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Implicit QBF Encodings for Positional Games

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Advances in Computer Games (ACG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14528))

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Abstract

We address two bottlenecks for concise QBF encodings of maker-breaker positional games, like Hex and Tic-Tac-Toe. We improve a baseline QBF encoding by representing winning configurations implicitly. The second improvement replaces variables for explicit board positions by a universally quantified symbolic board position. The paper evaluates the size of these lifted encodings, depending on board size and game depth. It reports the performance of QBF solvers on these encodings. We study scalability up to 19\(\times \)19 boards, played in human Hex tournaments.

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Notes

  1. 1.

    Precise encoding and measurement details are available in a technical report [29].

  2. 2.

    [29, Appendix B] also provides lifted and stateless encodings with implicit-board encodings and explicit-goal conditions, for completeness.

  3. 3.

    Available at https://github.com/vale1410/positional-games-qbf-encoding.

  4. 4.

    Available at https://github.com/irfansha/Q-sage.

  5. 5.

    We display an asymptotically equivalent function. [29, App. C] shows the exact size.

  6. 6.

    From https://www.littlegolem.net. See [29, Appendix A] for the selection process.

  7. 7.

    [29, Appendix D] gives detailed measurements per solver/preprocessor. [29, App. E] shows memory plots, and [29, Appendix F] separates True and False instances.

  8. 8.

    Available at http://www.qbflib.org/qbfeval22.php.

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Author information

Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Irfansha Shaik & Jaco van de Pol

  2. University of Potsdam, Potsdam, Germany

    Valentin Mayer-Eichberger

  3. The University of New South Wales, Sydney, Australia

    Abdallah Saffidine

Authors
  1. Irfansha Shaik
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  2. Valentin Mayer-Eichberger
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  3. Jaco van de Pol
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  4. Abdallah Saffidine
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Corresponding author

Correspondence to Jaco van de Pol .

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Editors and Affiliations

  1. University of Siegen, Siegen, Germany

    Michael Hartisch

  2. Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan

    Chu-Hsuan Hsueh

  3. University of Alberta, Edmonton, AB, Canada

    Jonathan Schaeffer

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Shaik, I., Mayer-Eichberger, V., van de Pol, J., Saffidine, A. (2024). Implicit QBF Encodings for Positional Games. In: Hartisch, M., Hsueh, CH., Schaeffer, J. (eds) Advances in Computer Games. ACG 2023. Lecture Notes in Computer Science, vol 14528. Springer, Cham. https://doi.org/10.1007/978-3-031-54968-7_12

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  • DOI: https://doi.org/10.1007/978-3-031-54968-7_12

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