The Image of the Process Interpretation of Regular Expressions Is Not Closed under Bisimulation Collapse

Milner (1984) defined a process semantics for regular expres- sions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Mil- ner’s system is complete, by motivating and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner’s system to ‘1-free’ regular expressions. As a cru- cial complication we recognize that process graphs with emp- ty-step transitions that satisfy the layered loop-existence and elimination property LLEE are not closed under bisim- ulation collapse (unlike process graphs with LLEE that only have proper-step transitions). We circumnavigate this ob- stacle by defining a LLEE-preserving ‘crystallization proce- dure’ for such process graphs. By that we obtain ‘near-col- lapsed’ process graphs with LLEE whose strongly connected components are either collapsed or of ‘twin-crystal’ shape. Such near-collapsed process graphs guarantee provable so- lutions for bisimulation collapses of process interpretations of regular expressions.