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On counting ac 0 circuits with negative constants

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Mathematical Foundations of Computer Science 1998 (MFCS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1450))

Abstract

Continuing the study of the relationship between TC 0, AC 0 and arithmetic circuits, started by Agrawal et al. [1], we answer a few questions left open in this paper. Our main result is that the classes DiffAC 0 and GapAC 0 coincide, under poly-time, log-space, and log-time uniformity. From that we can derive that under logspace uniformity, the following equalities hold: etse409-01etse

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References

  1. M. Agrawal, E. Allender, S. Datta, On TC 0, AC 0 and Arithmetic Circuits. In Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pp:134–148, 1997.

    Google Scholar 

  2. E. Allender, R. Beals, M. Ogihara, The complexity of matrix rank and feasible systems of linear equations. In Proceedings of the 28th ACM Symposium on Theory of Computing (STOC), pp:161–167, 1996.

    Google Scholar 

  3. C. álvarez, B. Jenner, A very hard logspace counting class. Theoretical Computer Science, 107:3–30, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. A. M. Barrington, N. Immerman, Time, Hardware, and Uniformity. In L. A. Hemaspaandra and A. L. Selman, eds., Complexity Theory Retrospective II, Springer Verlag, pp:1–22, 1997.

    Google Scholar 

  5. D. A. M. Barrington, N. Immerman, and H. Straubing, On Uniformity Within NC 1. Journal of Computer and System Science, 41:274–306, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Beame, S. Cook, H. J. Hoover, Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994–1003, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Caussinus, P. McKenzie, D. Thérien, H. Vollmer, Nondeterministic NC 1. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pp:12–21, 1996.

    Google Scholar 

  8. S. A. Fenner, L. J. Fortnow, S. A. Kurtz, Gap-definable counting classes. Journal of Computer and System Science, 48(1):116–148, 1995.

    Article  MathSciNet  Google Scholar 

  9. J. Köbler, U. Schöning, J. Torán, On counting and approximation. Acta Informatica, 26:363–379, 1989.

    MATH  MathSciNet  Google Scholar 

  10. B. Litow, On iterated integer product. Information Processing Letters, 42(5):269–272, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Mahajan, V. Vinay, Determinant: Combinatorics, Algorithms and Complexity. In Proceedings of SODA'97. ftp://ftp.eccc.unitrier.de/pub/eccc/reports/1997/TR97-036/index.html

    Google Scholar 

  12. A. A. Razborov, Lower bound on size of bounded depth networks over a complete basis with logical addition. Mathematicheskie Zametli, 41:598–607, 1987. English translation in Mathematical Notes of the Academy of Sciences of the USSR, 41:333–338, 1987.

    MATH  MathSciNet  Google Scholar 

  13. R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on the Theory of Computing (STOC), pp:77–82, 1987.

    Google Scholar 

  14. S. Toda, Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions, Informations and Systems, E75-D:116–124, 1992.

    Google Scholar 

  15. S. Toda, Counting problems computationally equivalent to the determinant. Manuscript.

    Google Scholar 

  16. L. Valiant, The complexity of computing the permanent. Theoretical Computer Science, 8:189–201, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Venkateswaran, Circuit definitions of non-deterministic complexity classes. SIAM Journal on Computing, 21:655–670, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. Vinay, Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pp:270–284, 1991.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Science Division, University of California at Berkeley, USA

    Andris Ambainis

  2. Computer Science Department, University of Massachusetts, USA

    David Mix Barrington

  3. Laboratoire de Recherche en Informatique, Université de Paris-Sud, France

    Huong LêThanh

Authors
  1. Andris Ambainis
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  2. David Mix Barrington
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  3. Huong LêThanh
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Editor information

Luboš Brim Jozef Gruska Jiří Zlatuška

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© 1998 Springer-Verlag Berlin Heidelberg

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Ambainis, A., Barrington, D.M., LêThanh, H. (1998). On counting ac 0 circuits with negative constants. In: Brim, L., Gruska, J., Zlatuška, J. (eds) Mathematical Foundations of Computer Science 1998. MFCS 1998. Lecture Notes in Computer Science, vol 1450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055790

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  • DOI: https://doi.org/10.1007/BFb0055790

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64827-7

  • Online ISBN: 978-3-540-68532-6

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