs’ catalogues) into Le Charlier’s layered framework for regular languages—comprising normalized expressions with unique integer identifiers and the background of equivalence classes and equations—so that such renaming-equivalent expressions are not redundantly represented. Specify concrete data structures and algorithms within the identifier-based architecture to support this integration and evaluate the resulting efficiency trade-offs for the core operations (union, concatenation, iteration, equation management, and DFA construction).

Background

The paper presents a two-layer framework that assigns unique integer identifiers to normalized regular expressions and maintains a background of equivalence classes and equations to unify expressions denoting the same language and to connect expressions with deterministic finite automata. This enables efficient manipulation and reuse of results across large sets of expressions.

In related work, Kahrs proposes using catalogues with a renaming strategy to avoid duplicating representations of expressions that are equivalent under permutations of alphabet letters. The author notes that, unlike tags and other features that can be mapped into the framework, this renaming-based symmetry reduction poses a challenge for integration with the current identifier-centric design and might incur significant efficiency costs. A concrete approach to incorporate such renaming equivalence without undermining the framework’s performance remains unspecified.