Skip to main content
SpringerLink
Log in

Hard Satisfiable Formulas for Splittings by Linear Combinations

  • Conference paper
  • First Online:
Theory and Applications of Satisfiability Testing – SAT 2017 (SAT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10491))

Abstract

Itsykson and Sokolov in 2014 introduced the class of \(\mathrm {DPLL}(\oplus )\) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of \(\mathrm {DPLL}\) algorithms that split by variables. \(\mathrm {DPLL}(\oplus )\) algorithms solve in polynomial time systems of linear equations modulo 2 that are hard for \(\mathrm {DPLL}\), \(\mathrm {PPSZ}\) and \(\mathrm {CDCL}\) algorithms. Itsykson and Sokolov have proved first exponential lower bounds for \(\mathrm {DPLL}(\oplus )\) algorithms on unsatisfiable formulas.

In this paper we consider a subclass of \(\mathrm {DPLL}(\oplus )\) algorithms that arbitrary choose a linear form for splitting and randomly (with equal probabilities) choose a value to investigate first; we call such algorithms drunken \(\mathrm {DPLL}(\oplus )\). We give a construction of a family of satisfiable CNF formulas \(\varPsi _n\) of size \(\mathrm {poly}(n)\) such that any drunken \(\mathrm {DPLL}(\oplus )\) algorithm with probability at least \(1 - 2^{-\varOmega (n)}\) runs at least \(2^{\varOmega (n)}\) steps on \(\varPsi _n\); thus we solve an open question stated in the paper [12]. This lower bound extends the result of Alekhnovich, Hirsch and Itsykson [1] from drunken \(\mathrm {DPLL}\) to drunken \(\mathrm {DPLL}(\oplus )\).

The research is partially supported by the Government of the Russia (grant 14.Z50.31.0030).

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Alekhnovich, M., Hirsch, E.A., Itsykson, D.: Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas. J. Autom. Reason. 35(1–3), 51–72 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Ben-Sasson, E.: Hard examples for bounded depth frege. In: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pp. 563–572. ACM (2002)

    Google Scholar 

  3. Beame, P., Kautz, H.A., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. (JAIR) 22, 319–351 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Beame, P., Pitassi, T.: An exponential separation between the parity principle and the pigeonhole principle. Ann. Pure Appl. Logic 80(3), 195–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cook, J., Etesami, O., Miller, R., Trevisan, L.: Goldreich’s one-way function candidate and myopic backtracking algorithms. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 521–538. Springer, Heidelberg (2009). doi:10.1007/978-3-642-00457-5_31

    Chapter  Google Scholar 

  6. Dantchev, S.S., Riis, S.: Tree resolution proofs of the weak pigeon-hole principle. In: Proceedings of the 16th Annual IEEE Conference on Computational Complexity, Chicago, Illinois, USA, pp. 69–75. IEEE Computer Society, 18–21 June 2001

    Google Scholar 

  7. Davis, M., Logemann, G., Loveland, D.W.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garlík, M., Kołodziejczyk, L.A.: Some subsystems of constant-depth Frege with parity (2017, Preprint)

    Google Scholar 

  10. Itsykson, D.: Lower bound on average-case complexity of inversion of Goldreich’s function by drunken backtracking algorithms. Theor. Comput. Syst. 54(2), 261–276 (2014)

    Article  MathSciNet  Google Scholar 

  11. Itsykson, D., Sokolov, D.: The complexity of inverting explicit Goldreich’s function by DPLL algorithms. J. Math. Sci. 188(1), 47–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Itsykson, D., Sokolov, D.: Lower bounds for splittings by linear combinations. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 372–383. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44465-8_32

    Google Scholar 

  13. Krajiček, J.: Randomized feasible interpolation and monotone circuits with a local oracle. CoRR, abs/1611.0 (2016)

    Google Scholar 

  14. Oparin, V.: Tight upper bound on splitting by linear combinations for pigeonhole principle. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 77–84. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_6

    Google Scholar 

  15. Pudlák, P., Impagliazzo, R.: A lower bound for DLL algorithms for k-SAT (preliminary version). In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, pp. 128–136, 9–11 January 2000

    Google Scholar 

  16. Pudlák, P., Scheder, D., Talebanfard, N.: Tighter Hard Instances for PPSZ. CoRR, abs/1611.0 (2016)

    Google Scholar 

  17. Razborov, A.A.: Resolution lower bounds for perfect matching principles. J. Comput. Syst. Sci. 69(1), 3–27 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Scheder, D., Tang, B., Chen, S., Talebanfard, N.: Exponential Lower Bounds for the PPSZ k-SAT Algorithm. In: Khanna, S. (ed.) Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, pp. 1253–1263. SIAM, 6–8 January 2013

    Google Scholar 

  19. Urquhart, A.: Hard examples for resolution. J. ACM 34(1), 209–219 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Dmitry Sokolov for fruitful discussions.

Author information

Authors and Affiliations

  1. Steklov Institute of Mathematics, Saint Petersburg, Russia

    Dmitry Itsykson & Alexander Knop

Authors
  1. Dmitry Itsykson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Knop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Knop .

Editor information

Editors and Affiliations

  1. UNSW Sydney, Sydney, New South Wales, Australia

    Serge Gaspers

  2. University of New South Wales , Sydney, New South Wales, Australia

    Toby Walsh

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Itsykson, D., Knop, A. (2017). Hard Satisfiable Formulas for Splittings by Linear Combinations. In: Gaspers, S., Walsh, T. (eds) Theory and Applications of Satisfiability Testing – SAT 2017. SAT 2017. Lecture Notes in Computer Science(), vol 10491. Springer, Cham. https://doi.org/10.1007/978-3-319-66263-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66263-3_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66262-6

  • Online ISBN: 978-3-319-66263-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation