∑II∑ threshold formulas

Abstract

A ∑II∑ formula has the form

, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by ∑II∑ formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every ∑II∑ formula computing the threshold functionT nk has size at least exp\((\Omega (\sqrt {k/\ln k} ))nlogn\). Fork andn large andkn 2/3, we show that there exist ∑II∑ formulas for computingT nk with size at most exp\((2(\sqrt {k/\ln k} ))nlogn\).

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References

  1. [1]

    R. B. Boppana: Optimal separations between concurrent-write parallel machines.Proceedings of the 23rd ACM STOC, 1989, 320–326.

  2. [2]

    R. B. Boppana: Amplification of probabilistic boolean formulas,Advances in Computing Research,5 (1989), 27–45.

    Google Scholar 

  3. [3]

    R. B. Boppana, andMichael Sipser:The Complexity of Finite Functions, Chapter 14, The Handbook of Theoretical Computer Science, (J. van Leeuven, ed.), Elsevier Science Publishers B. B., 1990, 759–804.

  4. [4]

    I. Csiszár, andJ. Körner:Information Theory, Coding Theorems for Discrete Memoryless Systems, Akadémiai Kiadó, Budapest, 1981.

    Google Scholar 

  5. [5]

    M. Fredman, andJ. Komlós: On the size of separating systems and perfect hash functions,Siam J. Alg. Disc. Meth., 1984, 61–68.

  6. [6]

    J. Hastad:Computational Limitations for Small Depth Circuits, MIT Press, 1986.

  7. [7]

    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.

    Google Scholar 

  8. [8]

    L. S. Khasin: Complexity bounds for the realization of monotone symmetrical functions by means of formulas in the basis ∨, ∧, ⌍,Sov. Phys. Dokl. 14 (1970), 1149–1151.

    Google Scholar 

  9. [9]

    V. M. Khrapchenko: A method of obtaining lower bounds for the complexity π-schemes,Math. Notes Acad. Sci. USSR 11 (1972), 474–479.

    Google Scholar 

  10. [10]

    J. Körner: Fredman-Komlós bound and information theory,Siam J. Alg. Disc. Meth., 1986, 560–570.

  11. [11]

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

    Google Scholar 

  12. [12]

    I. Newman, P. Ragde, andA. Wigderson: Perfect hashing, graph entropy and circuit complexity,Proceedings of the 5th Annual Conference on Structure in Complexity Theory, 1990, 91–99.

  13. [13]

    M. S. Paterson, N. Pippenger, andU. Zwick: Optimal carry save networks,Boolean Function Complexity: selected papers for the LMS symposium Durham 1990., Cambridge Univ. Press, 1992, 174–201.

  14. [14]

    J. Radhakrishnan: Better bounds for threshold formulas.Proceedings of the 32nd IEEE FOCS, 1991, 314–323.

  15. [15]

    J. Radhakrishnan: Improved bounds for covering complete uniform hypergraphs.Information Processing Letters 41 (1992), 203–207.

    Google Scholar 

  16. [16]

    M. Snir: The covering problem of complete uniform hypergraphs,Discrete Math. 27 1979, 103–105.

    Google Scholar 

  17. [17]

    L. G. Valiant: Short monotone formulae for the majority function,Journal of Algorithms 5 (1984), 363–366.

    Google Scholar 

  18. [18]

    I. Wegener:The Complexity of Boolean Functions, Wiley-Teubner Series in Computer Science, 1987.

Download references

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Radhakrishnan, J. ∑II∑ threshold formulas. Combinatorica 14, 345–374 (1994). https://doi.org/10.1007/BF01212982

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AMS subject classification code (1991)

  • 68 R 05
  • 94 C 10