The Complexity of Complexity

  • Eric AllenderEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)


Given a string, what is its complexity? We survey what is known about the computational complexity of this problem, and describe several open questions.


Complexity Class Kolmogorov Complexity Graph Automorphism Universal Machine Turing Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author acknowledges the support of NSF grant CCF-1555409, and thanks Diptarka Chakraborty (for helpful comments on an earlier draft of this work), Shuichi Hirahara (for allowing mention of his recent unpublished results), and Toni Pitassi (for helpful discussions).


  1. 1.
    Agrawal, M., Allender, E., Impagliazzo, R., Pitassi, R., Rudich, S.: Reducing the complexity of reductions. Comput. Complex. 10, 117–138 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allender, E.: NL-printable sets and nondeterministic Kolmogorov complexity. Theor. Comput. Sci. 355(2), 127–138 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allender, E.: Curiouser and curiouser: the link between incompressibility and complexity. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 11–16. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-30870-3_2 CrossRefGoogle Scholar
  4. 4.
    Allender, E.: Investigations concerning the structure of complete sets. In: Agrawal, M., Arvind, V. (eds.) Perspectives in Computational Complexity. PCSAL, vol. 26, pp. 23–35. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-05446-9_2 Google Scholar
  5. 5.
    Allender, E., Buhrman, H., Friedman, L., Loff, B.: Reductions to the set of random strings: the resource-bounded case. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 88–99. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32589-2_11 CrossRefGoogle Scholar
  6. 6.
    Allender, E., Buhrman, H., Friedman, L., Loff, B.: Reductions to the set of random strings: the resource-bounded case. Logical Methods Comput. Sci. 10(3), 1–18 (2014). CiE 2012 Special IssueMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Allender, E., Buhrman, H., Koucký, M.: What can be efficiently reduced to the Kolmogorov-random strings? Ann. Pure Appl. Logic 138, 2–19 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM J. Comput. 35, 1467–1493 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Allender, E., Das, B.: Zero knowledge and circuit minimization. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 25–32. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44465-8_3 Google Scholar
  10. 10.
    Allender, E., Davie, G., Friedman, L., Hopkins, S.B., Tzameret, I.: Kolmogorov complexity, circuits, and the strength of formal theories of arithmetic. Chicago J. Theor. Comput. Sci. 5, 1–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Allender, E., Friedman, L., Gasarch, W.: Exposition of the Muchnik-Positselsky construction of a prefix free entropy function that is not complete under truth-table reductions. Technical report TR10-138, Electronic Colloquium on Computational Complexity (ECCC) (2010)Google Scholar
  12. 12.
    Allender, E., Friedman, L., Gasarch, W.: Limits on the computational power of random strings. Inf. Comput. 222, 80–92 (2013). ICALP 2011 Special IssueMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Allender, E., Grochow, J., Moore, C.: Graph isomorphism and circuit size. Technical report TR15-162, Electronic Colloquium on Computational Complexity (ECCC) (2015)Google Scholar
  14. 14.
    Allender, E., Holden, D., Kabanets, V.: The minimum oracle circuit size problem. In: Symposium on Theoretical Aspects of Computer Science (STACS), LIPIcs, pp. 21–33. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)Google Scholar
  15. 15.
    Allender, E., Koucký, M., Ronneburger, D., Roy, S.: The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci. 77, 14–40 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Allender, E., Hellerstein, L., McCabe, P., Pitassi, T., Saks, M.E.: Minimizing disjunctive normal form formulas and AC\(^{\text{0 }} \) circuits given a truth table. SIAM J. Comput. 38(1), 63–84 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach, vol. 1. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Buhrman, H., Fortnow, L., Koucký, M., Loff, B.: Derandomizing from random strings. In: 25th IEEE Conference on Computational Complexity (CCC), pp. 58–63. IEEE (2010)Google Scholar
  19. 19.
    Buhrman, H., Mayordomo, E.: An excursion to the Kolmogorov random strings. J. Comput. Syst. Sci. 54, 393–399 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cai, J.: S\(_2^p \subseteq \rm {ZPP}^{NP}\). J. Comput. Syst. Sci. 73(1), 25–35 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Cai, M., Downey, R., Epstein, R., Lempp, S., Miller, J.: Random strings and tt-degrees of Turing complete c.e. sets. Logical Methods Comput. Sci. 10(3), 1–24 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Day, A.: On the computational power of random strings. Ann. Pure Appl. Logic 160, 214–228 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Grollmann, J., Selman, L.: Complexity measures for public-key cryptosystems. SIAM J. Comput. 17(2), 309–335 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gutfreund, D., Vadhan, S.: Limitations of hardness vs. randomness under uniform reductions. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX/RANDOM -2008. LNCS, vol. 5171, pp. 469–482. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-85363-3_37 CrossRefGoogle Scholar
  26. 26.
    Hirahara, S.: Personal Communication (2015)Google Scholar
  27. 27.
    Hirahara, S., Kawamura, A.: On characterizations of randomized computation using plain kolmogorov complexity. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 348–359. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44465-8_30 Google Scholar
  28. 28.
    Hirahara, S., Watanabe, O.: Limits of minimum circuit size problem as oracle. In: 31st Conference on Computational Complexity, CCC, LIPIcs, pp. 18:1–18:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  29. 29.
    Hitchcock, J.M.: Limitations of efficient reducibility to the kolmogorov random strings. Computability 1(1), 39–43 (2012)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Hitchcock, J.M., Pavan, A.: On the NP-completeness of the minimum circuit size problem. In: Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS). LIPIcs, vol. 45, pp. 236–245. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  31. 31.
    Hyde, K., Kjos-Hanssen, B.: Nondeterministic automatic complexity of overlap-free and almost square-free words. Electr. J. Comb. 22(3), P3.22 (2015)Google Scholar
  32. 32.
    Kabanets, V., Cai, J.-Y.: Circuit minimization problem. In: ACM Symposium on Theory of Computing (STOC), pp. 73–79. ACM (2000)Google Scholar
  33. 33.
    Kummer, M.: On the complexity of random strings. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 25–36. Springer, Heidelberg (1996). doi: 10.1007/3-540-60922-9_3 CrossRefGoogle Scholar
  34. 34.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kushilevitz, E., Weinreb, E.: On the complexity of communication complexity. In: ACM Symposium on Theory of Computing (STOC), pp. 465–474 (2009)Google Scholar
  36. 36.
    Levin, L.A.: Randomness conservation inequalities; information and independence in mathematical theories. Inf. Control 61, 15–37 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Masek, W.J.: Some NP-complete set covering problems (1979) (Unpublished manuscript)Google Scholar
  39. 39.
    Muchnik, A.A., Positselsky, S.: Kolmogorov entropy in the context of computability. Theor. Comput. Sci. 271, 15–35 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Murray, C., Williams, R.: On the (non) NP-hardness of computing circuit complexity. In: 30th Conference on Computational Complexity (CCC), LIPIcs, pp. 365–380. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)Google Scholar
  41. 41.
    Orlin, J.: Contentment in graph theory: covering graphs with cliques. Indagationes Math. (Proceedings) 80(5), 406–424 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Raviv, N.: Truth table minimization of computational models (2013)Google Scholar
  43. 43.
    Razborov, A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55, 24–35 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Regan, K.W.: A uniform reduction theorem extending a result of J. Grollmann and A. Selman. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 324–333. Springer, Heidelberg (1986). doi: 10.1007/3-540-16761-7_82 CrossRefGoogle Scholar
  45. 45.
    Rudow, M.: Discrete logarithm and minimum circuit size. Technical report TR16-108, Electronic Colloquium on Computational Complexity (ECCC) (2016)Google Scholar
  46. 46.
    Schöning, U.: A low and a high hierarchy within NP. J. Comput. Syst. Sci. 27(1), 14–28 (1983)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Shallit, J., Wang, M.: Automatic complexity of strings. J. Autom. Lang. Comb. 6(4), 537–554 (2001)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Spira, P.M.: On time-hardware complexity tradeoffs for Boolean functions. In: Proceedings of the 4th Hawaii Symposium on System Sciences, pp. 525–527 (1971)Google Scholar
  49. 49.
    Trakhtenbrot, B.A.: A survey of Russian approaches to perebor (brute-force searches) algorithms. IEEE Ann. Hist. Comput. 6(4), 384–400 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer-Verlag New York Inc., New York (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA

Personalised recommendations