Bisimulation Checking in Bigraph Tools

• Bisimulation checking in bigraph-based tools is a method for verifying behavioral equivalence using minimal contextual transition systems derived from bigraphical reactive systems.
• It models both spatial and non-spatial relationships by applying reaction rules to agents, thereby enabling a structured approach to agent verification.
• Advanced algorithms, such as the adapted McSplit for solving the Maximum Common Bigraph Problem, offer efficient construction and comparison of minimal transition systems.

Bisimulation checking within bigraph-based tools focuses on determining behavioral equivalence between agents modeled as bigraphs, particularly via their minimal contextual transition systems (MCTSs). Bigraphical reactive systems (BRSs) provide a compositional formalism for modeling both spatial and non-spatial relationships, facilitated by agents and reaction rules operating on them. Recent methodological advances have developed automated approaches for identifying bisimilar agents by efficiently constructing and comparing MCTSs; central to this is the Maximum Common Bigraph Problem (MCBP) and the adaptation of advanced induced subgraph algorithms for its resolution (Burns et al., 7 Jan 2026).

1. Formalization of Bisimulation in Bigraphical Reactive Systems

BRSs deploy the pair $(A, \mathcal{R})$ where $$A : \langle m, X \rangle \rightarrow \langle n, Y \rangle$$ denotes the initial agent bigraph and $\mathcal{R}$ a finite set of reaction rules $R \rightarrow R'$. The notion of a raw transition system is established by direct application of reaction rules, $A \to_0 G'$, capturing matches of $R$ within $A$ and subsequent rewrite. However, this approach does not account for arbitrary environmental composition.

Milner’s Contextual Transition System (CTS) introduces context-sensitive transitions, formalized as $A \xrightarrow{(f, j)} A'$, reflecting the action of first embedding $A$ into a context $f$ and then applying the reaction. Notably, the number of possible contexts $f$ is unbounded, making computation intractable if naively enumerated. The refinement to Minimal CTS (MCTS) restricts to minimal contexts sufficient for enabling transitions, defined by the absence of any smaller context preserving the match.

Bisimulation is defined by the existence of a relation $\sim$ such that $A \sim B$ implies for every minimal transition out of $A$, there is a matching transition out of $B$ (and vice versa) under the same minimal context, with resulting agents again related. Milner (2009) established that bisimilarity in this setting corresponds to isomorphism of the agents' MCTSs as labeled trees, enabling polynomial-time bisimulation checking once the MCTSs are constructed.

2. The Maximum Common Bigraph Problem (MCBP)

Bisimulation checking and MCTS construction depend on identifying minimal contexts, whose computation requires finding the largest substructure shared between the reaction redex $R$ and the agent state $A$. This is formalized as the Maximum Common Bigraph Problem (MCBP).

A concrete bigraph $$B = (V_B, E_B, \mathrm{ctrl}_B, \mathrm{prnt}_B, \mathrm{link}_B) : \langle m, X \rangle \rightarrow \langle n, Y \rangle$$ splits into (i) a place graph—a directed forest over entities expressing parental structure and controls, and (ii) a link graph—a hypergraph wiring ports to names and closed edges.

Given two solid bigraphs $G_1$, $G_2$, an $(G_M, M) \in \mathrm{MCB}(G_1, G_2)$ is defined such that $G_M$ is a solid bigraph of maximal support size, with embeddings decomposing both $G_1$ and $G_2$ via $G_M$ and their respective contexts. No strictly larger or more composed $G'_M$ admits such a decomposition.

MCBP generalizes the classic maximum common induced subgraph problem and is NP-hard, following the NP-completeness of the bigraph matching problem.

3. Algorithmic Solution: Adapting McSplit to MCBP

MCBP is algorithmically addressed by reduction to a maximum common induced subgraph (MCIS) problem on suitably encoded graphs, operationalized via an adaptation of the McSplit branch-and-bound algorithm.

• Place Graph Encoding: Sites and regions are discarded; only parent-child entity structure and control labels are encoded. The directed graph $\varphi_P(G)$ represents the place graph with labels $\ell(v)$.
• Link Graph Encoding: Each closed hyperedge is represented by a closure-node, ports become explicit nodes, and edges encode both entity-port and port-closure relations. Outer-name links are handled post-matching. Labels distinguish entity, port, and closure types.

Composition constraints are enforced during search:

1. Connectivity/Tensor-Product: Future matches must correspond either to adjacency in the place graph (maintaining connectivity) or be entirely disjoint (allowing tensor-product composition). Descendant relations are statically precomputed as bit-parallel matrices.
2. Closure Discipline: A closure-node (i.e., closed hyperedge) can be matched only after all its corresponding ports are matched, implemented by annotating degree classes and toggling visibility based on match coverage.

Modified scoring addresses artificial inflation of the vertex set due to port- and closure-nodes. The objective function is redefined as $$\mathrm{score}(M) = |\{(u, v) \in M \mid \ell(u) \neq port\}|$$, discarding port-only matches for support size. Search branches are pruned when the upper bound on this score does not exceed known solutions.

Algorithmically, McSplit-MCB proceeds by initial encoding, dynamic label-class partitioning, recursive branch generation (subject to visibility and closure checks), bound computation, and match decoding via relative pushout (RPO).

Complexity is $O(k \cdot n!)$ for $n$ vertices, but empirical pruning makes the approach feasible for bigraphs with several hundred encoded vertices.

4. MCBP in MCTS Construction and Bisimulation Checking

Construction of the MCTS for an agent state $A$ under a reaction rule $R$ involves computing $G_M \in \mathrm{MCB}(R, A)$. The decomposition $R = C_R \circ (G_M \otimes \mathrm{id})$, $A = C_A \circ (G_M \otimes D_A)$ directly yields the minimal context $f = C_R$ for the labeled transition $A \xrightarrow{(f, j)} A'$. Exhaustive iteration over all rules and matches produces the full MCTS.

Bisimilarity is then determined by searching for a tree isomorphism (via a standard polynomial-time algorithm) between the respective MCTSs of two agents, pairing nodes and ensuring matching transitions under identical minimal contexts.

5. Integration with Bigraph-Based Verification Toolchains

Integration of MCBP solving within practical bigraph-based tools involves:

• Data Structures: Place graphs represented as adjacency lists with control labeling; link graphs via flattened vertices, adjacency lists, and bit-parallel descendant matrices.
• Label Classes and Visibility: Implemented as contiguous array ranges with bitsets for fast label-class partitioning and candidate filtering.
• Optimization Strategies: Core branch-and-bound reimplemented in C/C++ using bitsets achieves $10$–$100\times$ speedup. An iterative McSplit$^\downarrow$ variant supports interactive queries with dynamic size targets. Heuristic ordering by degree or PageRank accelerates branching; MCB queries for multiple agent-rule pairs execute in parallel. Caching and reuse of MCB results further reduce computation in repeated bisimulation queries.
• Performance Characteristics: The Python/NetworkX prototype efficiently solves up to $\approx 300$-vertex agents in less than 1s, and $400$/2000 conference-call instances within 1s each. RPO reconstruction time is under $1\%$ of match time.

6. Experimental Evaluation and Properties

Application to several case studies demonstrates the practicality and expressiveness of the approach:

• Conference-Call Model: Over 2000 rule-agent pairs from the canonical example, with encoded sizes $59$–$1581$; the prototype solves $\sim 400$ instances in $<1$ s, and scaling is roughly linear to agent size up to $\approx 300$.
• Drone-Swarm Scenario: The MCBP-based MCTS reveals subtle transitions (e.g., a second drone outside a region triggering an alarm) missed by raw transition systems.
• Symmetry and Identity Checks: Identity property confirmed by surfacing $R$ itself as an MCB when fully matched; inverse rule property validated by obtaining inverse mappings under MCBP argument swapping.

7. Significance, Limitations, and Future Prospects

Automating minimal context search using MCBP and an adapted McSplit MCIS algorithm enables practical, scalable construction of MCTSs and bisimulation checking in bigraph-based tools. The approach handles moderately sized bigraphs interactively, and with further integration—such as optimized C/C++ implementations, parallelized MCB queries, and result caching—promises practical enablement of reduction, verification, and equivalence checking in toolchains like BigraphER.

A plausible implication is that further algorithmic refinement or exploitation of structural properties in domain-specific bigraphs could significantly extend feasible instance sizes for bisimulation checking in practice (Burns et al., 7 Jan 2026).