TCS Journal 2026 Journal Article
Image reflection on process graphs of 1-free regular expressions modulo bisimilarity
- Yuanrui Zhang
- Xinxin Liu
We study a phenomenon called “image reflection” on a type of characterization graphs — LLEE charts — for 1-free regular expressions modulo bisimularity. This property, stating that the images of a bisimulation function from an LLEE chart actually impose a special LEE structure corresponding to the LLEE chart, is recognized by our proposed “well-structured looping-back charts” as a sub-LLEE-structure of LLEE charts. As an application, our study naturally leads to a novel proof for the completeness of the inference system BBP for 1-free regular expressions, due to the correspondence between 1-free regular expressions and the provable solutions of LEE/LLEE charts. Compared to the previous approach, our proof is more direct in the sense that it does not rely on a graph transformation procedure on LLEE charts in which at each step two bisimilar nodes have to be carefully selected and merged together according to selection rules. Our observation on LLEE charts is useful to understand the completeness problems of regular expressions modulo bisimilarity from a new angle, and can be also helpful for solving the completeness problems of other expressions that share similar graph structures.