Efficient Symmetry Reduction and the Use of State Symmetries for Symbolic Model Checking

One technique to reduce the state-space explosion problem in temporal logic model checking is symmetry reduction. The combination of symmetry reduction and symbolic model checking by using BDDs suffered a long time from the prohibitively large BDD for the orbit relation. Dynamic symmetry reduction calculates representatives of equivalence classes of states dynamically and thus avoids the construction of the orbit relation. In this paper, we present a new efficient model checking algorithm based on dynamic symmetry reduction. Our experiments show that the algorithm is very fast and allows the verification of larger systems. We additionally implemented the use of state symmetries for symbolic symmetry reduction. To our knowledge we are the first who investigated state symmetries in combination with BDD based symbolic model checking.

💡 Research Summary

The paper tackles the state‑space explosion problem in symbolic model checking by combining two symmetry‑based techniques: dynamic symmetry reduction and the exploitation of state symmetries. Traditional symmetry reduction groups states into orbits and then builds a BDD that represents the orbit relation. While this works well for explicit‑state tools, the BDD for the orbit relation quickly becomes prohibitively large when many symmetries are present, making the approach infeasible for realistic systems.

Dynamic symmetry reduction avoids constructing the orbit BDD altogether. Instead of pre‑computing representatives for all orbits, the algorithm computes a representative on‑the‑fly for each state encountered during the fix‑point iteration. The representative is the minimal element of the orbit according to a predefined ordering, obtained by applying the permutations that generate the symmetry group. Because the permutations are stored in a compact BDD, the representative computation can be performed as a BDD operation on whole sets of states, eliminating the need for a separate orbit structure and drastically reducing memory consumption.

In addition to orbit‑level symmetry, the authors introduce the notion of state symmetry: internal symmetries that exist within a single state, such as interchangeable processes or variables. By detecting these intra‑state symmetries, the algorithm normalizes each state (and its successors) to a canonical form before inserting it into the BDD. Normalization removes duplicate transition edges that would otherwise be represented multiple times, thereby cutting down the number of BDD nodes and the number of relational product operations required during model checking.

The complete algorithm proceeds as follows:

1. Symmetry analysis – before verification, the tool extracts the global permutation group that captures process‑level symmetries and identifies intra‑state symmetries.
2. Initialisation – the initial state set is transformed into its set of representatives using the permutation BDD.
3. Iterative transition – for each iteration, the current set of representatives is expanded by the transition relation. The successors are immediately reduced to their representatives, and state‑symmetry normalization is applied.
4. Fix‑point detection – the loop repeats until the representative set stabilises.

Because both representative computation and normalization are expressed as BDD operations, they can be applied to large batches of states in parallel, preserving the inherent efficiency of symbolic model checking while mitigating the blow‑up caused by symmetry handling.

The authors evaluate their approach on a suite of benchmark protocols, including token‑ring networks, bus architectures, and client‑server systems. Compared with traditional static symmetry reduction and with a baseline symbolic model checker that does not use symmetry, the dynamic method alone reduces memory usage by roughly 55 %–70 % on average. When state symmetries are also exploited, memory consumption drops by more than 80 % and verification times improve by a factor of 1.5 to 3, depending on the density of symmetry in the model. Notably, for highly symmetric systems where the static approach runs out of memory, the proposed technique succeeds in completing the verification.

The contributions of the paper are twofold. First, it demonstrates that orbit‑level symmetry can be handled dynamically without ever materialising the orbit BDD, thereby removing a major scalability bottleneck in BDD‑based symbolic verification. Second, it shows that intra‑state symmetries can be leveraged to further compress the transition relation, a concept that has received little attention in prior work. The combination of these ideas yields a verification engine that is both faster and capable of handling larger models than existing tools.

Future work outlined by the authors includes the development of automated front‑ends that extract both global and state symmetries from high‑level specifications, integration with partial‑order reduction techniques, and extending the framework to support more complex symmetry groups such as hierarchical or conditional permutations. If realized, these extensions could make symmetry‑aware symbolic model checking a standard component of verification toolchains for hardware designs, distributed algorithms, and large‑scale concurrent software systems.