Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

On the Complexity of the Clone Membership Problem

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We investigate the complexity of the Boolean clone membership problem (CMP): given a set of Boolean functions F and a Boolean function f, determine if f is in the clone generated by F, i.e., if it can be expressed by a circuit with F-gates. Here, f and elements of F are given as circuits or formulas over the usual De Morgan basis. Böhler and Schnoor (Theory Comput. Syst. 41(4):753–777, 2007) proved that for any fixed F, the problem is coNP-complete, with a few exceptions where it is in P. Vollmer (Theory Comput. Syst. 44(1): 82–90, 2009) incorrectly claimed that the full problem CMP is also coNP-complete. We prove that CMP is in fact \(\boldsymbol {\Theta }^{\mathbf {P}}_{2}\)-complete, and we complement Böhler and Schnoor’s results by showing that for fixed f, the problem is NP-complete unless f is a projection. More generally, we study the problem B-CMP where F and f are given by circuits using gates from B. For most choices of B, we classify the complexity of B-CMP as being \(\boldsymbol {\Theta }^{\mathbf {P}}_2\)-complete (possibly under randomized reductions), coDP-complete, or in P.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306 (1990)

    Article  Google Scholar 

  2. Bauland, M., Mundhenk, M., Schneider, T., Schnoor, H., Schnoor, I., Vollmer, H.: The tractability of model checking for LTL: the good, the bad, and the ugly fragments. ACM Trans. Comput. Logic 12(2). Article no. 13, 28 pp. (2011)

  3. Behrisch, M.: Clones with nullary operations. Electron. Notes Theor. Comput. Sci. 303, 3–35 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bergman, C., Slutzki, G.: Complexity of some problems concerning varieties and quasi-varieties of algebras. SIAM J. Comput. 30(2), 359–382 (2000)

    Article  MathSciNet  Google Scholar 

  5. Bergman, C., Juedes, D., Slutzki, G.: Computational complexity of term-equivalence. Int. J. Algebra Comput. 9(1), 113–128 (1999)

    Article  MathSciNet  Google Scholar 

  6. Beyersdorff, O., Meier, A., Thomas, M., Vollmer, H.: The complexity of reasoning for fragments of default logic. J. Logic Comput. 22(3), 587–604 (2012)

    Article  MathSciNet  Google Scholar 

  7. Böhler, E., Schnoor, H.: The complexity of the descriptiveness of Boolean circuits over different sets of gates. Theory Comput. Syst. 41(4), 753–777 (2007)

    Article  MathSciNet  Google Scholar 

  8. Bulatov, A.A.: A dichotomy theorem for nonuniform CSPs. In: Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science, pp. 319–330 (2017)

  9. Buss, S.R.: The Boolean formula value problem is in ALOGTIME. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 123–131 (1987)

  10. Buss, S.R., Hay, L.: On truth-table reducibility to SAT. Inf. Computat. 91(1), 86–102 (1991)

    Article  MathSciNet  Google Scholar 

  11. Cohen, G., Damgård, I.B., Ishai, Y., Kölker, J., Miltersen, P.B., Raz, R., Rothblum, R.D.: Efficient multiparty protocols via log-depth threshold formulae. In: Canetti, R., Garay, J.A. (eds.) Advances in Cryptology—CRYPTO 2013. Proceedings, Part II, Lecture Notes in Computer Science, vol. 8043, pp. 185–202. Springer (2013). Full version available at http://www.wisdom.weizmann.ac.il/~ranraz/publications/Pmajority_mpc-1.pdf

  12. Creignou, N., Schmidt, J., Thomas, M.: Complexity classifications for propositional abduction in Post’s framework. J. Logic Comput. 22(5), 1145–1170 (2012)

    Article  MathSciNet  Google Scholar 

  13. Gupta, A., Mahajan, S.: Using amplification to compute majority with small majority gates. Comput. Complex. 6(1), 46–63 (1996)

    Article  MathSciNet  Google Scholar 

  14. Jeřábek, E.: Root finding with threshold circuits. Theor. Comput. Sci. 462, 59–69 (2012)

    Article  MathSciNet  Google Scholar 

  15. Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)

  16. Kozik, M.: A finite set of functions with an EXPTIME-complete composition problem. Theor. Comput. Sci. 407, 330–341 (2008)

    Article  MathSciNet  Google Scholar 

  17. Ladner, R.E.: The circuit value problem is log space complete for \(\mathcal P\). ACM SIGACT News 7(1), 18–20 (1975)

    Article  Google Scholar 

  18. Lau, D.: Function algebras on finite sets: a basic course on many-valued logic and clone theory. Springer, New York (2006)

    MATH  Google Scholar 

  19. Lewis, H.R.: Satisfiability problems for propositional calculi. Math. Syst. Theory 13, 45–53 (1979)

    Article  MathSciNet  Google Scholar 

  20. Lukasiewicz, T., Malizia, E.: A novel characterization of the complexity class \({\Theta }_{k}^{\mathrm {P}}\) based on counting and comparison. Theor. Comput. Sci. 694, 21–33 (2017)

    Article  Google Scholar 

  21. Mašulović, D.: GenClo and TermEquiv are EXPTIME-complete. Int. J. Algebra Comput. 18(5), 901–909 (2008)

    Article  MathSciNet  Google Scholar 

  22. Meier, A., Schneider, T.: Generalized satisfiability for the description logic \(\mathcal {{ALC}}\). Theor. Comput. Sci. 505, 55–73 (2013)

    Article  MathSciNet  Google Scholar 

  23. Post, E.L.: The Two-Valued Iterative Systems of Mathematical Logic. No. 5 in Annals of Mathematics Studies. Princeton University Press, Princeton (1941)

    Google Scholar 

  24. Reith, S.: Generalized satisfiability problems. Ph.D. thesis, Julius-Maximilians-Universität, Würzburg (2001)

  25. Ruzzo, W.L.: On uniform circuit complexity. J. Comput. Syst. Sci. 22(3), 365–383 (1981)

    Article  MathSciNet  Google Scholar 

  26. Sannella, D., Tarlecki, A.: On observational equivalence and algebraic specification. J. Comput. Syst. Sci. 34(2–3), 150–178 (1987)

    Article  MathSciNet  Google Scholar 

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

  28. Szendrei, Á.: Clones in universal algebra, Séminaire de mathématiques supérieurs, vol. 99. Université de Montréal (1986)

  29. Valiant, L.G.: Short monotone formulae for the majority function. J. Algorithms 5(3), 363–366 (1984)

    Article  MathSciNet  Google Scholar 

  30. Vollmer, H.: The complexity of deciding if a Boolean function can be computed by circuits over a restricted basis. Theory Comput. Syst. 44(1), 82–90 (2009)

    Article  MathSciNet  Google Scholar 

  31. Wagner, K.W.: Bounded query classes. SIAM J. Comput. 19(5), 833–846 (1990)

    Article  MathSciNet  Google Scholar 

  32. Wegener, I.: The Complexity of Boolean Functions. Teubner, Stuttgart (1987)

    MATH  Google Scholar 

  33. Zhuk, D.: A proof of CSP dichotomy conjecture. In: Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science, pp. 331–342 (2017)

Download references

Acknowledgements

I would like to thank the anonymous referees for helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Emil Jeřábek.

Additional information

Publisher’s Note

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

Supported by grant 19-05497S of GA ČR. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO: 67985840.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeřábek, E. On the Complexity of the Clone Membership Problem. Theory Comput Syst 65, 839–868 (2021). https://doi.org/10.1007/s00224-020-10016-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-020-10016-7

Keywords

Mathematics Subject Classification (2010)

Navigation