Symbolic Partition Refinement with Dynamic Balancing of Time and Space

Bisimulation minimization is one of the classical means to fight the infamous state space explosion problem in verification. Particularly in stochastic verification, numerical algorithms are applied, which do not scale beyond systems of moderate size. To alleviate this problem, symbolic bisimulation minimization has been used effectively to reduce very large symbolically represented state spaces to moderate size explicit representations. But even this minimization may fail due to time or memory limitations. This paper presents a symbolic algorithm which relies on a hybrid symbolic partition representation. It dynamically converts between two known representations in order to provide a trade-off between memory consumption and runtime. The conversion itself is logarithmic in the partition size. We show how to apply it for the minimization of Markov chains, but the same techniques can be adapted in a straightforward way to other models like labeled transition systems or interactive Markov chains.