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Analyzing probabilistic pushdown automata

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Abstract

The paper gives a summary of the existing results about algorithmic analysis of probabilistic pushdown automata and their subclasses.

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Notes

  1. The “BPA” acronym stands for Basic Process Algebra and is used mainly for historical reasons.

  2. For every tS, we fix a fresh variable Y t . If tT, we put Y t =1. If t cannot reach T at all, we put Y t =0. Otherwise, we put \(Y_{t} = \sum_{t {}\mathchoice {\stackrel {x}{\rightarrow }} {\mathop {\smash \rightarrow }\limits ^{\vrule width 0pt height 0pt depth 4pt\smash {x}}} {\stackrel {x}{\rightarrow }} {\stackrel {x}{\rightarrow }} {} t'} x \cdot Y_{t'}\). The resulting system of linear equations has only one solution in ℝ|S| which is the tuple of all .

  3. An instance of Square-Root-Sum is a tuple of positive integers a 1,…,a n ,b, and the question is whether \(\sum_{i=1}^{n} \sqrt{a_{i}} \leq b\). The problem is obviously in PSPACE, because it can be encoded in the existential fragment of Tarski algebra (see Sect. 2.2), and the best upper bound currently known is CH (counting hierarchy; see Corollary 1.4 in [1]). It is not known whether this bound can be further lowered to some natural Boolean subclass of PSPACE, and a progress in answering this question might lead to breakthrough results in complexity theory.

  4. Here “almost every” is meant in the usual probabilistic sense, i.e., the probability of the remaining runs is zero.

  5. The last configuration of a jump does not contribute to the total accumulated reward.

  6. Formally, the “head” of a given pOC configuration p(k) is either (p,0) or (p,1), depending on whether k=0 or k>0, respectively. The input alphabet of the corresponding DRA is then Q×{0,1}.

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Acknowledgements

T. Brázdil and A. Kučera are supported by the Czech Science Foundation, grant No. P202/12/G061. S. Kiefer is supported by a postdoctoral fellowship of the German Academic Exchange Service (DAAD).

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Correspondence to Antonín Kučera.

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Brázdil, T., Esparza, J., Kiefer, S. et al. Analyzing probabilistic pushdown automata. Form Methods Syst Des 43, 124–163 (2013). https://doi.org/10.1007/s10703-012-0166-0

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