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Pi-sigma-pi threshold formulas

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Abstract

We present lower and upper bounds on the size of pi-sigma-pi (Π ∑ Π) formulas computing threshold functions for small thresholds. Our results show that the limitations of Π ∑ Π formulas for computing threshold functions for small thresholds are more pronounced than suggested by the lower bounds for small depth circuits computing the majority function.

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References

  1. R. B. Boppana. Threshold functions and bounded depth monotone circuits.J. Comput. System Sci. 32(2) (1986), 222–229.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. B. Boppana. Amplification of probabilistic Boolean formulas.Advances in Computing Research, Vol. 5, JAI Press, Greenwich, CT, 1989, pp. 27–45.

    Google Scholar 

  3. G. Hansel. Nombre minimal de contacts de fermature nécessaires pour réaliser une fonction booléenne symétrique den variables.C. R. Acad. Sci. Paris 258 (1964), 6037–6040.

    MATH  MathSciNet  Google Scholar 

  4. J. Håstad. Almost optimal lower bounds for small depth circuits.Proceedings of the 18thACM Symposium on the Theory of Computing, 1986, pp. 6–20. Final version inAdvances in Computing Research, Vol. 5, JAI Press, Greenwich, CT, 1989, pp. 143–170.

    Google Scholar 

  5. R. E. Krichevskii. Complexity of contact circuits realizing a function of logical algebra.Soviet Phys. Dokl. 8 (1964), 770–772.

    Google Scholar 

  6. J. Radhakrishnan. Better bounds for threshold formulas.Proceedings of the 32ndIEEE Symposium on the Foundations of Computer Science, 1991, pp. 314–323.

  7. J. Radhakrishnan. ∑Π∑ threshold formulas.Combinatorial 14(3) (1994), 345–374.

    Article  MATH  MathSciNet  Google Scholar 

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  1. J. Radhakrishnan

    Present address: Tata Institute of Fundamental Research, Homi Bhabha Road, 400005, Colaba, Bombay, India

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  1. School of Information Science, Japan Advanced Institute of Science and Technology-Hokuriku, Tatsunokuchi, 923-12, Ishikawa, Japan

    J. Radhakrishnan

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  1. J. Radhakrishnan
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Radhakrishnan, J. Pi-sigma-pi threshold formulas. Math. Systems Theory 29, 357–374 (1996). https://doi.org/10.1007/BF01192692

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  • DOI: https://doi.org/10.1007/BF01192692

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