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computational complexity

, Volume 5, Issue 2, pp 132–154 | Cite as

PI k mass production and an optimal circuit for the Nečiporuk slice

  • Alain P. Hiltgen
  • Mike S. Paterson
Article
  • 15 Downloads

Abstract

Letf: {0,1} n →{0,1} m be anm-output Boolean function inn variables.f is called ak-slice iff(x) equals the all-zero vector for allx with Hamming weight less thank andf(x) equals the all-one vector for allx with Hamming weight more thank. Wegener showed that “PI k -set circuits” (set circuits over prime implicants of lengthk) are at the heart of any optimum Boolean circuit for ak-slicef. We prove that, in PI k -set circuits, savings are possible for the mass production of anyFX, i.e., any collectionF ofm output-sets given any collectionX ofn input-sets, if their PI k -set complexity satisfiesSC m (FX)≥3n+2m. This PI k mass production, which can be used in monotone circuits for slice functions, is then exploited in different ways to obtain a monotone circuit of complexity 3n+o(n) for the Nečiporuk slice, thus disproving a conjecture by Wegener that this slice has monotone complexity θ(n3/2). Finally, the new circuit for the Nečiporuk slice is proven to be asymptotically optimal, not only with respect to monotone complexity, but also with respect to combinational complexity.

Key words

Combinational complexity mass production slice functions set circuits upper bounds 

Subject classifications

68Q15 94C10 

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References

  1. N. Alon andR. B. Boppana, The monotone circuit complexity of Boolean functions.Combinatorica 7:1 (1987), 1–22.Google Scholar
  2. A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions.Dokl. Akad. Nauk 282 (1985), 1033–1037. English Translation:Sov. Math. Dokl. 31 (1985), 530–534.Google Scholar
  3. S. J. Berkowitz, On some relationships between monotone and non-monotone circuit complexity.Technical Report, University of Toronto, 1982.Google Scholar
  4. S. W. Boyack,The Robustness of Combinatorial Measures of Boolean Matrix Complexity. Ph.D. thesis, Massachusetts Institute of Technology, 1985.Google Scholar
  5. P. E. Dunne,Techniques for the Analysis of Monotone Boolean Networks. Ph.D. thesis, University of Warwick, 1984.Google Scholar
  6. P. E. Dunne, The complexity of central slice functions.Theoret. Comput. Sci. 44 (1986), 247–257.Google Scholar
  7. P. E. Dunne, On monotone simulations of nonmonotone networks.Theoret. Comput. Sci. 66 (1989), 15–25.Google Scholar
  8. G. Galbiati andM. J. Fischer, On the complexity of 2-output Boolean networks.Theoret. Comput. Sci. 16 (1981), 177–185.Google Scholar
  9. A. P. Hiltgen,Cryptographically Relevant Contributions to Combinational Complexity Theory, vol. 3 ofETH Series in Information Processing, ed.J. L. Massey. Hartung-Gorre, Konstanz, 1994. Reprint of: Ph.D thesis no. 10382, Swiss Federal Institute of Technology, ETH-Zürich, 1993.Google Scholar
  10. E. A. Lamagna andJ. E. Savage, On the logical complexity of symmetric switching functions in monotone and complete bases.Technical Report, Brown University, Providence RI, 1973.Google Scholar
  11. E. A. Lamagna and J. E. Savage, Combinational complexity of some monotone functions.Proc. 15th Ann. Symp. Switching and Automata Theory (1974), 140–144.Google Scholar
  12. K. Mehlhorn, Some remarks on Boolean sums.Acta Informatica 12 (1979), 371–375.Google Scholar
  13. E. I. Nečiporuk, On a Boolean matrix.Probl. Kibern. 21 (1969), 237–240. English Translation:Systems Theory Res. 21 (1971), 236–239.Google Scholar
  14. W. J. Paul, Realizing Boolean functions on disjoint sets of variables.Theoret. Comput. Sci. 2 (1976), 383–396.Google Scholar
  15. N. Pippenger, On another Boolean matrix.Theoret. Comput. Sci. 11 (1980), 49–56. Reprint of: IBM Research Report, Yorktown Heights, 1977.Google Scholar
  16. A. A. Razborov, Lower bounds on monotone complexity of the logical permanent.Matemat. Zametki 37 (1985a), 887–900. English Translation:Math. Notes of the Academy of Sciences of the USSR 37 (1985), 485–493.Google Scholar
  17. A. A. Razborov, Lower bounds for the monotone complexity of some Boolean functions.Dokl. Akad. Nauk 281 (1985b), 798–801. English Translation:Sov. Math. Dokl. 31 (1985), 354–357.Google Scholar
  18. J. E. Savage,The Complexity of Computing. Krieger Publishing Co., Malabar FL, 1987. Reprint of: First edition published by Wiley, New York, 1976.Google Scholar
  19. L. J. Stockmeyer, On the combinational complexity of certain symmetric Boolean functions.Math. Systems Theory 10 (1977), 323–336.Google Scholar
  20. R. E. Tarjan, Complexity of monotone networks for computing conjunctions.Ann. Disc. Math 2 (1978), 121–133.Google Scholar
  21. D. Uhlig, On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements.Matemat. Zametki 15:6 (1974), 937–944. English Translation:Math. Notes of the Academy of Sciences of the USSR 15 (1974), 558–562.Google Scholar
  22. L. G. Valiant, Negation is powerless for Boolean slice functions.SIAM J. Comput. 15:2 (1986), 531–535.Google Scholar
  23. I. Wegener, A new lower bound on the monotone network complexity of Boolean sums.Acta Informatica 13 (1980), 109–114.Google Scholar
  24. I. Wegener, On the complexity of slice functions.Theoret. Comput. Sci. 38 (1985), 55–68.Google Scholar
  25. I. Wegener, More on the complexity of slice functions.Theoret. Comput. Sci. 43, (1986), 201–211.Google Scholar
  26. I. Wegener,The Complexity of Boolean Functions. New York: Wiley (Stuttgart: Teubner), 1987.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Alain P. Hiltgen
    • 1
  • Mike S. Paterson
    • 2
  1. 1.EDP Security SystemsCrypto AGZug
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK

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