Abstract
We introduce parallel symbolic algorithms for bisimulation minimisation, to combat the combinatorial state space explosion along three different paths. Bisimulation minimisation reduces a transition system to the smallest system with equivalent behaviour. We consider strong and branching bisimilarity for interactive Markov chains, which combine labelled transition systems and continuous-time Markov chains. Large state spaces can be represented concisely by symbolic techniques, based on binary decision diagrams. We present specialised BDD operations to compute the maximal bisimulation using signature-based partition refinement. We also study the symbolic representation of the quotient system and suggest an encoding based on representative states, rather than block numbers. Our implementation extends the parallel, shared memory, BDD library Sylvan, to obtain a significant speedup on multi-core machines. We propose the usage of partial signatures and of disjunctively partitioned transition relations, to increase the parallelisation opportunities. Also our new parallel data structure for block assignments increases scalability. We provide SigrefMC, a versatile tool that can be customised for bisimulation minimisation in various contexts. In particular, it supports models generated by the high-performance model checker LTSmin, providing access to specifications in multiple formalisms, including process algebra. The extensive experimental evaluation is based on various benchmarks from the literature. We demonstrate a speedup up to 95times for computing the maximal bisimulation on one processor. In addition, we find parallel speedups on a 48-core machine of another 17times for partition refinement and 24times for quotient computation. Our new encoding of the reduced state space leads to smaller BDD representations, with up to a 5162-fold reduction.
Highlights
One of the main challenges for model checking is that the space and time requirements of model checking algorithms increase exponentially in the size of the models
There, we demonstrated that specialised binary decision diagrams (BDDs) operations for signature refinement provide a major speedup of the sequential algorithm, and scale across multiple processors
Similar to computing the partition, we find that quotient computation benefits from specialised BDD operations
Summary
One of the main challenges for model checking is that the space and time requirements of model checking algorithms increase exponentially in the size of the models. Note that branching bisimulation preserves the branching structure of an LTS, T. van Dijk, J. van de Pol preserving all properties expressible in CTL*-X [14] These notions correspond to strong and branching lumping for IMCs. The reduced state space consists of (representatives of) the equivalence classes in the largest bisimulation, which is typically computed using partition refinement. We find a speedup of 2–10× by using specialised operations, and we find significantly smaller BDDs (up to 5162× smaller) when using a representative state rather than the block number to encode the new transition system
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