State Space Estimation for DPOR-based Model Checkers

We study the estimation problem for concurrent programs: given a bounded program $P$, estimate the number of Mazurkiewicz trace-equivalence classes induced by its interleavings. This quantity informs two practical questions for enumeration-based model checking: how long a model checking run is likely to take, and what fraction of the search space has been covered so far. We first show the counting problem is #P-hard even for restricted programs and, unless $P=NP$, inapproximable within any subexponential factor, ruling out efficient exact or randomized approximation algorithms. We give a Monte Carlo approach to obtain a poly-time unbiased estimator: we convert a stateless optimal DPOR algorithm into an unbiased estimator by viewing its exploration as a bounded-depth, bounded-width tree whose leaves are the maximal Mazurkiewicz traces. A classical estimator by Knuth, when run on this tree, yields an unbiased estimate. To control the variance, we apply stochastic enumeration by maintaining a small population of partial paths per depth whose evolution is coupled. We have implemented our estimator in the JMC model checker and evaluated it on shared-memory benchmarks. With modest budgets, our estimator yields stable estimates, typically within a 20% band, within a few hundred trials, even when the state space has $10^5$--$10^6$ classes. We also show how the same machinery estimates model-checking cost by weighting all explored graphs, not only complete traces. Our algorithms provide the first provable poly-time unbiased estimators for counting traces, a problem of considerable importance when allocating model checking resources.