Symmetries of Quantified Boolean Formulas

  • Manuel KauersEmail author
  • Martina Seidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10929)


While symmetries are well understood for Boolean formulas and successfully exploited in practical SAT solving, less is known about symmetries in quantified Boolean formulas (QBF). There are some works introducing adaptions of propositional symmetry breaking techniques, with a theory covering only very specific parts of QBF symmetries. We present a general framework that gives a concise characterization of symmetries of QBF. Our framework naturally incorporates the duality of universal and existential symmetries resulting in a general basis for QBF symmetry breaking.


  1. 1.
    Polya, G.: How to Solve It: A New Aspect of Mathematical Method. Princeton University Press, Princeton (1945)zbMATHGoogle Scholar
  2. 2.
    Sakallah, K.A.: Symmetry and satisfiability. In: Biere, A., Heule, M., Van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 289–338. IOS Press, Amsterdam (2009)Google Scholar
  3. 3.
    Gent, I.P., Petrie, K.E., Puget, J.: Symmetry in constraint programming. In: Rossi, F., Walsh, T., van Beek, P. (eds.) Handbook of Constraint Programming. Foundations of Artificial Intelligence, vol. 2, pp. 329–376. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  4. 4.
    Cohen, D.A., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M.: Constraint symmetry and solution symmetry. In: Proceedings of the 21st National Conference on Artificial Intelligence and the 18th Innovative Applications of Artificial Intelligence Conference (AAAI/IAAI 2006), pp. 1589–1592. AAAI Press (2006)Google Scholar
  5. 5.
    Kleine Büning, H., Bubeck, U.: Theory of quantified Boolean formulas. In: Biere, A., Heule, M., Van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 735–760. IOS Press, Amsterdam (2009)Google Scholar
  6. 6.
    Audemard, G., Mazure, B., Sais, L.: Dealing with symmetries in quantified Boolean formulas. In: Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), Online Proceedings (2004)Google Scholar
  7. 7.
    Audemard, G., Jabbour, S., Sais, L.: Symmetry breaking in quantified Boolean formulae. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), pp. 2262–2267 (2007)Google Scholar
  8. 8.
    Audemard, G., Jabbour, S., Sais, L.: Efficient symmetry breaking predicates for quantified Boolean formulae. In: Proceedings of Workshop on Symmetry and Constraint Satisfaction Problems (SymCon 2007) (2007). 7 pagesGoogle Scholar
  9. 9.
    Jabbour, S.: De la satisfiabilité propositionnelle aux formules booléennes quantifiées. Ph.D. thesis, CRIL, Lens, France (2008)Google Scholar
  10. 10.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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). Scholar
  12. 12.
    Egly, U., Lonsing, F., Widl, M.: Long-distance resolution: proof generation and strategy extraction in search-based QBF solving. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013). Scholar
  13. 13.
    Pulina, L., Seidl, M.: The QBFEval 2017.
  14. 14.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Monotone literals and learning in QBF reasoning. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 260–273. Springer, Heidelberg (2004). Scholar
  15. 15.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  16. 16.
    Artin, M.: Algebra. Pearson Prentice Hall, Upper Saddle River (2011)zbMATHGoogle Scholar
  17. 17.
    Sabharwal, A., Ansotegui, C., Gomes, C.P., Hart, J.W., Selman, B.: QBF modeling: exploiting player symmetry for simplicity and efficiency. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 382–395. Springer, Heidelberg (2006). Scholar
  18. 18.
    Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning (KR 1996), pp. 148–159. Morgan Kaufmann (1996)Google Scholar
  19. 19.
    Devriendt, J., Bogaerts, B., Bruynooghe, M., Denecker, M.: Improved static symmetry breaking for SAT. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 104–122. Springer, Cham (2016). Scholar
  20. 20.
    Egly, U., Seidl, M., Tompits, H., Woltran, S., Zolda, M.: Comparing different prenexing strategies for quantified Boolean formulas. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 214–228. Springer, Heidelberg (2004). Scholar
  21. 21.
    Goultiaeva, A., Seidl, M., Biere, A.: Bridging the gap between dual propagation and CNF-based QBF solving. In: Proceedings of the International Conference on Design, Automation and Test in Europe (DATE 2013), EDA Consortium, San Jose, CA, USA, pp. 811–814. ACM DL (2013)Google Scholar
  22. 22.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Janota, M.: QFUN: towards machine learning in QBF. CoRR abs/1710.02198 (2017)Google Scholar
  24. 24.
    Tentrup, L.: Non-prenex QBF solving using abstraction. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 393–401. Springer, Cham (2016). Scholar
  25. 25.
    Narodytska, N., Walsh, T.: Breaking symmetry with different orderings. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 545–561. Springer, Heidelberg (2013). Scholar
  26. 26.
    Devriendt, J., Bogaerts, B., Bruynooghe, M.: Symmetric explanation learning: effective dynamic symmetry handling for SAT. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 83–100. Springer, Cham (2017). Scholar
  27. 27.
    Blinkhorn, J., Beyersdorff, O.: Shortening QBF proofs with dependency schemes. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 263–280. Springer, Cham (2017). Scholar
  28. 28.
    Peitl, T., Slivovsky, F., Szeider, S.: Dependency learning for QBF. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 298–313. Springer, Cham (2017). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for AlgebraJKU LinzLinzAustria
  2. 2.Institute for Formal Models and VerificationJKU LinzLinzAustria

Personalised recommendations