Abstract
We provide an algorithm to efficiently compute bisimulation for probabilistic labeled transition systems, featuring non-deterministic choice as well as discrete probabilistic choice. The algorithm is linear in the number of transitions and logarithmic in the number of states, distinguishing both action states and probabilistic states, and the transitions between them. The algorithm improves upon the proposed complexity bounds of the best algorithm addressing the same purpose so far by Baier, Engelen and Majster-Cederbaum (Journal of Computer and System Sciences 60:187–231, 2000). In addition, experimentally, on various benchmarks, our algorithm performs rather well; even on relatively small transition systems, a performance gain of a factor 10,000 can be achieved.
Highlights
In [1], Larsen and Skou proposed the notion of probabilistic bisimulation
We provide a new algorithm for probabilistic bisimulation for probabilistic labeled transition system (PLTS) of time complexity O (m a + m p ) log n p + m p log n a ) and space complexity O m a + m p, where n a is the number of states, m a the number of transitions labelled with actions, n p the number of distributions and m p the cumulative support of the distributions
As the algorithm restricts the handling of distributions to the states in the support of the distributions, the running time of the algorithm compares favourably when the fan-out is low in the PLTS under consideration, a situation occurring frequently in practice
Summary
In [1], Larsen and Skou proposed the notion of probabilistic bisimulation. described for deterministic transition systems, the same notion is very suitable for probabilistic transition systems with nondeterminism [2,3], i.e. so-called PLTSs. When distributions only touch a limited number of states, as is often the common situation, the implementation of our algorithm outperforms our implementation of the algorithm in [4], both in time as well as in space complexity. Using a brilliant, yet simple argument, taken from [6], the number of times a probabilistic transition is sorted can be limited by the fan-out of the source state of the transition This leads to the observation that we can use straightforward sorting without the need of any tailored data structure such as augmented ordered balanced trees or similar as in [4,7]. Memory-wise the implementation of our algorithm outperforms the implementation in [4] when the sizes of the probabilistic state space are larger Both findings are in line with the theoretical complexity analyses of both algorithms. Both implementations have been incorporated in the open source mCRL2 toolset [9,10]
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