Graph-Based Algorithms for Boolean Function Manipulation
IEEE Transactions on Computers
Multi-Terminal Binary Decision Diagrams: An Efficient DataStructure for Matrix Representation
Formal Methods in System Design
Symbolic Bisimulation Minimisation
CAV '92 Proceedings of the Fourth International Workshop on Computer Aided Verification
A Data Structure for the Efficient Kronecker Solution of GSPNs
PNPM '99 Proceedings of the The 8th International Workshop on Petri Nets and Performance Models
Optimal state-space lumping in Markov chains
Information Processing Letters
Lumping Matrix Diagram Representations of Markov Models
DSN '05 Proceedings of the 2005 International Conference on Dependable Systems and Networks
Solution of large markov models using lumping techniques and symbolic data structures
Solution of large markov models using lumping techniques and symbolic data structures
Compositional Performability Evaluation for STATEMATE
QEST '06 Proceedings of the 3rd international conference on the Quantitative Evaluation of Systems
Interactive Markov chains: and the quest for quantified quality
Interactive Markov chains: and the quest for quantified quality
Symmetry reduction for probabilistic model checking
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
Sigref: a symbolic bisimulation tool box
ATVA'06 Proceedings of the 4th international conference on Automated Technology for Verification and Analysis
PRISM: a tool for automatic verification of probabilistic systems
TACAS'06 Proceedings of the 12th international conference on Tools and Algorithms for the Construction and Analysis of Systems
Probabilistic model checking of complex biological pathways
Theoretical Computer Science
Proceedings of the 2nd international conference on Performance evaluation methodologies and tools
Reflection symmetry detection to reduce the state space of Markovian models
Proceedings of the 47th Annual Southeast Regional Conference
Symbolic partition refinement with automatic balancing of time and space
Performance Evaluation
Correctness issues of symbolic bisimulation computation for markov chains
MMB&DFT'10 Proceedings of the 15th international GI/ITG conference on Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance
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Many approaches to tackle the state explosion problem of Markov chains are based on the notion of lumpability, which allows computation of measures using the quotient Markov chain, which, in some cases, has much smaller state space than the original one. We present, for the first time, a symbolic algorithm and its implementation for the lumping of Markov chains that are represented using Multi-Terminal Binary Decision Diagrams. The algorithm is optimal, i.e., generates the smallest possible quotient Markov chain. Our experiments on various configurations of two example models show that the algorithm (1) handles significantly larger state spaces than an explicit algorithm, (2) is in the best case, faster than an efficient explicit algorithm while not prohibitively slower in the worst case, and (3) generates quotient Markov chains that are several orders of magnitude smaller than ones generated by a model-dependent symbolic lumping algorithm.