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Automata Terms in a Lazy WSkS Decision Procedure

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Abstract

We propose a lazy decision procedure for the logic WS\(k\)S. It builds a term-based symbolic representation of the state space of the tree automaton (TA) constructed by the classical WS\(k\)S decision procedure. The classical decision procedure transforms the symbolic representation into a TA via a bottom-up traversal and then tests its language non-emptiness, which corresponds to satisfiability of the formula. On the other hand, we start evaluating the representation from the top, construct the state space on the fly, and utilize opportunities to prune away parts of the state space irrelevant to the language emptiness test. In order to do so, we needed to extend the notion of language terms (denoting language derivatives) used in our previous procedure for the linear fragment of the logic (the so-called WS1S) into automata terms. We implemented our decision procedure and identified classes of formulae on which our prototype implementation is significantly faster than the classical procedure implemented in the Mona tool.

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Notes

  1. Note that our definition of projection differs from the usual one, which would create a symbol over the alphabet \({\mathbb {X}}\setminus {\mathbb {Y}}\); in the example, it would produce a single symbol \(\{Y \mapsto 0\}\) over the alphabet of symbols over \(\{Y\}\).

  2. The implementation is available at https://github.com/vhavlena/lazy-wsks.

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Acknowledgements

We thank the anonymous reviewers, both of the conference and the journal version of the paper, for their careful reading of the drafts, the spotted bugs, and the helpful comments on how to improve the exposition in this paper. This work was supported by the Czech Science Foundation project 19-24397S, the FIT BUT internal project FIT-S-20-6427, and The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project IT4Innovations excellence in science—LQ1602.

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  1. IT4I Centre of Excellence, Faculty of Information Technology, Brno University of Technology, Božetěchova 2, 612 00, Brno, Czech Republic

    Vojtěch Havlena, Lukáš Holík, Ondřej Lengál & Tomáš Vojnar

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  1. Vojtěch Havlena
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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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Havlena, V., Holík, L., Lengál, O. et al. Automata Terms in a Lazy WSkS Decision Procedure. J Autom Reasoning 65, 971–999 (2021). https://doi.org/10.1007/s10817-021-09597-w

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