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Quantified Boolean Solving for Achievement Games

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KI 2021: Advances in Artificial Intelligence (KI 2021)

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Abstract

Recent developments in the propositional representation of achievement games have renewed interest in applying the latest advances in Quantified Boolean Formula technologies to solving these games. However, the number of quantifier alternations necessary to explore the solution space still impairs and limits the applicability of these methods. In this paper, we show that one can encode blocking strategies for the second player and express the last moves of the play with a single string of existential quantifiers, instead of the usual alternations of universal and existential quantifiers. We experimentally show that our method improves the performance of state-of-the-art Quantified Boolean Formula solvers on Harary’s Tic-Tac-Toe, a well-known achievement game.

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References

  1. Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)

    Google Scholar 

  2. Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 101–115. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22438-6_10

    Chapter  Google Scholar 

  3. Csernenszky, A., Martin, R.R., Pluhár, A.: On the complexity of chooser-picker positional games. INTEGERS: Electron. J. Comb. Number Theory 11(G2) (2011)

    Google Scholar 

  4. Demaine, E.D., Hearn, R.A.: Playing games with algorithms: algorithmic combinatorial game theory, pp. 3–56. Mathematical Sciences Research Institute Publications, Cambridge University Press (2009)

    Google Scholar 

  5. Diptarama, Narisawa, K., Shinohara, A.: Drawing strategies for generalized Tic-Tac-Toe \((p, q)\). In: AIP Conference Proceedings, vol. 1705, no. 1, 020021 (2016)

    Google Scholar 

  6. Diptarama, Yoshinaka, R., Shinohara, A.: QBF encoding of generalized Tic-Tac-Toe. In: Quantified Boolean Formulas, QBF 2016. CEUR Workshop Proceedings, vol. 1719, pp. 14–26 (2016)

    Google Scholar 

  7. Gardner, M.: Mathematical games. Sci. Am. 240(4), 18–28 (1979)

    Article  Google Scholar 

  8. Gent, I., Nightingale, P.: A new encoding of all different into SAT. In: Modelling and Reformulating Constraint Satisfaction Problems, pp. 95–110 (2004)

    Google Scholar 

  9. Gent, I., Rowley, A.R.: Encoding Connect-4 using quantified Boolean formulae. In: Modelling and Reformulating Constraint Satisfaction Problems, pp. 78–93 (2003)

    Google Scholar 

  10. Halupczok, I., Schlage-Puchta, J.C.: Achieving snaky. INTEGERS: Electron. J. Comb. Number Theory 7(G02) (2007)

    Google Scholar 

  11. Harary, F.: Achieving the Skinny animal. Eureka 42, 8–14 (1982)

    Google Scholar 

  12. Harborth, H., Seemann, M.: Snaky is an edge-to-edge loser. Geombinatorics V(4), 132–136 (1996)

    Google Scholar 

  13. Harborth, H., Seemann, M.: Snaky is a paving winner. Bull. Inst. Combin. Appl. 19, 71–78 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Heule, M.J.H., Szeider, S.: A SAT approach to clique-width. ACM Trans. Comput. Log. 16(3), 24 (2015)

    Article  MathSciNet  Google Scholar 

  15. Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: International Joint Conference on Artificial Intelligence, IJCAI, pp. 325–331. AAAI Press (2015)

    Google Scholar 

  16. Lonsing, F., Egly, U.: DepQBF 6.0: a search-based QBF solver beyond traditional QCDCL. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 371–384. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_23

    Chapter  Google Scholar 

  17. Lonsing, F., Egly, U.: QRATPre+: effective QBF preprocessing via strong redundancy properties. In: Janota, M., Lynce, I. (eds.) SAT 2019. LNCS, vol. 11628, pp. 203–210. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24258-9_14

    Chapter  Google Scholar 

  18. Mayer-Eichberger, V., Saffidine, A.: Positional games and QBF: the Corrective encoding. In: Pulina, L., Seidl, M. (eds.) SAT 2020. LNCS, vol. 12178, pp. 447–463. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51825-7_31

    Chapter  MATH  Google Scholar 

  19. Peitl, T., Slivovsky, F., Szeider, S.: Qute in the QBF evaluation 2018. J. Satisfiability Boolean Model. Comput. 11(1), 261–272 (2019)

    Article  MathSciNet  Google Scholar 

  20. Reisch, S.: Hex ist PSPACE-vollständig (Hex is PSPACE-complete). Acta Informatica 15, 167–191 (1981)

    Article  MathSciNet  Google Scholar 

  21. Reisch, S.: Gobang ist PSPACE-vollständig (Gomoku is PSPACE-complete). Acta Informatica 13, 59–66 (1980)

    Article  MathSciNet  Google Scholar 

  22. Shukla, A., Biere, A., Pulina, L., Seidl, M.: A survey on applications of Quantified Boolean Formulas. In: International Conference on Tools with Artificial Intelligence, ICTAI 2019, pp. 78–84. IEEE (2019)

    Google Scholar 

  23. Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time: preliminary report. In: ACM Symposium on Theory of Computing (STOC), pp. 1–9. ACM (1973)

    Google Scholar 

  24. Tentrup, L.: CAQE and QuAbS: abstraction based QBF solvers. J. Satisfiability Boolean Model. Comput. 11(1), 155–210 (2019)

    Article  MathSciNet  Google Scholar 

  25. Gelder, A.: Primal and dual encoding from applications into quantified Boolean formulas. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 694–707. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40627-0_51

    Chapter  Google Scholar 

  26. Wimmer, R., Reimer, S., Marin, P., Becker, B.: HQSpre – an effective preprocessor for QBF and DQBF. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 373–390. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54577-5_21

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, UQAM, Montreal, Canada

    Steve Boucher & Roger Villemaire

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  1. Steve Boucher
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  2. Roger Villemaire
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Correspondence to Roger Villemaire .

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Editors and Affiliations

  1. Czech Technical University in Prague, Prague, Czech Republic

    Stefan Edelkamp

  2. University of Lübeck, Lübeck, Germany

    Ralf Möller

  3. University of Leoben, Leoben, Austria

    Elmar Rueckert

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Boucher, S., Villemaire, R. (2021). Quantified Boolean Solving for Achievement Games. In: Edelkamp, S., Möller, R., Rueckert, E. (eds) KI 2021: Advances in Artificial Intelligence. KI 2021. Lecture Notes in Computer Science(), vol 12873. Springer, Cham. https://doi.org/10.1007/978-3-030-87626-5_3

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  • DOI: https://doi.org/10.1007/978-3-030-87626-5_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87625-8

  • Online ISBN: 978-3-030-87626-5

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