Skip to main content
SpringerLink
Log in

Existence versus exploitation: the opacity of backdoors and backbones

  • Regular Paper
  • Published:
Progress in Artificial Intelligence Aims and scope Submit manuscript

Abstract

Boolean formulas often have what are known as hidden structures. We study the complexity of whether such structures (of certain sizes) exist in a formula, the complexity of getting one’s hands on the structure’s information, and whether even when in hand the information can be efficiently exploited. In particular, backdoors and backbones of Boolean formulas are important hidden structural properties. A natural goal, already in part realized, is that solver algorithms seek better performance by exploiting these structures. However, the present paper is not intended to improve the performance of SAT solvers, but rather is a cautionary tale. The theme of this paper is that there is a potential chasm between the existence of such structures in the Boolean formula and being able to effectively exploit them. This does not mean that these structures are not useful to solvers. It does mean that one must be very careful not to assume that it is computationally easy to go from the existence of information to being able to get one’s hands on it and/or being able to exploit it. We construct backdoor- and backbone-based cases where, if \(\mathrm {P}\ne \mathrm {NP}\), such assumptions fail. For example, if \(\mathrm {P}\ne \mathrm {NP}\), then (a) there are easily recognizable families of Boolean formulas with strong backdoors that are easy to find, yet for which it is hard to determine whether the formulas are satisfiable, and (b) there are easily recognizable sets of Boolean formulas for which it is hard to determine whether they have a large backbone.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Recall that if F is unsatisfiable, then “A determines F” means, perhaps somewhat confusingly as regards the plain English meaning of “determines,” that A proclaims F’s unsatisfiability.

  2. Here we assume that a clause \(\{x\}\) precedes a clause \(\{y\}\) in lexicographical order if x precedes y in lexicographical order.

  3. We have not been able to find Corollary (to the Proof) 4.2 in the literature. Certainly, two things that on their surface might seem to be the claim we are making in Corollary (to the Proof) 4.2 are either trivially true or are in the literature. However, upon closer inspection they turn out to be quite different from our claim.

    In particular, if one removes the word “nontrivial” from Corollary (to the Proof) 4.2’s statement, and one is in the model in which every satisfiable formula is considered to have the empty collection of variables as a backbone and every unsatisfiable formula is considered to have no backbones, then the thus-altered version of Corollary (to the Proof) 4.2 is clearly true, since if one with those changes takes A to be the set of all Boolean formulas, then the theorem degenerates to the statement that if \(\mathrm {P}\ne \mathrm {NP}\), then SAT is (NP-complete, and) not in \(\mathrm {P}\).

    Also, it is stated in Kilby et al. [25] that finding a backbone of CNF formulas is “NP-hard.” However, though this might seem to be our result, their claim and model differ from ours in many ways, making this a quite different issue. First, their hardness refers to Turing reductions (and in contrast our paper is about many-one reductions and many-one completeness). Second, they are not even speaking of NP-Turing-hardness—much less NP-Turing-completeness—in the standard sense since their model is assuming a function reply from the oracle rather than having a set as the oracle. Third, even their notion of backbones is quite different as it (unlike the influential Williams, Gomes, and Selman 2003 paper [37] and our paper) in effect requires that the function-oracle gives back both a variable and its setting. Fourth, our claim is about nontrivial backbones.

References

  1. Ansótegui, C., Bonet, M., Giráldez-Cru, J., Levy, J., Simon, L.: Community structures in industrial SAT instances. J. Artif. Intell. Res. 66, 443–472 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bellare, M., Goldwasser, S.: The complexity of decision versus search. SIAM J. Comput. 23(1), 97–119 (1994)

    Article  MathSciNet  Google Scholar 

  3. Berman, P.: Relationship between density and deterministic complexity of NP-complete languages. In: Proceedings of the 5th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, July 1978, vol. 62, pp. 63–71. Springer (1978)

  4. Borodin, A., Demers, A.: Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR 76-284, Department of Computer Science, Cornell University, Ithaca, NY, July (1976)

  5. Buhrman, H., Hitchcock, J.: NP-hard sets are exponentially dense unless coNP\(\,\subseteq \,\)NP/poly. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, June 2008, pp. 1–7. IEEE Computer Society Press (2008)

  6. Cai, J., Chakaravarthy, V., Hemaspaandra, L., Ogihara, M.: Competing provers yield improved Karp–Lipton collapse results. Inf. Comput. 198(1), 1–23 (2005)

    Article  MathSciNet  Google Scholar 

  7. Chen, W., Whitley, D.: Decomposing SAT instances with pseudo backbones. In: Proceedings of the 17th European Conference on Evolutionary Computation in Combinatorial Optimization. Lecture Notes in Computer Science, March 2017, vol. 10197, pp. 75–90. Springer (2017)

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Dilkina, B., Gomes, C., Sabharwal, A.: Tradeoffs in the complexity of backdoors to satisfiability: dynamic sub-solvers and learning during search. Ann. Math. Artif. Intell. 70(4), 399–431 (2014)

    Article  MathSciNet  Google Scholar 

  12. Dowling, W., Gallier, J.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Log. Program. 1(3), 267–284 (1984)

  13. Friedrich, T., Krohmer, A., Rothenberger, R., Sutton, A.: Phase transitions for scale-free SAT formulas. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence, February 2017, pp. 3893–3899. AAAI Press (2017)

