Skip to main content
SpringerLink
Log in

ESampler: Efficient Sampling of Satisfying Assignments for Boolean Formulas

  • Conference paper
  • First Online:
Dependable Software Engineering. Theories, Tools, and Applications (SETTA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 13071))

  • 491 Accesses

Abstract

Boolean satisfiability (SAT) has played a key role in diverse areas spanning planning, inferencing, data mining, testing and optimization. Apart from the classical problem of checking Boolean satisfiability, generating random satisfying assignments has attracted significant theoretical and practical interests over the years. For practical applications, a large number of satisfying assignments for a given Boolean formula are needed, the generation of which turns out to be a hard problem in both theory and practice. In this work, we propose a novel approach to derive a large set of satisfying assignments from a given one in an efficient way. Our approach is orthogonal to the previous techniques for generating satisfying assignments and could be integrated into the existing SAT samplers. We implement our approach as an open-source tool ESampler and conduct extensive experiments on real-world benchmarks. Experimental results show that ESampler performs better than three state-of-the-art samplers on a large portion of the benchmarks, and is at least comparable on the others, showcasing the efficacy of our approach.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 62072309 and No. 61872340, an oversea grant from the State Key Laboratory of Novel Software Technology, Nanjing University (KFKT2018A16), and Birkbeck BEI School Project (ARTEFACT).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abed, S., Abdelaal, A.A., Alshayeji, M.H., Ahmad, I.: Sat-based and CP-based declarative approaches for top-rank-k closed frequent itemset mining. Int. J. Intell. Syst. 36(1), 112–151 (2021)

    Article  Google Scholar 

  2. Achlioptas, D., Hammoudeh, Z.S., Theodoropoulos, P.: Fast sampling of perfectly uniform satisfying assignments. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 135–147. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_9

    Chapter  MATH  Google Scholar 

  3. Angluin, D.: On counting problems and the polynomial-time hierarchy. Theoret. Comput. Sci. 12, 161–173 (1980)

    Article  MathSciNet  Google Scholar 

  4. Bacchus, F., Dalmao, S., Pitassi, T.: Algorithms and complexity results for #SAT and Bayesian inference. In: Proceedings of the 44th Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, USA, pp. 340–351 (2003)

    Google Scholar 

  5. Bellare, M., Goldreich, O., Petrank, E.: Uniform generation of np-witnesses using an NP-oracle. Inf. Comput. 163(2), 510–526 (2000)

    Article  MathSciNet  Google Scholar 

  6. 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 

  7. Bjørner, N., Phan, A.: \(\nu \)z - maximal satisfaction with Z3. In: Proceedings of the 6th International Symposium on Symbolic Computation in Software Science, pp. 1–9 (2014)

    Google Scholar 

  8. Chakraborty, S., Fremont, D.J., Meel, K.S., Seshia, S.A., Vardi, M.Y.: On parallel scalable uniform SAT witness generation. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 304–319. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_25

    Chapter  Google Scholar 

  9. Chakraborty, S., Meel, K.S., Vardi, M.Y.: A scalable and nearly uniform generator of SAT witnesses. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 608–623. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_40

    Chapter  Google Scholar 

  10. Chakraborty, S., Meel, K.S., Vardi, M.Y.: Balancing scalability and uniformity in SAT witness generator. In: Proceedings of the 51st Annual Design Automation Conference (DAC), pp. 60:1–60:6 (2014)

    Google Scholar 

  11. Chen, G., Zhao, Z., Song, F., Chen, S., Fan, L., Liu, Y.: SEC4SR: a security analysis platform for speaker recognition. CoRR abs/2109.01766 (2021)

    Google Scholar 

  12. Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158 (1971)

    Google Scholar 

  13. DIMACS: Clique and coloring problems graph format (1993). http://archive.dimacs.rutgers.edu/pub/challenge/graph/doc/ccformat.tex. Accessed 16 Sept 2021

  14. Dutra, R., Laeufer, K., Bachrach, J., Sen, K.: Efficient sampling of SAT solutions for testing. In: Proceedings of the 40th International Conference on Software Engineering, pp. 549–559 (2018)

    Google Scholar 

  15. Ermon, S., Gomes, C.P., Sabharwal, A., Selman, B.: Embed and project: discrete sampling with universal hashing. In: Proceedings of the 27th Annual Conference on Neural Information Processing Systems, pp. 2085–2093 (2013)

    Google Scholar 

  16. Ermon, S., Gomes, C.P., Selman, B.: Uniform solution sampling using a constraint solver as an oracle. In: Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence, pp. 255–264 (2012)

    Google Scholar 

  17. Gomes, C.P., Sabharwal, A., Selman, B.: Near-uniform sampling of combinatorial spaces using XOR constraints. In: Proceedings of the 2th Annual Conference on Neural Information Processing Systems, pp. 481–488 (2006)

    Google Scholar 

  18. Guralnik, E., Aharoni, M., Birnbaum, A.J., Koyfman, A.: Simulation-based verification of floating-point division. IEEE Trans. Comput. 60(2), 176–188 (2011)

    Article  MathSciNet  Google Scholar 

  19. Hubara, I., Courbariaux, M., Soudry, D., El-Yaniv, R., Bengio, Y.: Binarized neural networks. In: Proceedings of the Annual Conference on Neural Information Processing Systems, pp. 4107–4115 (2016)

    Google Scholar 

  20. Ivrii, A., Malik, S., Meel, K.S., Vardi, M.Y.: On computing minimal independent support and its applications to sampling and counting. Constraints 21(1), 41–58 (2015). https://doi.org/10.1007/s10601-015-9204-z

    Article  MathSciNet  MATH  Google Scholar 

  21. Bayardo Jr., R.J., Schrag, R.: Using CSP look-back techniques to solve real-world SAT instances. In: Proceedings of the Fourteenth National Conference on Artificial Intelligence and Ninth Innovative Applications of Artificial Intelligence Conference, 27–31 July 1997, Providence, Rhode Island, USA, pp. 203–208 (1997)

    Google Scholar 

  22. Kitchen, N.: Markov chain Monte Carlo stimulus generation for constrained random simulation. Ph.D. thesis, University of California, Berkeley, USA (2010)

    Google Scholar 

  23. Kitchen, N., Kuehlmann, A.: Stimulus generation for constrained random simulation. In: Proceedings of the 2007 International Conference on Computer-Aided Design, pp. 258–265 (2007)

    Google Scholar 

  24. Korneev, S., Narodytska, N., Pulina, L., Tacchella, A., Bjorner, N., Sagiv, M.: Constrained image generation using binarized neural networks with decision procedures. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 438–449. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_27

    Chapter  Google Scholar 

  25. Kukula, J.H., Shiple, T.R.: Building circuits from relations. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 113–123. Springer, Heidelberg (2000). https://doi.org/10.1007/10722167_12

    Chapter  Google Scholar 

  26. Meel, K.S.: Constrained counting and sampling: bridging the gap between theory and practice. CoRR abs/1806.02239 (2018)

    Google Scholar 

  27. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    Chapter  Google Scholar 

  28. Narodytska, N.: Formal analysis of deep binarized neural networks. In: Proceedings of the 27th International Joint Conference on Artificial Intelligence, pp. 5692–5696 (2018)

    Google Scholar 

  29. Naveh, R., Metodi, A.: Beyond feasibility: CP usage in constrained-random functional hardware verification. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 823–831. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40627-0_60

    Chapter  Google Scholar 

  30. Naveh, Y., et al.: Constraint-based random stimuli generation for hardware verification. In: Proceedings of the 21st National Conference on Artificial Intelligence and the 18th Innovative Applications of Artificial Intelligence Conference, pp. 1720–1727 (2006)

    Google Scholar 

  31. Naveh, Y., et al.: Constraint-based random stimuli generation for hardware verification. AI Mag. 28(3), 13 (2007)

    MathSciNet  Google Scholar 

  32. Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82(1–2), 273–302 (1996)

    Article  MathSciNet  Google Scholar 

  33. Silva, J.P.M., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  34. Silva, J.P.M., Sakallah, K.A.: Grasp–a new search algorithm for satisfiability. In: Kuehlmann, A. (ed.) The Best of ICCAD, pp. 73–89. Springer, Boston (2003). https://doi.org/10.1007/978-1-4615-0292-0_7

    Chapter  Google Scholar 

  35. Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 330–335 (1983)

    Google Scholar 

  36. Soos, M., Gocht, S., Meel, K.S.: Tinted, detached, and lazy CNF-XOR solving and its applications to counting and sampling. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 463–484. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53288-8_22

    Chapter  Google Scholar 

  37. Soos, M., Nohl, K., Castelluccia, C.: Extending SAT solvers to cryptographic problems. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 244–257. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_24

    Chapter  Google Scholar 

  38. Sörensson, N., Eén, N.: MiniSat: a SAT solver with conflict-clause minimization. Solver Description (2005)

    Google Scholar 

  39. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)

    Article  MathSciNet  Google Scholar 

  40. Vorobyov, K., Krishnan, P.: Combining static analysis and constraint solving for automatic test case generation. In: Proceedings of the 5th IEEE International Conference on Software Testing, Verification and Validation, pp. 915–920 (2012)

    Google Scholar 

  41. Wei, W., Erenrich, J., Selman, B.: Towards efficient sampling: exploiting random walk strategies. In: Proceedings of the 19th National Conference on Artificial Intelligence, 16th Conference on Innovative Applications of Artificial Intelligence, pp. 670–676 (2004)

    Google Scholar 

  42. Wei, W., Selman, B.: A new approach to model counting. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 324–339. Springer, Heidelberg (2005). https://doi.org/10.1007/11499107_24

    Chapter  Google Scholar 

  43. Yuan, J., Aziz, A., Pixley, C., Albin, K.: Simplifying boolean constraint solving for random simulation-vector generation. IEEE Trans. Comput. Aided Des. Integr. Circ. Syst. 23(3), 412–420 (2004)

    Article  Google Scholar 

  44. Zhang, Y., Zhao, Z., Chen, G., Song, F., Chen, T.: BDD4BNN: a BDD-based quantitative analysis framework for binarized neural networks. In: Proceedings of the 33rd International Conference on Computer Aided Verification, pp. 175–200 (2021)

    Google Scholar 

  45. Zhang, Y., Li, J., Zhang, M., Pu, G., Song, F.: Optimizing backbone filtering. In: Proceedings of the 11th International Symposium on Theoretical Aspects of Software Engineering, pp. 1–8 (2017)

    Google Scholar 

  46. Zhang, Y., Zhang, M., Pu, G., Song, F., Li, J.: Towards backbone computing: a greedy-whitening based approach. AI Commun. 31(3), 267–280 (2018)

    Article  MathSciNet  Google Scholar 

  47. Zhao, Y., Bian, J., Deng, S., Kong, Z.: Random stimulus generation with self-tuning. In: Proceedings of the 13th International Conference on Computers Supported Cooperative Work in Design, pp. 62–65. IEEE (2009)

    Google Scholar 

  48. Zhao, Z., Chen, G., Wang, J., Yang, Y., Song, F., Sun, J.: Attack as defense: characterizing adversarial examples using robustness. In: Proceedings of the 30th ACM SIGSOFT International Symposium on Software Testing and Analysis, pp. 42–55 (2021)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ShanghaiTech University, Shanghai, China

    Yongjie Xu & Fu Song

  2. Birkbeck, University of London, London, UK

    Taolue Chen

Authors
  1. Yongjie Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fu Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Taolue Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fu Song .

Editor information

Editors and Affiliations

  1. Teesside University, Middlesbrough, UK

    Shengchao Qin

  2. University of York, York, UK

    Jim Woodcock

  3. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Wenhui Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Xu, Y., Song, F., Chen, T. (2021). ESampler: Efficient Sampling of Satisfying Assignments for Boolean Formulas. In: Qin, S., Woodcock, J., Zhang, W. (eds) Dependable Software Engineering. Theories, Tools, and Applications. SETTA 2021. Lecture Notes in Computer Science(), vol 13071. Springer, Cham. https://doi.org/10.1007/978-3-030-91265-9_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-91265-9_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-91264-2

  • Online ISBN: 978-3-030-91265-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation