[Paper Review] On Bisimulation in Absence of Restriction

Xian Xu|arXiv (Cornell University)|Oct 19, 2022

Petri Nets in System Modeling4 citations

TL;DR

This paper investigates weak bisimilarity in restriction-free process calculi, showing it remains strictly coarser than strong bisimilarity due to the presence of replication, which enables state retention and internal action persistence. It introduces quasi-strong bisimilarity—requiring strong matching of internal actions and no trailing internal steps—and proves it coincides exactly with weak bisimilarity in CCS⁻ and branching bisimilarity, revealing that weak bisimilarity retains significant discriminatory power even without restriction.

ABSTRACT

We revisit the standard bisimulation equalities in process models free of the restriction operator. As is well-known, in general the weak bisimilarity is coarser than the strong bisimilarity because it abstracts from internal actions. In absence of restriction, those internal actions become somewhat visible, so one might wonder if the weak bisimilarity is still 'weak'. We show that in both CCScore (i.e., Milner's standard CCS without $τ$-prefix, summation and relabelling) and its higher-order variant (named HOCCScore), the weak bisimilarity indeed remains weak, i.e., still strictly coarser than the strong bisimilarity, even without the restriction operator. These results can be extended to other first-order or higher-order process models. Essentially, this is due to the direct or indirect existence of the replication operation, which can keep a process retaining its state (i.e., capacity of interaction). By virtue of these observations, we examine a variant of the weak bisimilarity, called quasi-strong bisimilarity. This quasi-strong bisimilarity requires the matching of internal actions to be conducted in the strong manner, as for the strong bisimilarity, and the matching of visible actions to have no trailing internal actions. We exhibit that in CCScore without the restriction operator, the weak bisimilarity exactly collapses onto this quasi-strong bisimilarity, which is moreover shown to coincide with the branching bisimilarity. These results reveal that in absence of the restriction operation, some ingredient of the weak bisimilarity indeed turns into strong, particularly the matching of internal actions.

Motivation & Objective

• To investigate whether weak bisimilarity remains strictly coarser than strong bisimilarity in process models without the restriction operator.
• To analyze the role of replication in preserving the distinction between weak and strong bisimilarity in the absence of restriction.
• To introduce and formalize a new variant of bisimilarity—quasi-strong bisimilarity—that strengthens weak bisimilarity by enforcing strong matching of internal actions and eliminating trailing internal actions.
• To establish the coincidence of weak bisimilarity, quasi-strong bisimilarity, and branching bisimilarity in CCS⁻ without restriction.
• To explore the potential for similar results in higher-order process models like HOCCS⁻ and to identify conditions under which weak and strong bisimilarities may coincide.

Proposed method

• Formalizing CCS⁻ and HOCCS⁻ as restriction-free variants of standard CCS and CHOCS, excluding τ-prefix, summation, and relabelling.
• Defining quasi-strong bisimilarity as a refinement of weak bisimilarity that enforces strong simulation for internal (τ) actions and prohibits trailing τ-steps in visible action matching.
• Proving that in CCS⁻, weak bisimilarity coincides exactly with quasi-strong bisimilarity, using structural and operational reasoning on process terms.
• Establishing the equivalence of quasi-strong bisimilarity and branching bisimilarity in CCS⁻ through a series of simulation and coinduction arguments.
• Extending the analysis to HOCCS⁻, where replication is derived, and discussing the technical challenges in defining a comparable quasi-strong bisimilarity.
• Using the replication operation as a key mechanism to maintain process state and enable persistent internal behavior, thereby preserving the gap between weak and strong bisimilarity.

Experimental results

Research questions

• RQ1 Does weak bisimilarity remain strictly coarser than strong bisimilarity in process models without the restriction operator?
• RQ2 Can the replication operation in restriction-free calculi like CCS⁻ preserve the distinction between weak and strong bisimilarity despite the absence of hiding mechanisms?
• RQ3 Is there a refined form of weak bisimilarity that strengthens its requirements on internal actions while preserving its discriminating power?
• RQ4 Does quasi-strong bisimilarity coincide with weak bisimilarity and branching bisimilarity in restriction-free CCS?
• RQ5 Can a similar quasi-strong bisimilarity be defined and shown to coincide with weak bisimilarity in higher-order process models like HOCCS⁻?

Key findings

• In CCS⁻ without restriction, weak bisimilarity remains strictly coarser than strong bisimilarity, despite the absence of hiding mechanisms.
• The replication operation enables processes to retain state and sustain internal behavior, which preserves the distinction between weak and strong bisimilarity.
• Quasi-strong bisimilarity, which enforces strong matching of internal actions and no trailing internal steps, is introduced as a refined form of weak bisimilarity.
• In CCS⁻, weak bisimilarity exactly coincides with quasi-strong bisimilarity, demonstrating that the weak bisimilarity's discriminatory power is preserved under stricter matching conditions.
• Quasi-strong bisimilarity also coincides with branching bisimilarity in CCS⁻, revealing a tight structural relationship between these notions.
• The results suggest that replication can act as a form of implicit information hiding, maintaining the gap between weak and strong bisimilarity even without explicit restriction.