Document Open Access Logo

Randomness and Intractability in Kolmogorov Complexity

Author Igor Carboni Oliveira

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2019.32.pdf
  • Filesize: 0.51 MB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • computational complexity
  • randomness
  • circuit lower bounds
  • Kolmogorov complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability. 
This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time. 
Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.

Cite As Get BibTex

Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.32

Author Details

Igor Carboni Oliveira
  • Department of Computer Science, University of Oxford, UK

Acknowledgements

I am grateful to Ján Pich, Eric Allender, Ryan Williams, Shuichi Hirahara, Michal Koucký, Rahul Santhanam, and Jan Krajíček for discussions. Part of this work was completed while the author was visiting the Simons Institute for the Theory of Computing. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 615075.

References

  1. Eric Allender. Applications of time-bounded Kolmogorov complexity in complexity theory. In Kolmogorov complexity and computational complexity, pages 4-22. Springer, 1992. Google Scholar
  2. Eric Allender. When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity. In Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 1-15, 2001. URL: http://dx.doi.org/10.1007/3-540-45294-X_1.
  3. Eric Allender. The complexity of complexity. In Computability and Complexity, pages 79-94. Springer, 2017. Google Scholar
  4. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from Random Strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: http://dx.doi.org/10.1137/050628994.
  5. Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci., 77(1):14-40, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.004.
  6. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993. URL: http://dx.doi.org/10.1007/BF01275486.
  7. Boaz Barak. A Probabilistic-Time Hierarchy Theorem for "Slightly Non-uniform" Algorithms. In International Workshop on Randomization and Approximation Techniques (RANDOM), pages 194-208, 2002. URL: http://dx.doi.org/10.1007/3-540-45726-7_16.
  8. André Berthiaume, Wim van Dam, and Sophie Laplante. Quantum Kolmogorov Complexity. J. Comput. Syst. Sci., 63(2):201-221, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1765.
  9. Harry Buhrman, Lance Fortnow, and Rahul Santhanam. Unconditional Lower Bounds against Advice. In International Colloquium on Automata, Languages and Programming (ICALP), pages 195-209, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02927-1_18.
  10. Harry Buhrman, Lance Fortnow, and Thomas Thierauf. Nonrelativizing Separations. In Conference on Computational Complexity (CCC), pages 8-12, 1998. URL: http://dx.doi.org/10.1109/CCC.1998.694585.
  11. Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning Algorithms from Natural Proofs. In Conference on Computational Complexity (CCC), pages 10:1-10:24, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10.
  12. Gregory J. Chaitin. Information-Theoretic Limitations of Formal Systems. J. ACM, 21(3):403-424, 1974. URL: http://dx.doi.org/10.1145/321832.321839.
  13. Lance Fortnow. Kolmogorov complexity and computational complexity. Complexity of Computations and Proofs. Quaderni di Matematica, 13, 2004. Google Scholar
  14. Lance Fortnow, Rahul Santhanam, and Luca Trevisan. Hierarchies for semantic classes. In Symposium on Theory of Computing (STOC), pages 348-355, 2005. URL: http://dx.doi.org/10.1145/1060590.1060642.
  15. Eran Gat and Shafi Goldwasser. Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications. Electronic Colloquium on Computational Complexity (ECCC), 18:136, 2011. Google Scholar
  16. Shuichi Hirahara. Non-Black-Box Worst-Case to Average-Case Reductions within NP. In Symposium on Foundations of Computer Science (FOCS), pages 247-258, 2018. URL: http://dx.doi.org/10.1109/FOCS.2018.00032.
  17. Shuichi Hirahara and Rahul Santhanam. On the Average-Case Complexity of MCSP and Its Variants. In Computational Complexity Conference (CCC), pages 7:1-7:20, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.7.
  18. Russell Impagliazzo and Avi Wigderson. Randomness vs Time: Derandomization under a Uniform Assumption. J. Comput. Syst. Sci., 63(4):672-688, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1780.
  19. Marek Karpinski and Rutger Verbeek. On the Monte Carlo Space Constructible Functions and Seperation Results for Probabilistic Complexity Classes. Inf. Comput., 75(2):178-189, 1987. URL: http://dx.doi.org/10.1016/0890-5401(87)90057-5.
  20. Makoto Kikuchi. Kolmogorov complexity and the second incompleteness theorem. Archive for Mathematical Logic, 36(6):437-443, 1997. Google Scholar
  21. Adam R. Klivans and Dieter van Melkebeek. Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL: http://dx.doi.org/10.1137/S0097539700389652.
  22. Shira Kritchman and Ran Raz. The surprise examination paradox and the second incompleteness theorem. Notices of the AMS, 57(11):1454-1458, 2010. Google Scholar
  23. Leonid A. Levin. Universal sequential search problems. Problemy Peredachi Informatsii, 9(3):115-116, 1973. Google Scholar
  24. Leonid A. Levin. Randomness Conservation Inequalities; Information and Independence in Mathematical Theories. Information and Control, 61(1):15-37, 1984. URL: http://dx.doi.org/10.1016/S0019-9958(84)80060-1.
  25. Ming Li and Paul M. B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Texts in Computer Science. Springer, 2008. URL: http://dx.doi.org/10.1007/978-0-387-49820-1.
  26. Caterina E. Mora and Hans J. Briegel. Algorithmic Complexity and Entanglement of Quantum States. Phys. Rev. Lett., 95:200503, 2005. URL: http://dx.doi.org/10.1103/PhysRevLett.95.200503.
  27. Noam Nisan and Avi Wigderson. Hardness vs Randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80043-1.
  28. Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness magnification near state-of-the-art lower bounds. Electronic Colloquium on Computational Complexity (ECCC), 25:158, 2018. URL: https://eccc.weizmann.ac.il/report/2018/158.
  29. Igor Carboni Oliveira and Rahul Santhanam. Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness. In Computational Complexity Conference (CCC), pages 18:1-18:49, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.18.
  30. Igor Carboni Oliveira and Rahul Santhanam. Pseudodeterministic constructions in subexponential time. In Symposium on Theory of Computing (STOC), pages 665-677, 2017. URL: http://dx.doi.org/10.1145/3055399.3055500.
  31. Igor Carboni Oliveira and Rahul Santhanam. Hardness Magnification for Natural Problems. In Symposium on Foundations of Computer Science (FOCS), pages 65-76, 2018. URL: http://dx.doi.org/10.1109/FOCS.2018.00016.
  32. Karl Svozil. Quantum Algorithmic Information Theory. J. UCS, 2(5):311-346, 1996. URL: http://dx.doi.org/10.3217/jucs-002-05-0311.
  33. Luca Trevisan and Salil P. Vadhan. Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity, 16(4):331-364, 2007. URL: http://dx.doi.org/10.1007/s00037-007-0233-x.
  34. Christopher Umans. Pseudo-random generators for all hardnesses. J. Comput. Syst. Sci., 67(2):419-440, 2003. URL: http://dx.doi.org/10.1016/S0022-0000(03)00046-1.
  35. Paul M. B. Vitányi. Quantum Kolmogorov complexity based on classical descriptions. IEEE Trans. Information Theory, 47(6):2464-2479, 2001. URL: http://dx.doi.org/10.1109/18.945258.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact