## Algorithmic verification of recursive probabilistic state machines (2005)

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Venue: | In Proc. 11th TACAS |

Citations: | 40 - 7 self |

### BibTeX

@INPROCEEDINGS{Etessami05algorithmicverification,

author = {Kousha Etessami and Mihalis Yannakakis},

title = {Algorithmic verification of recursive probabilistic state machines},

booktitle = {In Proc. 11th TACAS},

year = {2005},

pages = {253--270},

publisher = {Springer}

}

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### Abstract

Abstract. Recursive Markov Chains (RMCs) ([EY04]) are a natural abstract model of procedural probabilistic programs and related systems involving recursion and probability. They succinctly define a class of denumerable Markov chains that generalize multi-type branching (stochastic) processes. In this paper, we study the problem of model checking an RMC against a given ω-regular specification. Namely, given an RMC A and a Büchi automaton B, we wish to know the probability that an execution of A is accepted by B. We establish a number of strong upper bounds, as well as lower bounds, both for qualitative problems (is the probability = 1, or = 0?), and for quantitative problems (is the probability ≥ p?, or, approximate the probability to within a desired precision). Among these, we show that qualitative model checking for general RMCs can be decided in PSPACE in |A | and EXPTIME in |B|, and when A is either a single-exit RMC or when the total number of entries and exits in A is bounded, it can be decided in polynomial time in |A|. We then show that quantitative model checking can also be done in PSPACE in |A|, and in EXPSPACE in |B|. When B is deterministic, all our complexities in |B | come down by one exponential. For lower bounds, we show that the qualitative model checking problem, even for a fixed RMC, is already EXPTIME-complete. On the other hand, even for simple reachability analysis, we showed in [EY04] that our PSPACE upper bounds in A can not be improved upon without a breakthrough on a well-known open problem in the complexity of numerical computation. 1

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Citation Context ...nlysin natural language processing (NLP) (see [MS99]). Multi-Type Branching Processes (MTBPs), are an important family of stochastic processes with many applications in a variety of areas (see, e.g., =-=[Har63]-=-). Both SCFG’s and MT-BP’s are essentially equivalent to singleexit RMC’s: the special case of RMC’s in which all components have one exit. Probabilistic models of programs and systems are of interest... |

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(Show Context)
Citation Context ...s of our results, we assume henceforth that Σ = Q, the vertices of A. This is w.l.o.g. since the problem can be reduced to this case by relabelling the RMC A and modifying the automaton B (see, e.g., =-=[CY95]-=-), however care must be taken when measuring complexity separately in the RMC, A, and in the BA, B, since typically B and Σ are small in relation to A. Our complexity results hold with respect to the ... |

160 | Efficient algorithms for model checking pushdown systems - Esparza, Hansel, et al. - 2000 |

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- 2000
(Show Context)
Citation Context ... component that does not contain any exits. Any RMC can readily be converted to an “equivalent” one in this form, while preserving relevant probabilities. Before describing M ′ A , let us recall from =-=[AEY01]-=-, the construction of a “summary graph”, HA = (Q, EHA), which ignores probabilities and is based only on information about reachability in the underlying RSM of A. Let R be the binary relation between... |

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72 | Recursive Markov chains, stochastic gram mars, and monotone systems of non-linear equations
- Etessami, Yannakakis
- 2005
(Show Context)
Citation Context ...te machines Kousha Etessami 1 and Mihalis Yannakakis 2 1 School of Informatics, University of Edinburgh 2 Department of Computer Science, Columbia University Abstract. Recursive Markov Chains (RMCs) (=-=[EY04]-=-) are a natural abstract model of procedural probabilistic programs and related systems involving recursion and probability. They succinctly define a class of denumerable Markov chains that generalize... |

68 | Mayr: Model checking probabilistic pushdown automata
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- 2006
(Show Context)
Citation Context ...f probabilistic Pushdown Systems (pPDS) was introduced and studied recently in [EKM04,BKS04]. They largely focus on model checking against branchingtime properties, but they also study deterministic (=-=[EKM04]-=-) and non-deterministic ([BKS04]) Büchi automaton specifications. There are efficient (linear time) translations between RMCs and pPDSs, similar to translations between RSMs and PDSs (see [AEY01,BGR01... |

49 | Model checking for probability and time: From theory to practice - Kwiatkowska - 2003 |

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35 | On the decidability of temporal properties of probabilistic pushdown automata
- Brázdil, Kučera, et al.
- 2005
(Show Context)
Citation Context ... (pPDS) was introduced and studied recently in [EKM04,BKS04]. They largely focus on model checking against branchingtime properties, but they also study deterministic ([EKM04]) and non-deterministic (=-=[BKS04]-=-) Büchi automaton specifications. There are efficient (linear time) translations between RMCs and pPDSs, similar to translations between RSMs and PDSs (see [AEY01,BGR01]). Our upper bounds, translated... |

14 | A problem that is easier to solve on the unit-cost algebraic RAM - Tiwari - 1992 |

13 | Verifying probabilistic procedural programs
- Esparza, Etessami
- 2004
(Show Context)
Citation Context ...ss of pBPAs (equivalent to single-exit RMCs). Also, the class of Bd-RMCs has no direct analog in pPDSs, as the total number of entries and exits of an RMC gets lost in translation to pPDSs. Reference =-=[EE04]-=- is a survey paper that predates this paper and summarizes only the results in prior papers [EKM04, EY05, BKS05]. The paper is organized as follows. Section 2 gives necessary definitions and backgroun... |