  14. Gasarch, W.: The third P =? NP poll. SIGACT News 50(1), 38–59 (2019)

    Article  MathSciNet  Google Scholar 

  15. Gaspers, S., Misra, N., Ordyniak, S., Szeider, S., Živný, S.: Backdoors into heterogeneous classes of SAT and CSP.  J. Comput. Syst. Sci. 85, 38–56 (2017)

  16. Hartmanis, J., Hemachandra, L.: Complexity classes without machines: on complete languages for UP. Theor. Comput. Sci. 58(1–3), 129–142 (1988)

    Article  MathSciNet  Google Scholar 

  17. Hemaspaandra, E., Hemaspaandra, L., Menton, C.: Search versus decision for election manipulation problems. ACM Trans. Comput. 12, 1–42 (2020)

    MATH  Google Scholar 

  18. Hemaspaandra, L.: Computational social choice and computational complexity: BFFs? In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, February 2018, pp. 7971–7977. AAAI Press (2018)

  19. Hemaspaandra, L., Narváez, D.: The opacity of backbones. Technical Report, Computing Research Repository, June 2016. Revised, January (2017). arXiv:1606.03634

  20. Hemaspaandra, L., Narváez, D.: The opacity of backbones. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence, February 2017, pp. 3900–3906. AAAI Press (2017)

  21. Hemaspaandra, L., Narváez, D.: Existence versus exploitation: the opacity of backbones and backdoors under a weak assumption. In: Proceedings of the 45th International Conference on Current Trends in Theory and Practice of Computer Science. Lecture Notes in Computer Science, January 2019, vol. 11376, pp. 247–259. Springer (2019)

  22. Hemaspaandra, L., Ogihara, M.: The Complexity Theory Companion. Springer, Berlin (2002)

    Book  Google Scholar 

  23. Hemaspaandra, L., Williams, R.: An atypical survey of typical-case heuristic algorithms. SIGACT News 43(4), 71–89 (2012)

    Article  Google Scholar 

  24. Hemaspaandra, L., Zimand, M.: Strong self-reducibility precludes strong immunity. Math. Syst. Theory 29(5), 535–548 (1996)

    Article  MathSciNet  Google Scholar 

  25. Kilby, P., Slaney, J., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proceedings of the 20th National Conference on Artificial Intelligence, July 2005, pp. 1368–1373. AAAI Press (2005)

  26. Kochemazov, S., Zaikin, O.: ALIAS: A modular tool for finding backdoors for SAT. In: Proceedings of the 21st International Conference on Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, June 2018, vol. 10929, pp. 419–427. Springer (2018)

  27. Mahaney, S.: Sparse complete sets for NP: solution of a conjecture of Berman and Hartmanis. J. Comput. Syst. Sci. 25(2), 130–143 (1982)

    Article  MathSciNet  Google Scholar 

  28. Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Informal Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing, May 2004, pp. 96–103 (2004)

  29. Papadimitriou, C.: Computational Complexity. Addison-Wesley, Boston (1994)

    MATH  Google Scholar 

  30. Rothe, J.: Complexity of certificates, heuristics, and counting types, with applications to cryptography and circuit theory. Habilitation thesis, Friedrich-Schiller-Universität Jena, Institut für Informatik, Jena, Germany, June (1999)

  31. Schaefer, T.: The complexity of satisfiability problems. In: Proceedings of the 10th ACM Symposium on Theory of Computing, May 1978, pp. 216–226. ACM Press (1978)

  32. Schöning, U.: Complete sets and closeness to complexity classes. Math. Syst. Theory 19(1), 29–42 (1986)

    Article  MathSciNet  Google Scholar 

  33. Semenov, A., Zaikin, O., Otpuschennikov, I., Kochemazov, S., Ignatiev, A.: On cryptographic attacks using backdoors for SAT. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, February 2018, pp. 6641–6648. AAAI Press (2018)

  34. Szeider, S.: Backdoor sets for DLL subsolvers. J. Autom. Reason. 35(1–3), 73–88 (2005)

    MathSciNet  MATH  Google Scholar 

  35. Tardos, G.: Query complexity, or why is it difficult to separate NP\({\rm }^{A}{}\cap \,\)coNP\({\rm }^{A}\) from P\({\rm {{}}}^{A}\) by random oracles \({A}\). Combinatorica 9, 385–392 (1989)

    Article  MathSciNet  Google Scholar 

  36. Valiant, L.: The relative complexity of checking and evaluating. Inf. Process. Lett. 5(1), 20–23 (1976)

    Article  MathSciNet  Google Scholar 

  37. Willams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence, August 2003, pp. 1173–1178. Morgan Kaufmann (2003)

Download references

Acknowledgements

An earlier version [21] appeared in the SOFSEM 2019 conference. We are extremely grateful to the anonymous conference and journal referees for valuable comments and suggestions that much improved this paper.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Rochester, Rochester, NY, 14627, USA

    Lane A. Hemaspaandra & David E. Narváez

  2. College of Computing and Information Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    David E. Narváez

Authors
  1. Lane A. Hemaspaandra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David E. Narváez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David E. Narváez.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported in part by NSF grant CCF-2006496, NSF grant CCF-2030859 to the Computing Research Association for the CIFellows Project, a Renewed Research Stay award from the Alexander von Humboldt Foundation, and sabbatical visit support from ETH Zürich’s Department of Computer Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hemaspaandra, L.A., Narváez, D.E. Existence versus exploitation: the opacity of backdoors and backbones. Prog Artif Intell 10, 297–308 (2021). https://doi.org/10.1007/s13748-021-00234-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13748-021-00234-6

Keywords

Search

Navigation