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Fast Computations on Ordered Nominal Sets

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Theoretical Aspects of Computing – ICTAC 2018 (ICTAC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11187))

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Abstract

We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the level of orbits, which allows for an accurate complexity analysis. The approach is implemented as the library Ons (Ordered Nominal Sets).

Our main motivation is nominal automata, which are models for recognising languages over infinite alphabets. We evaluate Ons in two applications: minimisation of automata and active automata learning. In both cases, Ons is competitive compared to existing implementations and outperforms them for certain classes of inputs.

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Notes

  1. 1.

    Other implementations of nominal techniques that are less directly related to our setting (Mihda, Fresh OCaml, and Nominal Isabelle) are discussed in Sect. 6.

  2. 2.

    Ons can be found at https://github.com/davidv1992/ONS.

  3. 3.

    The G-action on words is defined point-wise: \((w_1 \ldots w_n) g = (w_1 g) \ldots (w_n g)\).

  4. 4.

    Abstractly, an automaton is minimal if it has no proper quotients. Minimal deterministic automata are unique up to isomorphism.

  5. 5.

    See https://joshuamoerman.nl/papers/2017/17popl-learning-nominal-automata.html for a sketch of the polynomial algorithm.

  6. 6.

    Can be found on https://github.com/eryxcc/lois/blob/master/tests/learning.cpp.

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Acknowledgement

We would like to thank Szymon Toruńczyk and Eryk Kopczyński for their prompt help when using the Lois library. For general comments and suggestions we would like to thank Ugo Montanari and Niels van der Weide. At last, we want to thank the anonymous reviewers for their comments.

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Authors and Affiliations

  1. Institute for Computing and Information Sciences, Radboud Universiteit, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    David Venhoek, Joshua Moerman & Jurriaan Rot

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  1. David Venhoek
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Correspondence to Jurriaan Rot .

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  1. Stellenbosch University, Stellenbosch, South Africa

    Bernd Fischer

  2. Reykjavík University, Reykjavik, Iceland

    Tarmo Uustalu

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Venhoek, D., Moerman, J., Rot, J. (2018). Fast Computations on Ordered Nominal Sets. 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_26

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  • DOI: https://doi.org/10.1007/978-3-030-02508-3_26

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