New bounds on formula size

  • M. S. Paterson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 48)


A variety of theorems bounding the formula size of rather simple Boolean functions are described here for the first time. The principal results are improved lower and upper bounds for symmetric functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [FMP 75]
    M.J. Fischer, A.R. Meyer and M.S. Paterson. "Lower bounds on the size of Boolean formulas: preliminary report", Proc. 7th Ann. ACM Symp. on Th. of Computing (1975), 45–49.Google Scholar
  2. [HaS 72]
    L.H. Harper and J.E. Savage. "On the complexity of the marriage problem", Advances in Mathematics 9, 3 (1972), 299–312.CrossRefGoogle Scholar
  3. [Hod 70]
    L. Hodes. "The logical complexity of geometric properties in the plane", J. ACM 17, 2 (1970), 339–347.CrossRefGoogle Scholar
  4. [HoS 68]
    L. Hodes and E. Specker. "Lengths of formulas and elimination of quantifiers I", in Contributions to Mathematical Logic, K. Schutte, ed., North Holland Publ. Co., (1968), 175–188.Google Scholar
  5. [Hot 75]
    G. Hotz. "Untere Schranken für das Analyseproblem kontext-freier Sprachen", Techn. Bericht, Univ. des Saarlandes, 1976.Google Scholar
  6. [Kha 69]
    L.S. Khasin. "Complexity bounds for the realization of monotone symmetrical functions by means of formulas in the basis ⋁, &, ⌜.", Eng. trans. in Soviet Physics Dokl., 14 12 (1970), 1149–1151; orig. Dokl. Akad. Nauk SSSR, 189, 4 (1969), 752–755.Google Scholar
  7. [Klo 66]
    B.M. Kloss. "Estimates of the complexity of solutions of systems of linear equations", Eng. trans. in Soviet Math Dokl. 7, 6 (1966), 1537–1540; orig. Dokl. Akad. Nauk SSSR, 171, 4 (1966), 781–783.Google Scholar
  8. [McC 76]
    W.F. McColl. "Some results on circuit depth", Ph.D. dissertation, Computer Science Dept., Warwick University, 1976.Google Scholar
  9. [Meh 76]
    K. Mehlhorn. "An improved bound on the formula complexity of context-free recognition". Unpublished report, 1976.Google Scholar
  10. [Nec 66]
    E.I. Neciporuk. "A Boolean function", Soviet Math. Dokl. 7, 4 (1966), 999–1000, orig. Dokl. Akad. Nauk SSSR 169, 4 (1966), 765–766.Google Scholar
  11. [Pat 76]
    M.S. Paterson. "An introduction to Boolean function complexity". Stanford Computer Science Report STAN-CS-76-557 Stanford University, 1976; to appear in Astérisque.Google Scholar
  12. [Pau 75]
    W. Paul. "A 2.5 N lower bound for the combinational complexity of Boolean functions", Proc. 7th Ann. ACM Symp. on Th. of Comp. Albuquerque (1975), 27–36.Google Scholar
  13. [Pip 74]
    N. Pippenger. "Short formulae for symmetric functions", IBM Research Report RC-5143, Yorktown Hts., 1974.Google Scholar
  14. [Pip 75]
    N. Pippenger. "Short monotone formulae for threshold functions". IBM Research Report RC 5405, Yorktown Hts., 1975.Google Scholar
  15. [Pra 75]
    V.R. Pratt. "The effect of basis on size of Boolean expressions". Proc. 16th Annual IEEE Symposium on Foundations of Computer Science, 119–121.Google Scholar
  16. [Sav 76]
    J.E. Savage. The Complexity of Computing, Wiley-Interscience, New York, 1976.Google Scholar
  17. [Sha 49]
    C.E. Shannon. "The synthesis of two-terminal switching circuits", Bell System Technical Journal 28 (1949), 59–98.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • M. S. Paterson

There are no affiliations available

Personalised recommendations