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Local Restrictions from the Furst-Saxe-Sipser Paper

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Abstract

In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13–27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.

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Notes

  1. It should be noted at this point that Ajtai independently obtained the corresponding result by using a similar notion in a different context [6]. Also we note here that Subbotovskaya [24] used random restrictions in computational complexity in the early 60’s.

  2. The technical exposition part of this article is mainly from this survey.

  3. We abuse the notation here; NC0 should be defined as a complexity class, that is, the class of Boolean functions computable by circuits described above.

References

  1. Agrawal, M.: The isomorphism conjecture for NP. In: Computability in Context: Computation and Logic in Real World, pp 19–48. World Scientific (2011)

  2. Agrawal, M.: The isomorphism conjecture for constant depth reductions. J. Comput. Syst. Sci. 77, 3–13 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrawal, M., Allender, E.: An isomorphism theorem for circuit complexity. In: Proceedings of 11th Conference on Computational Complexity, pp 2–11. IEEE (1996)

  4. Agrawal, M., Allender, E., Rudich, S.: Reductions in circuit complexity: An isomorphism theorem and a gap theorem. J. Comput. Syst. Sci. 57, 127–143 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agrawal, M., Allender, E., Impagliazzio, R., Pitassi, T., Rudich, S.: Reducing the complexity of reductions. Comput. Complex. 10(2), 117–138 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ajtai, M.: \({{\Sigma }_{1}^{1}}\)-formulae on finite structures. Ann. Pure Appl. Logic 24(1), 1–48 (1983)

    Article  MathSciNet  Google Scholar 

  7. Beame, P.: A Switching Lemma Primer, Department of Computer Science and Engineering, University of Washington, UW-CSE–95-07-01 (1994)

  8. Berman, L.: Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University (1977)

  9. Berman, L., Hartmanis, J.: On isomorphism and density of NP and other complete sets. SIAM J. Comput. 1, 305–322 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boppana, R., Lagarias, J.: One-way functions and circuit complexity. In: Proceedings of Structure in Complexity Theory, Lecture Notes in Computer Science, vol. 223, pp 51–65 (1986)

  11. O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press (2014)

  12. Furst, M., Saxe, J., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17(1), 13–27 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Håstad, J.: Computational Limitations for Small Depth Circuits. MIT Press, Cambridge (1986)

    Google Scholar 

  14. Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing, pp 6–20 (1986)

  15. Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science, pp 68–80 (1988)

  16. Kurtz, S., Mahaney, S., Royer, J.: The structure of complete degrees. In: Selman, A. (ed.) Complexity Theory Retrospective, pp 108–146. Springer-Verlag (1990)

  17. Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40(3), 607–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lupanov, O.: Implementing the algebra of logic functions in terms of constant-depth formulas in the basis +,∗,−. Sov. Phys. Dokl. 6(2), 107–108 (1961)

    MATH  Google Scholar 

  19. Rossman, B.: The average sensitivity of bounded-depth formulas. In: Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, pp 424–430. IEEE (2015)

  20. Rossman, B., Servedio, R., Tan, L.: An average-case depth hierarchy theorem for Boolean circuits. In: Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, pp 1030–1048. IEEE (2015)

  21. Rossman, B., Servedio, R., Tan, L.: Complexity theory column 89: The polynomial hierarchy, random oracles, and Boolean circuits. SIGACT News 46(4), 50–68 (2015)

    Article  MathSciNet  Google Scholar 

  22. Santhanam, R., perebor, Fighting: New and improved algorithms for formula and QBF satisfiability. In: Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer Science, pp 183–192. IEEE (2010)

  23. Seto, K., Tamaki, S.: A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Comput. Complex. 22(2), 245–274 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Subbotovskaya, B.: Realizations of linear functions by formulas using +,∗,−. Sov. Phys. Dokl. 2, 110–112 (1961)

    MathSciNet  MATH  Google Scholar 

  25. Watanabe, O.: On one-one polynomial time equivalence relations. Theor. Comput. Sci. 38, 157–165 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

On this occassion we would like to express our deep appreciation to Alan Selman for his continuous effort in running Mathematical Systems Theory and Theory of Computing Systems as the editor-in-chief.

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Authors and Affiliations

  1. Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Suguru Tamaki

  2. Tokyo Institute of Technology, Meguro-ku Ookayama, Tokyo, 152-8552, Japan

    Osamu Watanabe

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  1. Suguru Tamaki
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  2. Osamu Watanabe
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Correspondence to Osamu Watanabe.

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This article is part of the Topical Collection on 50th Anniversary

The first author is supported in part by MEXT KAKENHI (24106003); JSPS KAKENHI (26330011, 16H02782), and the second author is supported in part by MEXT KAKENHI (24106008).

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Tamaki, S., Watanabe, O. Local Restrictions from the Furst-Saxe-Sipser Paper. Theory Comput Syst 60, 20–32 (2017). https://doi.org/10.1007/s00224-016-9730-0

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