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Nested antichains for WS1S

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Abstract

We propose a novel approach for coping with alternating quantification as the main source of nonelementary complexity of deciding WS1S formulae. Our approach is applicable within the state-of-the-art automata-based WS1S decision procedure implemented e.g. in Mona. The way in which the standard decision procedure processes quantifiers involves determinization, with its worst case exponential complexity, for every quantifier alternation in the prefix of a formula. Our algorithm avoids building the deterministic automata—instead, it constructs only those of their states needed for (dis)proving validity of the formula. It uses a symbolic representation of the states, which have a deeply nested structure stemming from the repeated implicit subset construction, and prunes the search space by a nested subsumption relation, a generalization of the one used by the so-called antichain algorithms for handling nondeterministic automata. We have obtained encouraging experimental results, in some cases outperforming Mona, and some of the other recently proposed approaches, by several orders of magnitude.

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Notes

  1. Results for the other families are very similar and hence skipped here. An interested reader is referred to [27].

  2. Note that the HornSub family is not supported by Toss and Coalg, and thus we chose a comparably complex family of SetClosed to present the overall comparison.

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Acknowledgements

We thank the anonymous reviewers for their helpful comments on how to improve the presentation in this paper. The work in this paper was supported by the Czech Science Foundation project 16-24707Y, the IT4IXS: IT4Innovations Excellence in Science project (LQ1602), and the FIT BUT internal project FIT-S-17-4014.

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  1. FIT, IT4Innovations Centre of Excellence, Brno University of Technology, Božetěchova 2, 612 66, Brno, Czech Republic

    Tomáš Fiedor, Lukáš Holík, Ondřej Lengál & Tomáš Vojnar

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  1. Tomáš Fiedor
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  2. Lukáš Holík
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  3. Ondřej Lengál
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  4. Tomáš Vojnar
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Correspondence to Ondřej Lengál.

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An extended abstract of this paper was first presented in [1]. The current paper extends [1] with a more detailed presentation of the approach, the needed proofs, an illustrating example, and an extended experimental evaluation of the approach.

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Fiedor, T., Holík, L., Lengál, O. et al. Nested antichains for WS1S. Acta Informatica 56, 205–228 (2019). https://doi.org/10.1007/s00236-018-0331-z

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