computational complexity

, Volume 26, Issue 2, pp 469–496 | Cite as

The Minimum Oracle Circuit Size Problem

Article
  • 73 Downloads

Abstract

We consider variants of the minimum circuit size problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MSCPQBF is known to be complete for PSPACE under ZPP reductions. We show that it is not complete under logspace reductions, and indeed it is not even hard for TC0 under uniform AC0 reductions. We obtain a variety of consequences that follow if oracle versions of MCSP are hard for various complexity classes under different types of reductions. We also prove analogous results for the problem of determining the resource-bounded Kolmogorov complexity of strings, for certain types of Kolmogorov complexity measures.

Subject classification

F.1.3 Complexity Measures and Classes 

Keywords

Kolmogorov complexity Minimum circuit size problem PSPACE NP-intermediate sets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Manindra Agrawal (2011) The isomorphism conjecture for constant depth reductions. Journal of Computer and System Sciences 77(1): 3–13MathSciNetCrossRefMATHGoogle Scholar
  2. Manindra Agrawal (2014). Personal Communication.Google Scholar
  3. Eric Allender, Harry Buhrman, Michal Kouckỳ, Dieter van Melkebeek & Detlef Ronneburger (2006). Power from random strings. SIAM Journal on Computing 35(6), 1467–1493.Google Scholar
  4. Eric Allender & Bireswar Das (2014). Zero Knowledge and Circuit Minimization. In Mathematical Foundations of Computer Science (MFCS), volume 8635 of Lecture Notes in Computer Science, 25–32. Springer.Google Scholar
  5. Eric Allender, Vivek Gore (1991) Rudimentary Reductions Revisited. Information Processing Letters 40(2): 89–95MathSciNetCrossRefMATHGoogle Scholar
  6. Eric Allender & Vivek Gore (1993). On strong separations from AC0. In Advances in Computational Complexity Theory, Jin-Yi Cai, editor, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 21–37. AMS Press.Google Scholar
  7. Eric Allender, Dhiraj Holden & Valentine Kabanets (2015). The Minimum Oracle Circuit Size Problem. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, Ernst W. Mayr & Nicolas Ollinger, editors, volume 30 of LIPIcs, 21–33. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. ISBN 978-3-939897-78-1. http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21.
  8. Eric Allender, Michal Koucký, Detlef Ronneburger & Sambuddha Roy (2010). The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences 77, 14–40.Google Scholar
  9. José L Balcázar, Antoni Lozano & Jacobo Torán (1992). The complexity of algorithmic problems on succinct instances. In Computer Science, 351–377. Springer.Google Scholar
  10. Stephen A. Fenner, Lance Fortnow & Stuart A. Kurtz (1994). Gap-Definable Counting Classes. Journal of Computer and System Sciences 48(1), 116–148.Google Scholar
  11. Lance Fortnow (2000) Time-Space Tradeoffs for Satisfiability. Journal of Computer and System Sciences 60(2): 337–353MathSciNetCrossRefMATHGoogle Scholar
  12. Hana Galperin, Avi Wigderson (1983) Succinct representations of graphs. Information and Control 56(3): 183–198MathSciNetCrossRefMATHGoogle Scholar
  13. Oded Goldreich, Amit Sahai & Salil Vadhan (1998). Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, 399–408. ACM.Google Scholar
  14. Johan Håstad (1989). Almost optimal lower bounds for small depth circuits. In Randomness and Computation, S. Micali, editor, 143–170. Advances in Computing Research, vol. 5, JAI Press, Greenwich, Connecticut.Google Scholar
  15. Valentine Kabanets & Jin-Yi Cai (2000). Circuit minimization problem. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, 73–79. ACM.Google Scholar
  16. Leonid Levin (1984) Randomness Conservation Inequalities; Information and Independence in Mathematical Theories. Inf. and Control 61: 15–37MathSciNetCrossRefMATHGoogle Scholar
  17. Pierre McKenzie, Michael Thomas, Heribert Vollmer (2010) Extensional uniformity for boolean circuits. SIAM Journal on Computing 39(7): 3186–3206MathSciNetCrossRefMATHGoogle Scholar
  18. Cody Murray & Ryan Williams (2015). On the (Non) NP-Hardness of Computing Circuit Complexity. In 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, volume 33 of LIPIcs, 365–380. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365.
  19. Tatsuaki Okamoto (2000) On Relationships between Statistical Zero-Knowledge Proofs. Journal of Computer and System Sciences 60(1): 47–108MathSciNetCrossRefMATHGoogle Scholar
  20. Christos H. Papadimitriou (2003). Computational complexity. John Wiley and Sons Ltd.Google Scholar
  21. Christos H. Papadimitriou & Mihalis Yannakakis (1986). A note on succinct representations of graphs. Information and Control 71(3), 181–185.Google Scholar
  22. Joel I. Seiferas, Michael J. Fischer & Albert R. Meyer (1978). Separating Nondeterministic Time Complexity Classes. J. ACM 25(1), 146–167.Google Scholar
  23. Raymond M. Smullyan (1961). Theory of Formal systems. In Annals of Math. Studies 47. Princeton University Press.Google Scholar
  24. Seinosuke Toda (1991) PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5): 865–877MathSciNetCrossRefMATHGoogle Scholar
  25. Jacobo Torán (1991). Complexity classes defined by counting quantifiers. J. ACM 38(3), 752–773.Google Scholar
  26. Boris A. Trakhtenbrot (1984). A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms. IEEE Annals of the History of Computing 6(4), 384–400.Google Scholar
  27. Klaus W. Wagner (1986). The complexity of combinatorial problems with succinct input representation. Acta Informatica 23(3), 325–356.Google Scholar
  28. Celia Wrathall (1978) Rudimentary predicates and relative computation. SIAM Journal on Computing 7(2): 194–209MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Eric Allender
    • 1
  • Dhiraj Holden
    • 2
  • Valentine Kabanets
    • 3
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.CSAILMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.School of Computing ScienceSimon Fraser UniversityBurnabyCanada

Personalised recommendations