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Convex Language Semantics for Nondeterministic Probabilistic Automata

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Theoretical Aspects of Computing – ICTAC 2018 (ICTAC 2018)

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Abstract

We explore language semantics for automata combining probabilistic and nondeterministic behaviors. We first show that there are precisely two natural semantics for probabilistic automata with nondeterminism. For both choices, we show that these automata are strictly more expressive than deterministic probabilistic automata, and we prove that the problem of checking language equivalence is undecidable by reduction from the threshold problem. However, we provide a discounted metric that can be computed to arbitrarily high precision.

This work was partially supported by ERC starting grant ProFoundNet (679127), ERC consolidator grant AVS-ISS (648701), a Leverhulme Prize (PLP-2016-129), and an NSF grant (1637532).

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Notes

  1. 1.

    In prior work [7], the monad was defined to take all convex subsets rather than just the finitely generated ones. However, since all the monad operations preserve finiteness of the generators, the restricted monad we consider is also well-defined.

  2. 2.

    All concrete automata considered in this paper will have a finite state space, but this is not required by Definition 6. The distribution monad, for example, does not preserve finite sets in general.

  3. 3.

    The \(\mathsf {max}\) semantics is perhaps preferable since it recovers standard nondeterministic finite automata when there is no probabilistic choice and the output weights are in \(\{0, 1\}\), but this is a minor point.

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Acknowledgements

We thank Nathanaël Fijalkow and the anonymous reviewers for their useful suggestions to improve the paper. The semantics studied in this paper has been brought to our attention in personal communication by Filippo Bonchi, Ana Sokolova, and Valeria Vignudelli. Their interest in this semantics is mostly motivated by its relationship with trace semantics previously proposed in the literature. This is the subject of a forthcoming publication [8].

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Authors and Affiliations

  1. Department of Computer Science, University College London, London, UK

    Gerco van Heerdt & Alexandra Silva

  2. Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA

    Justin Hsu

  3. Max Planck Institute for Software Systems, Saarbrücken, Germany

    Joël Ouaknine

  4. Department of Computer Science, Oxford University, Oxford, UK

    Joël Ouaknine

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  1. Gerco van Heerdt
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  4. Alexandra Silva
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Correspondence to Justin Hsu .

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Editors and Affiliations

  1. Stellenbosch University, Stellenbosch, South Africa

    Bernd Fischer

  2. Reykjavík University, Reykjavik, Iceland

    Tarmo Uustalu

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van Heerdt, G., Hsu, J., Ouaknine, J., Silva, A. (2018). Convex Language Semantics for Nondeterministic Probabilistic Automata. In: Fischer, B., Uustalu, T. (eds) Theoretical Aspects of Computing – ICTAC 2018. ICTAC 2018. Lecture Notes in Computer Science(), vol 11187. Springer, Cham. https://doi.org/10.1007/978-3-030-02508-3_25

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